Spherical D-modules and
Representations of Reductive Lie Groups
Freddric Vincent Bien
Licence en mathematiques, Universite Libre de Bruxelles
(1983)
Submitted to the
Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
Massachusetts Institute of Technology
June 1986
Frederic V. Bien 1986
The author hereby grants to M.I.T. permission to reproduce and
to distribute copies of this thesis document in whole or in part.
Signature redacted
Signature redacted
Signature of author
Certified by
Professor Joseph N. Bernstein
Thesip*ugervisor
Signature redacted
Accepted b3
p.-
Professor Nesmith C. Ankeny, Chairman
Departmental Graduate Committee
Department of Mathematics
ssACUSE1TS
ISTUTE
AUG 0 4 1986
Spherical D-modules and
Representations of Real Reductive Groups
by
Fr6ddric Bien
Submitted to the Department of Mathematics
on May first, 1986 in partial fulfillment of the
requirements for the Degree of Doctor of Philisophy in
Mathematics
ABSTRACT
We study the representations of reductive Lie groups which
occur in the space of smooth functions on indefinite symmetric
spaces. We characterize these representations in the theory of Dmodules by a condition on the support and a condition on the
fibers. This enables us to simplify Oshima-Matsuki's theorem on
the discrete series of indefinite symmetric spaces, and to prove an
L2 -multiplicity one theorem. We also interpret the C*-multiplicity
of standard representations as the dimension of a cohomology space
of smooth algebraic varieties.
By microlocalization, we exhibit the global sections and localization functors as direct and inverse images of the moment map.
We prove that the global sections of irreducible D-modules always
break up into components with the same associated variety, and
may stay irreducible in some singular cases.
Thesis Supervisor: Professor Joseph Bernstein
2
Contents
Page
Introduction
2
Localization theory
8
1.
Basic Notions on D-modules
8
2.
D-modules with a group action
9
3.
Parabolic subgroups and flag spaces
11
4.
Differential operators on flag
13
I.
spaces
5.
Global sections of D
15
6.
D-affine varieties
19
7.
Holonomic modules and
22
Harish-Chandra modules
II.
Spherical D-modules
26
1.
K-orbits in a flag space
26
2.
(D,K)-modules with a K-
30
fixed vector
3.
Relations with the analytic theory
31
4.
Going from (D, K)-modules to (D,H)-
36
modules
5.
H-spherical (D,K)-modules
42
6.
D-modules and hyperfunctions
48
7.
Closed orbits and discrete series
54
8.
An algebraic Poisson transform
59
III.
Microlocalization and Singularities
1.
Basic Microlocal notions
62
2.
Microlocal study of the moment map
68
3.
Nilpotent varieties
75
3
Acknowledgements
It is a pleasure to express my deep gratitude to J. Bernstein who supervised
this thesis. I learned from him the theory of D-modules, and he generously shared
with me his understanding of mathematics in a very stimulating atmosphere. Our
many conversations were the major inspiration of this work.
I thank sincerely D. Vogan for teaching me the foundations of representation
theory. He also gave me the benefit of his insight on the subject on many later
occasions. I am grateful to B. Kostant and G. Lusztig for several helpful suggestions
and for showing interest in this work.
This research was facilitated by discussions with several friends. In particular
P-Y. Gaillard introduced me to the Poisson transform, K. Vilonen provided my
initiation to perverse and microlocal objects and A. Vistoli helped me find my way
through singularities. To all of them, I express my warmest thanks.
I am also grateful to the MIT Mathematics Department for allowing me to
work with J. Bernstein and for its financial support in 1984-1985. I received grants
from the Belgian American Educational Foundation in 1983-84 and from the Sloan
Foundation in 1985-86. Poincar6 and Fermat Sun workstations did a beautiful job
as typesetters. M-E. Butts commanded them skillfully and I thank her for typing
two-thirds of this report in a record time.
This thesis is dedicated to my former teacher G. Hirsch who will certainly be
pleased with the pervasive influence of topology on these pages.
4
Introduction.
The goal of this work is to study the representations of reductive Lie groups
which occur in the space of smooth functions on an indefinite symmetric space.
The representations realized by square integrable functions were constructed by
Flensted-Jensen, Oshima and Matsuki. We prove that the discrete series has multiplicity one, and we present a cohomological formula for the multiplicities of standard
representations. We have chosen to study these representations by Beilinson and
Bernstein's theory of differential operators on complex flag manifolds. We find a
canonical map between certain D-modules and the sheaf of hyperfunctions along a
real flag manifold. Combined with previous work of Helgason and Flensted-Jensen,
this exhibits a natural intertwining operator between these D-modules and functions
on symmetric spaces. By microlocalization, we also make a finer study of the moment map, and decide the irreducibility of the global sections of certain D-modules
considered as representations of Lie algebras.
More precisely, let G be a complex connected reductive linear algebraic group,
and let GR be a real form of G. Consider the fixed point subgroup H of an involution
a of G and its corresponding real form HR. Then GR/HR is a symmetric space. The
problem is to find which representations of GR can be imbedded in CO(GR/HR),
which ones can be imbedded in L 2 (GR/HR) and when they can be so imbedded, in
how many different ways.
To formulate the answer, one should first understand that the building blocks of
C(GR/HR) are generally not the irreducible representations of GR. The standard
representations are better designed for this purpose but not all of them will appear.A
smooth representation V of GR which does appear is called HR-spherical and is
characterized by: HomHR (V, C) 4 0, i.e. its continuous dual V* contains an HRfixed functional vo. If we want V to be imbedded into C*(GR/HR), then vo must
generate V*. The duality involved is important because in general V itself does
not contain any HR-invariant vector.
By a result of Casselman and [Wallach],
one can replace the H-spherical condition by Hom(h,KH)(V
0,
C) : 0 where V 0
is the Harish-Chandra module of V. Using the inductive version of Zuckerman's
5
functor, this is equivalent to HOmH(L V 0 , C) : 0, i.e. (L VO)H : 0 since H acts
semisimply on this space.
We pass to the theory of D-modules. The Helgason isomorphism for the Riemannian symmetric spaces shows that the spherical representations have a particular
shape: they can be realized on a partial flag variety. The Iwasawa decomposition
associates to G and H a complex flag space X and a real flag space XR imbedded
in X. To fix the ideas, consider a real form Gd of G for which Hd = H n Gd is a
maximal compact subgroup. Let PR be a minimal parabolic subgroup of Gd with
complexification P. Then X ~ G/P and XR ~ GR/PR. We also need the fixed point
subgroup K of an involution 0 of G commuting with a, such that KR = K n GR is
a maximal compact subgroup of GR. Let Tp = P/(P,P) be the Cartan factor of P.
To every A E t*,, one can associate a G-sheaf DX of twisted differential operators on
X. To define the notion of H-spherical (DA, K)-modules, we will use the functor L'
introduced by Bernstein: it maps (DA, K)-modules to (PA, H)-modules in a fashion
analogous to the inductive Zuckerman's functor. Assume henceforth that X E t* is
dominant, so that the global section functor is exact. We say that a (DA, K)-module
is H-spherical if
I'(X, 'CHM)H 7 0
The standard (DA, K)-modules on X are constructed as follows. Take a K-orbit
Y with a K-homogeneous line bundle Oy (r) such that (A - pp, r) is a module for
the Harish-Chandra pair (py, Ku), y E Y. To simplify the exposition we suppose
in this introduction that i : Y c-+ X is a closed imbedding; a more general case is
worked out in the text. Let i, be the direct image in the category of DX-modules.
Then M (Y, r) = i. Oy (r) is a standard (DA, K)-module. Now let X0 denote the open
H-orbit in X and let L be the Levi factor of P.
Theorem 1: Let A + PL be B-dominant and regular. M (Y, r) is H-spherical if and
0 and (h,, K n H,) acts trivially on the fiber Oy (r)y, y E Y n X0
.
only if Y n X 0
Let us call H-square-integrable a (DA, K)-module whose global sections can be
imbedded as a discrete summand of L'
fn(GR/HR).
Using [Oshima and Matsuki)
L 2 -estimates, we can simplify their result on the discrete series of GR/HR. Suppose
6
M(Y,-r) is H-spherical and G is semisimple.
Theorem 2: M (Y, r) is H-square integrable if and only if A E t* is dominant regular
and rank G/H = rank K/H n K.
Let t be the canonical Cartan space of G. If A + pt E t* is dominant, then
r (X, MN(Y, r)) is an irreducible
(9, K)-module.
But in general there is a small strip
of dominant A E * for which A + pt is not dominant. D. Vogan has checked by
coherent continuation that P(X, M(Y, r)) remains irreducible in this strip. This
hard result has a pretty consequence.
Theorem 3: If G is a classical group or G 2 , the discrete series in L 2 (GR/HR) is
multiplicity free.
In the case of compact groups, this is the well-known fact that (GR, HR) is a
Gelfand pair. Let us note that one should distinguish between the multiplicity of an
irreducible representation V of G as subquotient of C*(GR/HR, X) and as submodule, here X denotes an eigencharacter of the algebra of invariant differential operators
on the symmetric space. The former can be quite big although finite, cf. [van den
Ban] and we do not know a good way to determine it. The latter is dim (V*)H.
Theorem 3 does not affirm that this dimension is 1 for V = r(X, m(Y, r)) as above.
If this were true, it would imply easily the irreducibility of V. In fact the search
for some multiplicity one theorem motivates this whole thesis. We are able to interpret (V*)H as the cohomology space of a smooth algebraic variety with values in
a local system. This formula holds for all standard modules obtained from affinely
imbedded K-orbits. If V is reducible, it simply determines the dimension of the
(G/H)). For any x in the open H-orbit X0
,
space of morphisms: HomoR (V, C
let M be the isotropy group of z in H. The invertible sheaf Oy (r) gives rise to a
n X0
.
K n H-homogeneous local system C(A, r) on Y
Theorem 4: Suppose that A + PL is B-dominant and regular. Then
r (X, Z J.4(Y,
r))H
~ HO(Y n XO, C(A, r))KnH
where s = dimY + dimM - dimK n H.
7
This formula gives easily the uniqueness of the HR-invariant functional in many
nontrivial cases. It also shows that the even principal series for SL 2 (R) admits
generically two R -invariant functionals.
Now we describe the content of the chapters.
In Chapter 1, we summarize Beilinson-Bernstein theory of D-modules on flag
spaces and its relations to the representations of g. We take a slightly more general
viewpoint than is usual, for we work over the variety X of parabolic subgroups of G
conjugate to a given one, say P. The vanishing theorem for DA-modules is true as
long as A is dominant in t*. But P (X, DA) need not be generated by the enveloping
algebra U (g) for all A E t,, as it is for the space of Borel subgroups. This is due to
the fact that the closure of the Richardson orbit Cp C g* of P need not be a normal
variety. However, if A + PL E t* is dominant, then U (g) generates P (X, DA). If in
addition A + PL is regular, the category of (DA, K)-modules on X is equivalent to
the category of (g, K)-modules with infinitesimal character A + PL and associated
variety contained in ~Cp. To study P(X, DA), we generalize a method of Militi6. The
non-vanishing of the global sections of a (Dx, K)-module depends on the Borel-Weil
theorem. To include entirely this classical result in the theory of D-modules, one
should consider modules over a sheaf of matrix differential oeprators. One could
generalize the theory much further than we have done here.
In Chapter 2, after recalling some properties of the K-orbits in X, we study the
case K = H of Riemannian symmetric space. We classify the K-spherical (DA, K)modules: they all come from trivial line bundles on the open K-orbit. In particular,
there is no square integrable module and they all have a unique K-invariant vector.
These results have been known since [Kostant]. In Section 2.3 we explain what is
the foundation of our approach for the reader with an analytic background. Next
we describe Bernstein's definition of Zuckerman's functor for D-modules. At this
point we are ready to study the H-spherical (DA, K)-modules. The idea is again
that they are related to the open H-orbit and to homogeneous line bundles which
are trivial for (h, K n H). We also explain how to deal with the spaces GR/HR. We
prove the formula giving the number of (0, H)-morphisms from Ox into
8
Ki.M(Y,
r)
for a standard (DA, K)-module .M(Y, r), in terms of the cohomology of Y n X 0 . A
better understanding of this formula would give a purely algebraic classification,
but at the present time we are obliged to use as an intermediate step the real flag
variety XR of G' which is imbedded in the middle of X0 . In some sense this is
fortunate because we can construct a bijection between K-orbits Y in X such that
Y n XR 7 0 and Norm(G , K)-orbits in XR. Then we answer a question raised
by Flensted-Jensen in a lecture at Utrecht in August 1985: what is the relation
between the square-integrable standard H-spherical (DA)-modules and the sheaf of
hyperfunctions sections of a line bundle on XR? It turns out that the first can
be imbedded into the second, thanks to the existence of a canonical morphism
of functions
f*
f! 0
-+
w[-rdf] which exists for any map f. Let us remark that
tracing back through Helgason and Flensted-Jensen isomorphisms, we obtain the
unique 'imbedding' of some square integrable standard H-spherical
into
L 2 (GR/HR).
(DP,
K) module
This shows that the D-module realization of a Harish-Chandra
module is in fact quite natural to study harmonic analysis on symmetric spaces.
-
Then we focus on closed K-orbits in the equal rank case, and using a deep L2
estimate of Oshima and Matsuki, we prove that the discrete series of GR/HR has
multiplicity one. Finally, we define an algebraic Poisson transform from (DX, K)module on G/P to
(Dx,, K)-modules
on G/H. [De Concini and Procesi] have con-
structed a very nice compactification of G/H which exhibits G/P as a piece of
the boundary of G/H. I expect that the nearby cycle functor is the inverse of the
Poisson transform. I hope to be able to prove it soon.
In Chapter 3, we solidify the bridge between g-modules and DA-modules on an
arbitrary flag variety X. Except for its last section, chapter 3 is independent of
chapter 2. It consists of an attempt to understand when does the global section
functor preserves irreducibility. Let Ux be the quotient of U(g) by the ideal determined by A E t*. The image of the moment map A : T*X -+ g* is the closure of
the Richardson orbit Cp C g* of P. Following a method introduced by Gabber and
[Ginzburg], we define the formal microlocalizations ex, Ux of the algebras DA, Ux.
eA is a G-sheaf of twisted formal microdifferential operators on T*X, and Ux is a
G-sheaf supported on Cp. The microlocalization is an exact and faithful functor
9
which sends a D-module M to an e-module whose support is the characteristic
variety CharM of M. The functors r and A of global sections and localizations
between DA- and Ux-modules become the functors
images between
AX
1
. and i* of direct and inverse
and Vx-modules. One result of this approach is the following.
Proposition 1: Suppose M is an irreducible DX-module on X and A is dominant in
t*,. Then all the irreducible (g, K)-submodules of r(X, M) have the same associated
variety.
Concerning irreducibility itself we obtain the following improvement when the
moment map g is birational.
Proposition 2: Let M be an irreducible (Dx, K) module with A dominant in t*
Suppose pt(CharM) contains at least one normal point of
'.
Then P(X, M) is
(g, K) irreducible (or zero).
[Kraft and Procesi] have essentially determined when Cp is a normal variety for
the classical groups. Combining their results with the above proposition proves the
irreducibility of certain (g, K) modules-known under the name of Ap(A)-for some
non-trivial cases, like for SPn(R).
The microlocal approach and the study of examples suggests that the unibranchness of an open dense subset of pt(CharM) should suffice to imply the irreduciblity
of r(X, M) over (g, K) when M is (DA, K)-irreducible and 1z birational. The reader
will find in Section 111.3 a formula to detect the unibranchness of a point in Op.
More precisely, using [Borho-MacPherson] theory, we show that the number of connected components in
1r(z), z E Cp, is the multiplicity of the special Weyl group
representation attached to Cp, in a cohomology space of the Springer fiber of z. To
express a quantity related to X in terms of the full flag variety - as this formula
does - simplifies the problem. Still, this multiplicity remains mysterious to us.
10
Chapter I. Localization Theory
1. Basic notions on P-modules
Let X be a smooth complex algebraic manifold. Let 0 = Ox be the sheaf of
regular functions on X and D = DX be the sheaf of differential operators on X. A
D-module M is a sheaf on X which is quasi-coherent as an 0-module and which has
a structure of module over D. Let M(D) denote the category of left D-modules. We
shall deal with some sheaves of rings slightly more general than D. Consider the
category of pairs (A, iA) where A is a sheaf on X of C-algebras and ig : 0 -+ A is a
morphism of C-algebras. The pair (D, i) where i : 0 -+ D is the natural inclusion
will be called the standard pair.
Definition 1.1: A sheaf of twisted differential operators on X (tdo for short) is any
pair (A, iA) which locally isomorphic to the standard pair.
Lemma 1.2: The group of automorphisms of (i : 0 -+ D) is naturally isomorphic
to ZI(X) the group of closed 1-forms on X. The set of isomorphism classes of tdo's
on X is in bijection with the 6ech cohomology space H1(X, Z1).
In particular tdo's form a linear space; for a proper variety X H'(X, Z') is the
C-subspace of Hd (X) generated by algebraic cycles.
Example 1.3: Let L be an invertible sheaf of 0-modules on X,then the sheaf
D (L) of differential operators from
into itself is a tdo. Let
corresponds to the
element c in the Picard group H'(X, 0*), then D(L) corresponds to the logarithmic
differential of c, i.e. dc/c in H1(X, Z').
Example 1.4: If X = P" is the projective n-space, then H1(X, 0*) = Z, hence,
the invertible sheaves on P' form a lattice. But H1(P", Z1) = C: the sheaves of
twisted differential operators on P" form a vector space.
Remark 1.5: If V is a locally free sheaf of 0-modules on X of rank bigger than
one, then the ring of differential operators of V into itself is not a tdo. It consists
of matrix differential opreators.
Let
f
: X -+
Y be a morphism of smooth complex algebraic varieties.
11
Let
Oy, Ox be the sheaves of regular functions on Y and X. We will denote by f*, f"
the inverse and direct images functors between the categories of 0-modules. There
is an induced map f* : H1 (Y, Z') -+ H'(X, Z'). Starting from a tdo (Dy,A, i) on
E
Y corresponding to w
H1 (Y, Z 1 ), we can construct the tdo. (Dx,pA, f*i) on X
which by definition is associated to the data f*w E H'(X, Z'). Let M(Dy,) (resp.
M(Dxf .A)) be the category of Dyx-modules on Y (resp. Dx,p x-modules on X). Then
one can define two functors :
* inverse image:
* direct image:
f
f.
: M(Dy,A) -+ M(Dx,pA)
: M(Dx,-x) -+ M(Dy,x)
In case f is not affine, the direct image functor has good properties only between
the derived categories of bounded complexes of D-modules. In fact at the level of
derived categories, the functor f' is simply the inverse image of 0-modules shifted
by the relative dimension of f: rd(f) = dimY - dimX. One can transform a left Dmodule into a right one by tensorization with 0 the sheaf of top degree differential
forms. Then the direct image of a right Dx,..A-module M in the derived category is
the direct image as 0-module of .M &DX
fA.
f*Dy,,%. For material of this section we
refer to [Bernstein] and [Beilinson - Bernstein 19831.
2. D-modules with a group action
Let G be a complex algebraic group with Lie algebra g, and suppose G acts on
a smooth variety X. Let Y be a sheaf on X. By definition, a weak action of G on 7
is an action of G on Y which extends the action of G on X. When .7 is a Dx-module,
there is however more structure involved. If a denote the action of G on X, then we
have a morphism da : g -+ r(X, Dx) and P(X, Dx) acts on itself by the commutator
action ad. So there is a map ad -da : g -+ Endr(X, Dx). On the other hand Dx has
obviously a weak action of G, say P, which yields the map d,8 : g -+ EndP(X, Dx).
It is natural to require that the map da be G-equivariant and that ad - da coincide
with dp. More generally, let (D, i) be a sheaf of twisted differential operators on X.
Definition 2.1: An action of G on D is a weak action 3 together with a morphism
12
7r
(X, D) such that:
: g -+
1.
7r
is G-equivariant with respect to Ad on g and P on P(X, D),
2. for
E g : ad7r( ) = dfl( ).
Suppose now that D is a tdo with a G-action given by P and
7r.
Definition 2.2: A (D, G)-module .7 on X is a quasi-coherent Ox -module with a
structure -y of D-module and a weak action b of G such that:
1. -y is G-equivariant with respect to 3 on D and b on 7,
2. for e G g : -)r(e) = db(e)
The difference between a weak action and an action can also be illustrated as
follows. Consider the diagram
X
-G x X
x
(g, X) -+ a(g)X
<-
a+ X
where a is the action morphism and p is the projection on the second factor. Then
D has a G-action if and only if p*D
a*D and a G-action on the D-module 7 is
equivalent to the data of an isomorphism p*7
a*7.
Decoding the definitions, one gets:
Proposition 2.3: Suppose that G acts transitively on X and let H be the stabilizer
of a point x E X.
1. The tdo on X with a G-action are in bijection with the tdo on {x} with an
H-action, that is the 1-dimensional representations r of h = Lie H.
2. The (D, G)-modules on X are in bijection with the (C, H)-modules on {x},
that is the representations (V,,3) of H such that for e E h : 7r(e) = d3(e).
Remark 2.4: From this proposition, we see that if H is connected, the sheaf V of
sections of a G-homogenous vector bundle on X will not be a (D, G)-module when
rankV > 1. In fact in this case, V is a module over a sheaf of matrix differential
operators, as in remark 1.5. We could enlarge the theory of D-modules to include
the case of matrix differential operators acting on matrix valued functions. But we
will not need this generality here.
13
3. Parabolic subgroups and flag spaces
The structure theory of linear algebraic groups is very clearly explained in
Springer's book, so here we will only review certain facts from a point of view
suited to the later developments.
The unipotent radical RuG of a linear algebraic group G is the largest closed,
connected, unipotent normal subgroup of G. G is called reductive if RuG = {e}.
Henceforth, G shall denote a complex, connected, reductive, linear algebraic group.
The connectedness assumption is not at all necessary for the study of D-modules
but it simplifies the exposition here.
A parabolic subgroup P of G is a closed subgroup such that the quotient variety
G/P is complete. A Borel subgroup B is a connected and solvable subgroup of G
which is maximal for these properties.
One proves that a closed subgroup of G
is parabolic if and only if it contains a Borel subgroup. Since G is connected, a
parabolic subgroup is its own normalizer. We prefer to view the flag space X
G/P as the as the variety of all parabolic subgroups of G conjugate to P.
Let P be a parabolic subgroup of G and N = Np its unipotent radical. A Levi
subgroup L of P is a closed subgroup such that the product map LxN -+ P is an
isomorphism of varieties. Then L is reductive, normalizes N and is the centralizer
in G of a connected torus in P. We shall also say that L is a Levi subgroup of G.
If T is a maximal torus in P, there is a unique Levi subgroup of P containing T.
All Levi subgroups of P are conjugate. The projection P -+ P/N gives an isomorphism of any Levi subgroup of P with P/N. Therefore we call P/N the Levi factor of
P and we denote it by Lp; it is canonically attached to P. The Levi factors Lp and
Lp, of two conjugate parabolic subgroups P and P' are not necessarily canonically
conjugate because we can twist a given isomorphim by an inner automorphism of
Lp or LP,.
.
Let P 1 := (P,P) denote the commutator subgroup of P and put Tp = P/P1
Tp is isomorphic to a maximal torus in the center of Lp and we call it the Cartan
factor of P. For two conjugate parabolic subgroups P and P', the Cartan factors Tp
14
and Tt are canonically conjugate. When P is a Borel subgroup, its Cartan factors
coincides with its Levi factor and is isomorphic to a Cartan subgroup of G. This
special case is good to bear in mind for understanding the sequel easily.
Let us denote by boldface letters the Lie algebras of the groups considered. To
define the set of roots of t, in g, we use the following trick. Choose a Levi subgroup
L of P and let C be the connected component of its center. Then C acts by the
adjoint action on g and we obtain the roots of C in g. Note that these roots do
,
not always form a root system. Now C is isomorphic to Tp by the map P -+ P/P
1
hence to every root of C in g corresponds a unique linear form on t,. We call these
linear forms the roots of t, in g; they are independent of the choice of L.
Let R(t,) c t* be the set of roots of t, in g. R(t,) is naturally divided into the
set of roots whose root spaces are contained in n and its complement. Let R+(tP)
be the set of roots of t, in g/p. If a is a root of t, in g, the corresponding root
space g, need not have dimension 1; dim g, is called the multiplicity of a. Let pp
be the half sum of the roots contained in R+(t,) counted with their multiplicities.
Let B be a Borel subgroup of G, contained in P. The map B/B1 -+ P/P1 gives a
canonical surjective homomorphism TB -- Tp which dualizes to an inclusion t*
tb. Hence we may think of R(t,) as a subset of t.
R(t,) is always a root system.
If we had chosen a invariant bilinear form on g*, then R(t,) could be viewed as the
projection of R(tb) into t*, and the multiplicity of a E R(t,) would be the number
of roots in R(tb) which project onto a.
We can also define the roots in the Levi factor
4, as follows.
Choose a maximal
torus T in B. There is a unique Levi subgroup L of P containing T, so we have
the roots of t in E. From the canonical isomorphism t
4, which
R(tep) of roots of t,
in
R(tb) c t*. Set R+(tb,
4,) =
roots in R+(tb,ep). We have p,
-~+
t*, we obtain the set
is independent of the choice of T. R(t,4,)
g
R+(tb) n R(tb,4,) and let p, be the half sum of the
= Pb -
Pp
The outcome of this stylistic exercise is that we have a canonical comparison
between the various root systems considered.
To a Borel subgroup B of G, we
can associate the triple (tb,R+(tp),R(tb)) where tb = b/b and R+(t,) c t.
15
For
different Borel subgroups, these triples are canonically conjugate. So we identify
them with one abstract triple (t,R+,R) called the Cartantriple of G. The reductivity
of G implies that R = R+ - R+. Similarly to a pair B C P we can associate the
quadruple (t,t,,R+(t,,), R+(t,)) where t, = p/pi, R+(t,) C t* '-+ t* D R+(te,).
Note that there is a canonical element p, in t.
Let D(t) be the set of simple roots in R+(t). The set D(e,) = D(t) n R(t,ep)
of simple roots of L characterizes the conjugacy class of the parabolic subgroup P.
Via this correspondence, the subsets of D(t) are in bijection with the G-conjugacy
classes of parabolic subgroups of G.
If P is a parabolic subgroup of G corresponding to the subset I of D(t), let X
XI the space of all subgroups of G conjugate to P. X is called the flag space of
G of type P or of type I; we will also use the words flag manifold or flag variety of
type P. When P is a Borel subgroup of G, we call X the full flag variety of G. X
is always a complete smooth complex projective algebraic variety on which G acts
transitively by conjugation. If ICJ are two subsets of D(t), and if P is a parabolic
subgroup of G of type I, there exists a unique parabolic subgroup
that P C
Q.
flag variety of
Q of type
J such
This yields a smooth fibration Xr --+Xr with fiber isomorphic to the
Q of type
P.
4. Differential operators on flag spaces
Let X be the flag space of G of type P. Let 0 = Ox be the structure sheaf
of X and TX be its tangent sheaf. We denote by a:
g -+ TX the morphism of
Lie algebras defined by the action of G on X. Let U be the enveloping algebra of
g. Put U* := OfcU and endow this sheaf with a multiplication extending the
ring structure of U, and the structure of 0-module on U*. Explicitely for f, g E 0,
A,B E U:
[f 0 A,g 0 BI = fg
[A,B+ fa(A)g 0 B - ga(B)f 0 A
A direct computation shows that this bracket induces a Lie algebra structure
16
on go := 00cg C U*. Set:
p0 = Ker(a : g* --+ Tx) = {Eg*
E px VX E X}
Here p. is the parabolic subalgebra corresponding to the point z in X and 2 is
the value of the local section e at the point x. Since p0 is the kernel of a morphism
of Lie algebras, it is an ideal in g*, and hence p* is also an ideal in g*. Moreover
the restriction of the bracket to p0 is 0-linear and p*/p* ~ Ooct,.
We can use proposition 2.3 to classify the sheaves of twisted differential operators on X which have a G-action. They correspond precisely to the linear forms
A :p --+ C trivial on pi, i.e. to the elements of t.
As is customary the center of
symmetry of the picture is not o but ppEt*. So to avoid further normalizations, we
set D.X to be the tdo on X corresponding to the weight A - pp E t.
We can describe
more explicitly the tdo DA. Every weight A Et* determines a morphism A* : p*
Ox. Let I.X be the ideal of U* generated by the elements (
-
-+
(A - p,)o(e) where e
is a local section of p0 . Then DA = U*/Ih.
Example 4.1: Let A Et*, i.e. A EHom(Tp,C*), and let 0 (A) be the corresponding
invertible G-sheaf of 0-modules. Then Diff 0 (A) = DA+p,. In particular DX = Dp.
Let us denote by DA the global sections on X of Dx. The center of DA is C:
the constant functions. We have a morphism 7r : g -+ DA, since DX is a tdo. It
extends to a morphism 7r : U -+ DA which must send the center Z of U to CED
by a certain character 8. On the other hand the Harish-Chandra isomorphism
4
identifies Z with the ring Z(t)' of polynomials on t invariant by the Weyl group W
of G. One should think of 0 first as a collection of isomorphisms, one for every point
factors of all Borel subgroups of G with t. By transposition, we obtain b* :
SpecZ : A
'-4
-
of the full flag space, which turns out to be constant once we identify the Cartan
XX. It follows that XA = X1 if and only if A = wit for some w E W.
Lemma 4.2: Let X be the flag space of type P with Levi factor L, and let A E t*.
Then the character 0 :Z -+ CC DX coincides with XA+p,.
The proof of this lemma is a standard gymnastic exercise with p-shift. The shift
by p, reflects the difference in the action of T on the volume forms of the full flag
17
variety and of X.
5. Global sections of 9A
Our goal in this section is to describe '(X, DA) in terms of the enveloping algebra
U. In the case of the full flag variety, this question has a nice answer, but for partial
flag varieties the general situation is not clear yet. We shall first construct some
bigger sheaves of algebras D, and Dt, which will give us some insight in the problem.
Recall that po = {
I
Ego I e,, Ep., Vx E X}, and similarly define n* = {e Ego
e En,, Vx E X}. These are both G-invariant subsheaves of g*, hence these are
ideals in g*.
Definition 5.1:
Pt,
DP = U*/Uon*
=
U*/U 0 p*
The center of D is isomorphic to the center Z(f) of U(e), while the center of Ae,
is isomorphic to Z(tp) = U(t,). We can view these algebras as living respectively
on SpecZ(f) = t*/WL and SpecZ(t,) = t.
into t.
Of course t* = (t*"L imbeds naturally
D, is the specialization of De along the subvariety t*, and for A Et*, DA is
the specialization of Dt, at the point A.
Geometrically we can interpret the sheaves D and Dt, as follows. Identify X
with G/P and let P=LN be a Levi decomposition of P. We have to fibrations 7r
=
G/N -+ G/P and r = G/P1 -+ G/P. Then P1
=
?ro(DG/N)L is the direct image
in the category of 0-modules of the sheaf of differential operators on G/N which
commute with the right action of L on the fibers of 7r. Similarly, Dt, = Tr(DG/PIp)C
where C ne Tp is the connected component of the center of L.
We have the Harish-Chandra isomorphism for L : '/4 : Z(e) --+ Z(t)W" which
involves only a shift by pl. We also have a homomorphism 0 : Z -+ Z(e) which
involves a shift by pp. Let us denote by H the space of harmonic polynomials on
t for the action of the Weyl group W of g; dim H(g) =
#
holds for L and H(e)_;H. Chevalley's theorem asserts that:
Z(t) = H 0 Z
Z(t) = H(t) ® Z(t)
18
W. A similar notation
#
It is readily seen that Z(e)= HWL®Z, so that Z(t) is a free Z-algebra on
W/WL generators. Indeed H is isomorphic to the regular representation of W and
HwL is isomorphic to the space of functions on W/WL.
Definition 5.2: Ut = U OzZ( E)
There is a natural map Z(t) -- D1, because although p*/n* is not necessarily
a trivial bundle over X, the ambiguity dissappears if we consider the center of the
enveloping algebra of 1,. Moreover the restriction this map to Z coincides with the
restriction of U -+ De to Z. Hence we obtain a well-defined morphism U1 --+ (X, DA).
Lemma 5.3:
U, -+
r (X, DI) is an isomorphism.
H'(X, D) = 0
for i > 0.
The proof we will sketch generalizes MilitiI's proof for the full flag variety,
De is an isomorphism
cf. Milicic's forthcoming book. First observe that Z(e)
-+
onto the center of DP. Next to show that
~+ I(X,.), it suffices to
U ®z Z(e)
prove that it is an isomorphism at the graded level.
Z. Put Y = G/N , X =G/P,
7r
Set S = S(g), then SG =
: T*Y -+ X and let OTky be the sheaf of regular
functions on T*Y which are right invariant under the action of L on Y and which are
homogenous in the fiber variables of the projection T*Y-+Y. Then grD -
7.
Oz
and F(X, 7r0 O#.y) = R(T*Y)L is the ring of regular functions on T*Y which are
right L-invariant. Since there is a natural inclusion grr(X, Di) c
I(X,grDj), it
suffices to prove that:
S 9z Z(1) -+
R(T*Y)L
To prove this, one resolves the sheaf 7r 0 .f
A
by a Koszul complex C = S* ®
no. C has an obvious structure of left g-module, but it has also a structure r of
right g -module via the formula:
r(x)(u 9 v) = -ux 0 v +u 9 [x,v]
for x eg, u e So and v e An 0. C is endowed with the usual derivation which
preserves the (g,g)-module structure. One proves that this C complex is acyclic.
19
Moreover, S*/S'n* is equal to grDe, thus C is a left resolution of the (g,g)-module
DP. There is a third quadrant spectral sequence whose term E"'- is HP(X, S* ®0
Aq n"*C) and which abuts to its term E;
=HP-(X,grDj).
Now we can compute HP(X, S 0 ®o Aq n*)
identify AqnO with
r4j
S ®HP(X, Aq no). Indeed we can
=
the sheaf of holomorphic q-forms on X via the Killing form on
g. By Dolbeault's theorem HP(X, 11q) ~H!' (X), and since X is a compact Kihler
smooth manifold, the Hodge theorem implies that H"(X, C) ~ ep+q=nHa'P(X). On
the other hand, the cohomology of a flag variety is generated by the fundamental
classes of the Schubert cycles. Algebraic cycles live only in degrees p=q. There is a
natural length function on the quotient W/WL; let us denote by 1, the number of
elements of length p in W/WL. Then we obtain:
HP(X,Aqno) =0
ifp : q
dimHP(X, APn*) = 1,
This implies that the spectral sequence degenerates, and that:
E,= Hn(X, grD) = 0
forn 7 0
grEo, = grr(X,grD) = S®H*(X,C)
H* (X, C) is nothing else than the space of WL-invariant harmonic polynomials
on t* for the full Weyl group W. It follows, using Chevalley's theorem recalled
above, that S ®z grZ(f) is isomorphic to grr(X,grD). And since the gradations
correspond, we obtain the desired result.
E
I have not found a simple description of F(X,DP,,), but at least the following can
be said. Let X1 = G/P1 , and let D(X1 ) denote the algebra of algebraic invariant
differential operators on X1 . The action of G on X1 gives a morphism op : U -*
D(X1 ) called the operator representation of U on X1 . Let I(X1 ) denote its kernel.
Proposition 5.4: (Borho-Brylinski)
I(X1 ) = Ann(U ®U(p) C) =
20
f
Ann(U ®u(p) CA)
The map 7r : X1 -+
X is a G-equivariant Tp-fibration.
The ring 7r 0OX, is
graded by the lattice of characters of Tp acting on the right of X1 and we have
the corresponding gradation on 7rDx,. The zero component is just
Dt,, and since
the G-action commutes with the right Tp-action, roop(U) lies in L(X,Dt,). On the
other hand the right action of Tp on every fiber of 7r gives a monomorphism r:Z(tp)
-+
D(X1 ), and by the commutativity of Tp, irerZ(t,) lies also in L(X,Dt,).
Now composing the Harish-Chandra homomorphism V) :Z -+Z(t) with the natu-
ral projection Z(t)-+Z(t,), we can view Z(tp) as a Z-module. Then it is not difficult
to see that there is a well-defined monomorphism:
U/I(X) (
Z
Z(t,)
Note that when X is the full flag variety, I(X1 ) = 0, and the above map is an
isomorphism by proposition 5.2.
Now we examine the global sections of D), A Et*. Recall that A determines a
character XX:Z-C.
Proposition 5.5: (Beilinson-Bernstein-Brylinski-Kashiwara)
If X is the full flag
variety, for any A Et*, we have:
U/U.Kerxx ~+ P(X, Dx)
For a proof, see [Militie].
Proposition 5.6: Let X be a flag space of type P and A Et*. If A+ PL is dominant
in t, then U -* P(X,DA) is surjective with kernel I(X 1) + U.KerXA.
Proof: Consider the map 7r : Y -+ X where Y is the full flag variety.
r'Dis a
DY,A+PL-module on X. By applying Dy,,\+p, to the section 7r'(1), we get a surjective
map of sheaves Dy,A+,p, -- 7r*D,\. Now the functor F(Y,-) of global sections is exact
because A + PL is dominant, c.f. theorem 1.6.3. Hence F(Y,Dy,A+P,)-+r(Y,y,A+P,)
is still surjective. But U surjects onto the first algebra by the previous result, and
r(y, 7rDA) = r(X,DA). Thus U surjects onto L(X,DA). The assertion on the kernel
is clear from the discussion above.
21
Remark 5.7: If A + pi is not dominant in t,
surjective, even though A is dominant in
t.
then U -- F(X,DA) may not be
A example with unitary highest weight
modules for SP 4 (R) (8 x 8 matrices)-communicated to me by D.Vogan- exhibits
this phenomenon. However it is always true that U/(I(X + U.Ker xA) injects into
J(X,DA).
6. P-affine varieties
The theory of Beilinson and Bernstein works as well over any algebraically
closed field k of characteristic zero. So let X be a scheme over k. Define an Oxring R to be a sheaf of rings on X together with a ring morphism Ox --+ R such
that R is quasicoherent as a left Ox-module. An R-module is then a sheaf of
left R-modules, quasicoherent as a sheaf of Ox-modules. Denote by M(k) the
category of R-modules.
F : M(R)
Put R := L(X, k). There are natural adjoint functors
MOk): A where P(M) := P(X, M) are the global sections of M and
A(N) := R OR N is called the localization of N. r is left exact, A is right exact
and we have the derived functors Rr and LA.
Definition 6.1: We say that X is R-affine if r and A are (mutually inverse) equivalence of categories.
Here is a criterion for R-affinity.
Proposition 6.2: If every R-module is generated by its global sections and H'(X, .M)
=
0 for i > 0, then X is R-affine.
This proposition says that if r is exact and faithful, then it is an equivalence
of categories. It is clear by Serre's theorem that any affine variety is R-affine. Dx
is an Ox-ring; we shall see that any flag space X is Dx-affine. A Et,, of twisted
differential operators on a flag space X of type P for G. For simplicity we consider
only the case k = C. The dA for A E Mor(Tp, CX) define a lattice in t* and hence a
real structure. We shall say that A Et* is P-dominant if for any root a E R+(t,) we
have < A, av >$ 0, -1, -2,....
We shall say that A Et* is P-regularif for any root
a E R+(tp), we have < A, av >$ 0. Recall that the positive roots are those which
22
are in g/p. Via the inclusion t, c
t*, we view the elements of t* as elements of
t*, and there is a well-defined element pt=
Pb - Pp
Et*.
Theorem 6.3: (Beilinson-Bernstein)
1. If A Et* is P-dominant, then the functor
r : M(A)
-+ M(DA) is exact.
2. If A Et* is P-dominant and A+ peE t* is B-regular, then the functor
r
is also
faithful.
Thus under the conditions of the theorem, X is DA-affine. The case of the full flag
variety is explained in [Beilinson-Bernstein 1981] and this theorem can be proved
in an similar way. We will only describe the changes for the key lemma.
Set
0
=0x. Let F be an irreducible G-module, Y = 0 0c F the corresponding
(.T),
G-sheaf. Let
for i = 1,... , k, be a filtration of 7 by G-sheaves of 0-modules
such that the quotients .j/.-
1
~ V (a,) correspond to irreducible representations
of the Levi factor L of P with highest weight Ii Et*. If IL Vt*, V(tt) is not a
D-module strictly speaking because it is not an invertible sheaf. It is the sheaf of
sections of a homogenous vector bundle over X and what really acts on V(It) are
twisted matrix differential operators. The ring of these operators is still generated
locally by 0 and the enveloping algebra U. Hence
(I)
is a U*-module and the
center Z of U acts on it by the character Xm+,'b. Let us call L-weights of F the
weights 1L Et* which appear in the Jordan-H6lder series of F. Let 1L be the highest
weight of 7 and v be the highest weight of F*. Set 7(pt) = V(g) Oo Y . Let
i:
71(v) -- F(v) and p : F -+
i
idm : F1(v)®)9 -+ F®l (v) and pm =p9idM : 1 9M -* M{(IL). Observe that if
Fk/7-1
~ V(ts). For any 0-module M, put iM =
V Et*, then 71(v) ~ 0, and iM : M --+ 7
M (v). Now let M be a DA-module, then
all the sheaves considered above have a structure of U*-modules by the Leibnitz
formula and ijM, pm are morphisms of U*-modules.
Lemma 6.4:
1. Take V Et*. If A Et* is P-dominant, then iM has a right inverse
jM (unique) in the category of U*-modules.
2. Take
,
Et *. If in addition A + pt Et* is B-regular, then pm has a right inverse
qg (unique) in the category of U*-modules.
Proof: Consider the filtration .5 0 M(v) of 7 0 M(v). It is easy to check that
23
the subquotients .F E M(v)/..1 ® .(v)
= .M(A + v) are U*-modules on which Z
acts by the characters Xi = Xx+A+v+p.
Claim: The weight A + Pt = A + Al + v + pt E t* is not conjugate by the Weyl
group of G to any weight A + It, + v + p, for i > 1.
To prove such a statement, we can assume that A is dominant in the analytic
sense,i.e. < ReA, av >;> 0 for all a E R+(t,). For the Bernstein-Gelfand principle
will make the passage from the analytic notion of dominance to the algebraic one,
cf. appendix in [Bernstein-Gelfand]. Now suppose that there exists some element
w E W such that
w(A + pe) = A + Ai + v + pt, for some i,
Since si Et* is the lowest weight of F with respect to t, -it
positive roots. Let us introduce a norm
product and let
resp. t-',
1
+ Ai is a sum of
on t*coming from a W-invariant scalar
I - IN,resp. - L denote the composition of the projections onto t*,
followed by the norm. We have I
I A 1=1 A IN and I Pt
1=1
1=
IN + IXIL for
x Et. Then
Pt IL. Since W acts by isometries, if A + pt is conjugate to
A + jyi + v + pt, they must have the same norm. But
i+v+p IN unless ui =kti,
I A + PIIL <;
I A + Ai + v + pt IL
.
I A+pt IN < IA+
The first inequality follows from the fact that V = -i
Et* is the lowest weight of
F and A is P-dominant, hence B-dominant. The second inequality comes from the
fact that izi is the highest weight of a representation of L. This proves the claim.
By Harish-Chandra's theorem, this means that M = Y, 9 .(v)
X1 -eigenspace of Z in 7 ® .(v).
7
is exactly the
Therefore there exists a unique projection j.M
M4(v) --+ M which is the right inverse of im in the category of U*-modules.
The proof of the second part of the lemma is the same as for the full flag variety
and we will skip it.
To finish the proof of the theorem, one uses the theorems A and B of CartanSerre, as is done for the full flag variety.
I
24
Remark 6.5: The localization functor Ax : M(DA) -+ M(Dx) : M -
DA OD, M
is left adjoint to P. A representation M of g which arises from a D-module on X
must satisfy certain conditions if X is not the full flag variety. First its infinitesimal
character must be of the form A + p for some A E t*, and also its associated variety,
cf. chapter 3, must be contained in the closure of the Richardson orbit of P. Let
x E X. One knows that the natural action of &4
on H.(n,,M) factorizes through t.
and that the spectrum of this action is contained in the intersection of the W-orbit
of A - p with t*. Denote by a subscript A - pp the corresponding eigenspace. Then
in the situation of the theorem, the geometric fiber of the sheaf Ax(M) at x is
CZ 0 0 AX (M) = Ho(nz, M)x,
(1)
7. Holonomic modules and Harish-Chandra modules
Let D be a tdo on X. A D-module is called smooth if it is coherent as a sheaf of
0-module or, equivalently, if after a local isomorphism D = Dx it becomes the sheaf
of local sections of a vector bundle with integrable connection. One says that a Dmodule is coherent if it is locally finitely generated. Let M,(D) denote the category
of coherent D-modules. Consider D as a (D, D)-bimodule a natural functor of duality
* : M,(D)* -* M,(D) by the formula *M = RHom(.M, D[dimX]). One has ** = id.
A smooth D-module V is of course coherent and *V = Homo (V, Qx) with an
obvious Do-module structure. Coherent D-modules correspond to finitely generated
representations and * corresponds to the functor *M := RHom(M, Ux[dimX]).
This duality is different from the usual contragredient functor for representations
and the precise relation between the two is not transparent.
Next consider a locally closed affine imbedding i : Y c--+ X with Y smooth.
We denote by D(y) the tdo on Y inverse image of D on X. Let M be a smooth
D(y)-module on Y, then i.M is a coherent D-module on X. Put si = *i. * M. We
have 'ii M = ili.M = M and there is a unique morphism f : il M ->
i .M such that
i'(f) = idM. Denote Imf by i!.M. If M is irreducible, then i!.M is the unique
25
irreducible submodule of i.M and the unique irreducible quotient of fiM (and the
unique irreducible subquotient of any of these D-modules whose restriction to Y is
non-zero). The modules i.M, i.M are called standard modules or also respectively
maximal and minimal extension of M, while i .M is called the irreducible module
corresponding to (Y, M) or also the middle extension of M.
By definition a D-module is holonomic if it has finite length and all its JordanH6lder components are irreducible modules of the type constructed above. One
says that a holonomic D-module (on compact X in the twisted case) has regular
singularities (RS for short) if all its components originate in bundles with regular
singularities at infinity. The basic property of holonomic modules is that the corresponding derived category of complexes with holonomic cohomology is stable under
the functors of type
f 1, f.;
if M is holonomic then * M is also holonomic. The same
applies to holonomic RS.
Let us return to representations of g. We are going to study (g,K)-modules for
certain algebraic groups K such that the connected component K' is a subgroup of
G. First we say that (g,K) is a Harish-Chandrapair if g is a complex Lie algebra,
K is a complex linear algebraic group (possibly disconnected) such that k is a subalgebra of g and there is a compatible map Ad: K -- Intg. A (g,K)-module M
is by definition a representation of g and an algebraic representation of K on the
same linear space M such that the representations coincide on
k
and the map g
xM -+ M is K-equivariant. (g,K)-modules correspond via the functor of localization to (D,K)-modules, i.e. D-modules M with an action of K such that
the imbedding
k -+D
k
acts via
and the map D x M --+ M is K-equivariant. The case where K
does not act by inner automorphisms on g is also interesting. To include it in this
framework, it suffices to assume that the group G' generated by G and the outer
automorphisms of g given by K, acts on the flag space X and on the tdo D.
To get an interesting theory one needs sufficiently large groups K. Say that K
is admissible if K acts on the full flag variety of G with finitely many orbits or,
equivalently, if k is transverse to some Borel subalgebra. Fix an admissible K. It is
not hard to see that any coherent (D,K)-module is smooth along the stratification
26
given by the orbits of K, so is holonomic and has regular singularities. The irreducible (D,K)-modules are in bijective correspondence via the if, construction with
the irreducible smooth (D(y),K)-modules on the various affinely imbedded K-orbits
Y, and these smooth modules are charaterized by representations of the stabilizers
of points. This readily gives a classification of irreducible (D,K)-modules and so
of (g,K)-modules. For any K-orbit Y, put Ty = K n Py/K n (Py, Py) 9 Tp where
y E Y (note that Ty does not depend on y E Y). Ty is the product of the torus Ty
and the finite abelian group Ty/Ty.
Theorem 7.1: (Beilinson-Bernstein) For A Et,* the irreducible (Dx, K)-modules
are in bijective correspondence with the set of pairs (Y,ry) where Y is a K-orbit in
X and Ty is an irreducible (t,Ty)-module on which t acts by A - p,. If X is the
full flag variety and if A Et is dominant regular, this is also the classification of
irreducible (g,K)-modules with infinitesimal character xA.
On a partial flag variety X of type P, under the hypothesis that A E t,*, is
P-dominant and A + p E t* is B-regular, then we obtain the classification of (g,K)modules with infinitesimal character Xx+p, and associated variety contained in the
closure of the Richardson orbit of P.
To work clearly with the standard modules one has to suppose that the Korbits Y are affinely imbedded. We see that the standard and irreducible modules
corresponding to an orbit Y form families with dim(tp/ty) continuous parameters
and dimTy discrete parameters. All standard modules are irreducible for generic
values of the parameters. If they are irreducible for all values of the parameters
then Y is a closed orbit.
The groups K we will consider are the fixed points of involutions of G. This
corresponds to the Harish-Chandra modules or representations of real reductive
groups. Then the standard modules i. correspond to the standard representations,
cf.[Vogan]. One could also take K=N or B where B is a Borel subgroup of G and
N is its unipotent radical. This corresponds to representations of g with highest
weights. These cases can be reduced to the first - although in general one prefers
to go the other way - thanks to the facts that N-orbits are equal to B-orbits and
27
B-orbits on X are in bijection with G-orbits on XxX. This yields the equivalence
between representations with highest weights and representations of complex reductive groups.
In general the hypothesis in the theorem 6.3.2. can be weakened. For a (DX,K)module M with K is reductive, if A Et* is P-dominant and A + pt Et* is regular
with respect to the roots of K, then I(X, x) 5 0. This follows from the Borel-Weil
theorem.
In the following chapters, we will only consider coherent (D,K)-modules and
finitely generated (g,K)-modules. So we add this hypothesis to the definitions. For
any map f : Y -+ X between smooth algebraic varieties, define the functor
*f!*.
28
f*
=
Chapter II. Spherical D-modules
1. K-orbits in a flag space.
The results of this section are known to specialists.
Let G be a complex connected reductive linear algebraic group. Let B be the
variety of Borel subgroups of G, and P the variety of subgroups of G conjugate to a
fixed parabolic subgroup P. Let K be an algebraic subgroup of G; it acts on B and
P. If the number of K-orbits is finite, then there is a Zariski open K-orbit which
is automatically unique and dense. Conversely:
1.1 Lemma:(Brion) If an algebraic group acts on a flag space with an open orbit,
then the number of orbits is finite.
Two subgroups K and B of G are said to be transversal if k + b = g. The
existence of an open K-orbit on B is equivalent to the existence of a Borel subgroup
B transversal to K. We have a G-equivariant fibration 7r : B -+ P which assigns
to a Borel subgroup B the parabolic subgroup P E P containing B. Therefore the
finiteness of the number of K-orbits in B implies this finiteness in P. The fiber of
7r over P is the variety of Borel subgroups of P. Note that if Y is the closure of one
K-orbit in P, then 7r- 1 Y is a closed K-stable subset of B with the same number of
components as Y hence, it is the closure of one K-orbit in B.
Let 0 be an involution of G and put K = Ge. Then any Iwasawa decomposition
of g with respect to k shows that K acts with finitely many orbits on B.
1.2 Lemma: The orbits of K are affinely imbedded in B.
Proof: Consider the map h : B -+ B x B : B -+ (B,OB). The diagonal action
of G on B x B decomposes this variety into #W orbits of G where W is the Weyl
group of G. To w E W corresponds the G-orbit C, of (B, wB) where B is any Borel
subgroup of G.
Claim 1: The G-orbits C, is affinely imbedded in B x B for any w E W.
Indeed, let us consider the projections pi and P2 : B x B --+ B on each factor.
For simplicity we fix a Borel subgroup B of G. The B-orbits in B are called the
29
Bruhat cells: they are indexed by W and we denote by B. the B-orbit of wB. B,,,
is an affine space of dimension (w). Let wO be the longest element of W and B the
corresponding cell. Then B is open in B.
Consider the open subset V = piI(B0) n pjl(B) C B x B; it is isomorphic
Since B =
to A21(wo).
U
wB*, the subsets
VW1,W
2
= p 1 (wiB0) nl p21 (w 2 B0), for
wEW
wi, w 2 E W, form an open cover of B x B by affine spaces. It suffices to check that
Cw n VW 1 ,W is affine for any w 1 , w 1 , w 2 E W. Consider the map pi : Cw n pi(B*) -+
B* ~ A(wo). It is surjective since C. = G - (B, wB) and the fiber over w 0B E B is
2
{(w,B, wObwB) I b E B} ~ B, ~ A-(w). Since it is B equivariant, it is a fibration
over an affine space with affine fibers. Restricting B* to a smaller open subset U of
B, if necessary, we see that Cw n pi 1(U) is affine. Now the complement of BO in B
is the set of orbits of non-maximal dimension; it is a connected hypersurface H of
B. Hence C, n pi 1 (U) n pil(B ) is the subset of B x B obtain from C,,, n pi 1 (U)
by removing the points of B x H. But C n pi I(U) n (B x H) is a connected
hypersurface in C n pj1(u). Hence C, n piL(U) n pi 1 (B ) is still affine. A similar
.
arguments applies to the other open sets VW,,W 2
We continue the proof of the lemma.
Every K-orbit in B is mapped by h into a single G-orbit. Consider one G-orbit
Y in B x B, h-1Y is a K-stable subset of B, which may be disconnected.
Claim 2: The K-orbits in h-'Y are all open subsets of h-1 Y.
Let us first show that this claim implies the lemma.
h-1Y
B
-+
h
Y
BxB
This is a Cartesian square and j is an affine morphism by claim 1. Since base
change preserves affinity, i is also an affine morphism. Claim 2 says that a K-orbit
in h-1 Y is a union of connected components of h'1Y. Hence it is affinely embedded
in B.
30
To prove claim 2, consider two points B and B' in h-1 Y which are close to
each other, so that we may write B' = exp x B for some x Eg.
Then h(B') =
(expxB,O(expx - B)) and h(B) = (B,OB). Since h(B) and h(B) belong to
the same G orbit Y, for B and B' close enough, there exists y Eg such that
(exp x B, O(exp x B)) = (exp yB,expy OB). But a Borel subgroup is its own normalizer, hence exp(-x) exp y E B and exp(-x) exp Gy E B. Since z and y are
small, (exp)~ 1 exp y ~ exp(-x + y) and exp(-x) exp Oy = exp(-x +
x - y Eb and x - Oy Eb. Therefore x -
Dy).
Hence
8 Eb and this implies
B' = exp (
2
B
Obviously exp(Yev) E K. Thus B' is in the same K-orbit as B. This proves the
claim and finishes the proof the Lemma 1.2. RNow let us consider the K-orbit on a general flag space P. The finiteness of
the number of orbits follows from that for B. But these K-orbits need no longer be
affinely embedded.
1.3 Example:
G=G1 3
P= 0
*
*
P=P 2 (C)
0~
O = Ad diag (1, -1, -1)
so that K = G1I x Gl 2 =
0)*
0*
*.
Then K has three
orbits on P: the point 0 corresponding to the line Iz, 0, olin C' and the hyperplane
H ~ P1 consisting of the lines contained in {[0, X 2 , Xs1 I
X 2 ,xi,
E C} and the
complement C of these two first orbits. C = P 2 \ {p1 U 0} = A 2 \ {0} is not an
affine variety. Note that P has two orbits on P: {O} and P 2 \ {0}. The latter is
again not affine.
If we analyze the proof of lemma 1.2 we first have to replace B x B by P x6 P
where 'P denotes the set of parabolic subgroups of G conjugate to OP. Note that
although there always exist 0 stable Borel subgroups, OP need not be conjugate
to P.
(Take for example GL 3 and K = 03). The G orbits on P x' P are still
31
paramelrized by W/WL where L is the Levi factor of P and so are the B-orbits on
P.
PxIP=
I
G-(P,6(wP)) P=
I
BwP
WEW/WL
WEW/WL
The B-orbits on P are affine spaces. Let Bw,P = P0 be the big cell and {C' =
G -(P,0(wP))}. Then Cf n p-1(P*)
--
P0 is a fibration but the fiber over wP is
isomorphic to P - 6(wP), i.e. a P-orbit on OP. The trouble is that the P orbits on
P or 'P need not be affinely embedded, cf. example above. The second part of the
proof of lemma 1.2 extends easily to the general case.
The fact that a K-orbit Y is affinely embedded means essentially that the
boundary 8Y of Y has codimension 1 in Y. However, a closed subset of an algebraic
variety is always affinely imbedded.
Hence the closed K-orbits in P are affinely
imbedded and smooth. The K-orbit of a Borel subgroup B is closed if and only if
B is 8-stable, [Matsuki, Springer]. Similarly:
1.4 Lemma: The K-orbit of P in P is closed if only if P contains a 6-stable Borel
subgroup.
Proof: Let Y = K - P and
7r
: B -*
P. If Y is closed then ir-'Y is the
closure of one K-orbit. Hence r- 1 Y contains a closed K-orbit say K - B in B. By
Matsuki-Springer's characterization, B is 0-stable and since 7r(B) E Y, there is a
K-conjugate of P which contains B. Hence P contains a 0 stable Borel subgroup.
Conversely, if P contains a 8-stable Borel subgroup B, then Y = r(K-B). Again
by Matsuki-Springer, K -B is closed, hence compact. Therefore Y is compact. [I
Note that if K - P is closed, then K n P is parabolic in K and K- P is isomorphic
to the flag space of K of type P n K.
The map h - B --+ B x B : B e-* (B,8B) can easily be related to Springer's
parametrization of K-orbits on B. Choose a 8-stable Borel subgroup B and a 0stable cartan subgroup T in B. Let A = {g E G I g 1 8g E NG(T)}. K acts on the
left of A and T on the right. Put V = K \ A/T.
1.5 Proposition:(Springer) B = f K - vB
vEV
32
On the other hand B x B = ]I G- (B - wB). Since h maps K-orbits into
wEW
G-orbits, it induces a map h : V -+ W
n
'-4
n-1 n. The image of h consists of
0-twisted involutions, i.e. elements w E W such that w - O(w) = 1.
II.2. (D, K)-modules with a K-fixed vector
Let g=keaen be an Iwasawa decomposition of g with respect to 0. It is not
unique but all choices are conjugate by K. At the group level KAN is only an open
dense subset of G. Let L = CentG(A) and P = LN; this is a Levi decomposition
of the parabolic subgroup P. The G-conjugacy class of P is uniquely determined
by 0, and we say that P is associated to K. The corresponding flag variety X = P
is also called associated to K and K - P is the open K orbit in X.
Note that if G(0, R) denotes a real form of G whose Cartan involution is 6 and
if a is chosen to be defined over R, then P is the complexification of a minimal
parabolic subgroup of G(O, R).
An (g, K)-module V is called K-spherical if the space VK of K-invariant vectors
in V is nonzero. As will be explained in the next section, the irreducible K-spherical
(g, K)-modules are those which are realizable as functions over the real symmetric
space G(0, R)/K(R). Kostant classified the irreducible K-spherical representations
of G(0, R): they all are quotients of principal series representations induced from
one-dimensional representations eA ®1N, A E t,, of a minimal parabolic subgroup
P(0, R) = L(, R)N(0, R) of G(0, R) cf. [Kostant]. Let D be a tdo on X.
We shall say that a (D, K)-module M on X is K-spherical if Hom(O,K)(0, 4)
0. This notion is interesting mainly for simple or standard (D, K)-modules. We
also say that M has trivial K-isotropy if for every K-orbit Y in the support of M,
the isotropy group Ky acts trivially on the fiber of M at y E Y.
2.1 Theorem: Let M be a simple or a standard (D, K) module on X. Then M
is K-spherical if and only if supp M = X and M has trivial isotropy. Moreover M
has at most one K-invariant section, up to scalar.
Proof: Hom(OK)(0, M) is a vector space and we want to compute its dimension. One can verify that the only K-orbit on a flag space which can support a
33
standard (D, K)-module containing the trivial representation of K is the open Korbit. The sections of 0 are determined by their restriction to the open K-orbit
X*. Le i :
c-
X0 be the inclusion of a point in X*. By K-equivariance, it suffices
to compute dim HomK, (1, i'M) where K. is the stabilizer of x in K and 1 is the
trivial module C. This expression shows that if M is K-spherical, then x E suppM,
and by K-equivariance X 0 C suppM, hence X = supp). Let j : X 0 c+ X. Since
M is simple or standard it can be written as jl, j.L or jI.L for some line bundle L
on X0 corresponding to a representation r of K. Now i'M = i'
dim hom(1, C,)
=
1 if r is trivial
=
0
= C,. Hence
otherwise.
The converse assertion is clear from the above discussion. LI
2.2 Remark: Harish Chandra has proved that if 6 is an irreducible representation
of K and V is an irreducible quasisimple Banach space representation of G(O, R),
then
mtp(6,V) < dim(6)
cf. [Godement]. In particular if 6 = 1 then this implies mtp(1, V) E 1, as in the
above proposition.
II.3. Relations with the analytic theory.
It is inspiring to bear in mind the relations between the D-module picture and
the analytic picture on symmetric spaces. In this section we adopt the notation of
the analysts. Let G denote a real reductive Lie group obtained as follows. Consider
the complex connected reductive algebraic group G, (that we previously denoted
by G) and let 0 be an involution of G, with fixed point group K, Take G = G,(6, R)
to be a real form of G, such that 0 is a Cartan involution of G, i.e. K = KC(R) =
K,(0, R) = K, n G is a maximal compact subgroup of G.
Consider another involution a of G, which commutes with 0. Let H, be the
fixed point set of a in GC and let H be H, n G. Using a we can also define another
real form of Ge, namely Gd := G, (a, R) and Kd := Kn Gd. Hd := Hn Gd. Observe
34
that by definition Hd is a maximal compact subgroup of Gd. The symmetric space
Gd/Hd is Riemannian and is called the dual of G/H.
Consider an Iwasawa decomposition of g, with respect to a : gc = hc E a e
n, and let W(a) be the Weyl group of a in g,. Let D(G/H) and D(Gd/Gd) be
the algebras of invariant differential operators on G/H and Gd/Hd respectively.
They are naturally via holomorphic differential operators on GC/H and there is
an isomorphism D(G,/H) ~ S(a)w(") or if you prefer a*/W(a) ~ MaxD(G,/H,)
where a shift by p(a, n) is included. In particular these algebras are commutative
and every A E a* defines a character XX of both algebras.
We are going to study the irreducible representations V of G which occur in
C**(G/H); as is customary we assume that V is quasi-simple, i.e. the center Z of
U acts by scalars on V. Note that Z is the algebra of left and right G-invariance
differential operators on G, hence there is a natural map Z -+ D(G/H). Thus we
reduce the problem to study the irreducible representations of G which occur in
C* (G/H, xX) = the XX-eigenspace of D(G/H) in C (G/H).
Next by a result of [Casselman - Wallach] there is a natural C* topology on
any irreducible Harish-Chandra module coming from a canonical globalization of
M which is an irreducible smooth representation M'" of G. In particular one can
deduce from their result that every K-invariant functional on M extends continuously to M', i.e. HoMK (M,
1)
= HoMK (M , IL) where the second Hom contains
only continuous functionals. This result implies that the study of the irreducible
G-submodules of C**(G/H) is equivalent to the study of the irreducible (g, K)submodules of C**(G/H) := the space of K-finite differentiable functions on G/H.
Assembling these two observations, the problem is reduced to the decomposition of the (g, K)-modules of C'*(G/H; Xx). Because of the K-finiteness these
eigenfunctions are automatically real analytic. Thus, the object of interest is really the space A(G/H; Xx) of real analytic K-finite functions on G/H which are
eigenfunctions of D(G/H) for the eigencharacter XX.
3.1 Lemma: Let V be an irreducible representationof G with infinitesimal character XX. Then the multiplicity of V as submodule of C*(G/H) is finite.
35
Observe that this multiplicity is the dimension of
Homc(V, C*(G/H)) = HomH(V,
by Fr5benius reciprocity.
1)
Let VH be the space of H-coinvariants of V.
VH =
L1HV , V/hV surjects onto VH. and (VH)* = HomH(V, 11) = (V*)H. By the above
result of Casselman-Wallach, V/hV = V*/hV* where V* is the Harish-Chandra
module of V. V 0 /hV 0 can be considered as a space of functions on G which are
eigenvectors of Z, left K-finite and right invariant by H . These conditions form
a holonomic system of differential equations with regular singularities. It is known
that its solution space is finite dimensional.
Let us mention that the multiplicity of V in C* (G/H) as submodule i.e. dimVH
can be quite smaller than the multiplicity of V has subquotient. However, they are
both finite and thanks to the following result we have an upper bound on the
multiplicity of V as subquotient. Let W be the complex Weyl group of g.
Let
g = h e r = k e s; let a, be a maximal abelian subspace of r n s and t = cent(a; g).
Let W1 be the complex Weyl group of t.
3.2 Proposition: [van den Ban] Let 6 be an irreducible representationof K, and
A E a*. Then:
mtp(6; AK(G/H; X,\)) <; dim(6) #W/WI
An advantage of the Riemannian symmetric space is the Helgason isomorphism
between A(G/K; XA) and a space B(XR; LA) of hyperfunctions.
More precisely,
consider the minimal parabolic subgroup P in G defined by P = CentG(A)N where
A, N correspond to an Iwasawa decomposition of G. Then put XR = the variety of
parabolic subgroup of G conjugate to P. As in Chapter 1 we can work invariantly at
every point P of XR, and define Cartan factors Tp = P/(pp) which are canonically
conjugate for different points P E XR. The 8-split component of Tp is A ~ (R*,
where t is the split rank of G and Mor(A, C') - a*. Every A E a* determines a line
bundle LA over XR corresponding to the representation A - p of P for p = p(a, n).
Then B (XR; LA) =
{
hyperfunctions sections of LAoverXR}
36
The Poisson transform is
S: B(XR; LA)
-+
ff
A(G/K; xA)
f (-k)dk
Here we have identified B(XR, LA) with hyperfunctions on G which transform
according to A - p for the right action of P. Helgason proved that S is an isomorphism between K-finite vectors and he conjectured that S is an isomorphism of
topological spaces. This was settled positively by Kashiwara, Kowata, Minemura,
Okamoto, Oshima and Tanaka, cf. [Schlichtkrull].
One should observe that B(XR, LA) depends on A E a*, while A(G/K : XA)
depends only on the orbit W(a) - A, and the definition of S is independent of A. In
this sense S has several inverse maps #,A : A(G/K; X\) -+ B(XR; LJA) for every
wA E W(a)A. #.x is a boundary value map since XR can be viewed as the boundary
of G/K, in the direction w, see [Oshima, Matsuki, p. 346]. If A is dominant, then
S o #A = 1 and 8,\ o S = 1 but if A is not dominant, this is false.
One of the most useful tools in the study of indefinite symmetric spaces is the
Flensted-Jensen isomorphism:
r7 : AK(G/H; x)-~+AKd (Gd/Hi; x)
which is obtained by analytic continuation to GC/H. The right hand side is the
space of real analytic functions on the Riemannian space Gd/Hd which are eigenfunctions of D(Gd/Hd) ~ S(a)"(a) for the same eigencharacter XA, and which are
Kd-algebraic, i.e. they transform under Kd according to the restriction of an algebraic representation of K,. This isomorphism is generally stated for connected
groups but one can extend its proof in [Schlichtkrull] to our set-up.
What we propose to do in this chapter is to describe the (g, K)-submodules of
AK(G/IH; XA) in terms of (DA, K0 )-modules on the complex flag variety X = GC/Pda
However one should understand that this project contains objects of two different
natures. When we speak of a (g, K)-submodule V of AK(g/H; X\), we are given
37
a concrete realization of V. Hence we should focus on (Dx, K)-modules M with a
concrete realization on X, for example the standard ones.
If we do not want to specify a concrete realization, we can work with H-spherical
objects. But, H does not act on a (g,K) or (D,K) module. So we will have to
use a functor which transforms (g, K)-modules into (g, H)-modules: this functor
is simply the inductive version of Zuckerman's functor. The construction of the
analogous functor for D-modules will be explained in
4, and used in
5.
On the other hand, XR is a real algebraic subvariety of X. If we have a standard
H-spherical (D, K)-module on X, we would like to compare it with the space of
hyperfunctions along XR. This question was raised by Flensted-Jensen, and we
will answer it in 6.
We can summarize these relations by a diagram:
H - spherical(Ux,,K) - modules
AK (G/ H; x)
H - spherical(D,K) - modules on X
r
T
S
BK(XR; LA)
The horizontal arrows are bijective when A is dominant, regular. The vertical
dotted arrows involve concrete realizations, and are not defined for all H-spherical
modules. The case of square integrable functions and closed Kg-orbits will be studied in 7.
The Poisson transform cannot be defined directly for an indefinite symmetric
space G/H, because H is not compact. We will see in 8, that one can define a
kind of map between (DA, K)-modules on X and (Dx,, K,)-modules on G/H,.
3.3 Remark: There is a duality involved in the passage from H-spherical representations of G to submodules of C**(G/H).
C*(G/H) if and only if VH
$
V is an irreducible submodule of
0, i.e. if and only if (V*)H
#
0 where V* is the
continuous dual of V, but VH may well be zero. However if M is a (g, Hc)-module,
then H acts algebraically on M and it is clear that
MH 5 0.'=> MH 0 O
38
We will see from the classification of H-spherical (D, K) modules that for a
representation V of G
VH$ 0 <-=* (V)H 00
where V is the contragredient representation. In this sense H carries well the name
of symmetric subgroup. At the other extreme, there is N a maximal unipotent
subgroup of G. Then
VN # 0
-
( )V$Q 0
where N is the subgr6up opposite to N.
II.4. Going from (D, K) modules to
(D, H) modules
The new results of this section are due to [J. Bernstein].
If B is a subgroup of H, (we will take B = H n K for the applications), one
can construct two functors r
and L H from the category of (g, B) modules to the
category of (g, K). The functor r was introduced by Zuckerman who showed that
the derived functors of r are quite meaningful for the theory of representations, see
[Vogan, 1 p. 325]. The functor L appears in [Enright-Wallach], where it is defined
using a double duality. It is the inductive version of r in the following sense.
Given two abelian categories one can define the notion of inductive functors and
projective functors. The inductive functors are those which are right exact, commute
with products and (inverse) limits, so can be left adjoint. The projective functors
are those which are left exact, commute with coproducts (sums) and colimits (direct
limits), so can be right adjoint. For example, given a parabolic subalgebra p of g
with Levi decomposition p = t E n and B = K n L, we may construct the following
39
functors. (For simplicity, assume C, =
(i, B) - mod
MA
--
MOCP
+
--
n - coinv: (p, B) - mod
(t, B)mod
M
(p,B) -mod
M
ind:
+
M/nM 0 C_,
--
(t, B) - mod
{x E M I nx = 0}®C-,
-+4
(p, B) - mod
(g, B) - mod
--
+
n -inv:
M
(p, B) - mod
U(g)OU(P)M
+
--
(g, B) - mod
--
+
pro:
is a representation of B).
(p, B) - mod
+
ext
(A*P n) i
M
Homp(U(g), M)B - finite
-4+
res:
(g, B) - mod
(p, B) - mod
-+4
M
M
-4+
r:
(g, B) - mod
(g, K) - mod
-+4
M
L:
F:
(g,B) - mod
(g, K) - mod
M
largest K - algebraic quotient of M
(g, K) - mod
ek
Fv:
largest K - algebraic submodule of M
-+4
(g, B) - mod
-+4
M6
G6Ek
(g, K) - mod
M6
(g, B) - mod
eSEk M6
(H,6EkM6)-lgebac
Moreover all of these categories are endowed with a contragredient functor ~ ; for
example if M E (g, K) - mod then M = K-finite dual of M.
Note: If K is disconnected FM is the module induced from B - K* to K from
the largest BK* finite submodule of M, similarly for L.
The classification goes as follows with adjoint functors facing each other:
40
inductive functors
projective functors
ext
-~n
n - inv
Mn
=
(M)
extM
=
extM
ext
ind
indM =
proM
res
res
resM =
resM
pro
Fv
n - coinv
L
LM =
rM
F
FM =
FM
r
Now put d = dimk/n. Let us denote by L' the i-left derived functor of L and by ri
the j-th right derived functor of P. To be rigorous L is defined for negative indices
and ri is defined for positive indices.
4.1 Proposition:
L-'
rd-i (
Ad(k/b)
The proof is at the end of this section.
When working with D-modules we should use inductive functors because they
will commute with the operation of taking the fiber of a sheaf. If K is not a subgroup
H, we can still define L~g by LflK a Fk nK. Let us describe more precisely what is
L in our situation. K, H and G are complex reductive groups.
Let R(H) be the ring of regular functions on H then for M E (g, K) - mod,
LKM considered as an (h, H)-module is simply R(H)®(h,KnH)M. Now to put a
structure of (g, H)-module on LM, observe that a (g, H) module M is simply
an (h, H)-module together with an H-map g
0M
-+ M. By definition for any
(g, H)-module V we want L(FV 0 M) = V 0 LM So we get a map g ® LM =
L(Fg 9 M) -+ LM as desired. It is possible to explicit th (g, H)-module structure
of LM. Let R(G) be the space of H-algebraic hyperfunctions on G supported on
H. In other words consider the inclusion i : H -+ G of complex algebraic groups
and put R'H(G) = P(G, i OH) = H" (G, OG) where c = codim(H; G), i. is the direct
image in the sense of D-modules on H and G and H' (G, OG) is a local cohomology
group. Then for a (g, K)-module M:
LH(M)=R'(G)
F
(g,KnH)
41
"KnH
(M)
Note to define r one can put rM = Hom(h,HK) (R (h), M) and use the same
trick as above.
We are going to construct a functor C
modules on X.
from (D,K) modules on X to (D, H)
As before we first use a forgetful functor from (D,K)-mod to
(D, K n H)-mod. Then consider the diagram
X
H x X -- > HxHnKX a
X
where p is the projection on the second factor and q is the quotient map given by the
diagonal action of H
n K on the right of H and on X, and a is the action morphism
of H on X.
4.2 Lemma:Let Y be a smooth variety on which the group B acts freely.
Put
q: Y-> Y/B. Then q* : DYIB - mod -+ (Dy, B) - mod is an equivalence of categories.
Recall that q* is the inverse image in the category of 0-modules. Denote by q+
the inverse of q*. It is related but in general different from the direct image functor.
4.3 Definition:Let M be a (D, K) module on X.Set
LHMX K:= a~q+pr FKnH
Ffl(M)
Kp
Ki M
is a well-defined (D, H)-module for D has a G-action. q+ enters the formula
because we want an H-action on LHM which is as compatible as possible with the
K-action on M. We will only need the case where H/K n H is an affine variety,
then a. is well-defined without recourse to derived categories.
4.4 Theorem: 2H o A)
A, o LK
for A dominant in t*,.
The proof of this result is explained below.
Proof of Proposition 4.1:
L-' ~ rd-i 0 Ad(k/b)
Let M be a (g, B)-module and V a (g,K)-module. We have to relate Hom(g,K)(V, PM)
and Hom(g,K) (V, LM).
A (g, K) module V is just a K-module with a map g
®
V -+ V compatible with
the representations of K on both sides. r and L commute with tensoring by the
42
finite dimensional representation g of K. So it suffices to relate the spaces
HomK(E,IM) and HomK(E,LM)
where E is now a finite dimensional representation of K. Moreover HomK (E, PM) =
HomK (1, E* o FM)and HomK(E, LM) = HomK(11, E* ® LM). Again, because
r and L commute with tensoring by algebraic representations of K, it suffices to
relate the spaces.
HomK(11,
M) and HOmK (IL, LM)
Now HomK(1, PM) = Hom(k,B) (11, M) and the right derivatives of this module
are the spaces H*(k, B; M). On the other hand HomK(I, L,M) = IL(kB)M and
the left derivatives of this module are the spaces H. (k, B; M).
Let I be the ideal generated by b in the exterior algebra
In
At k.
A* k
and put I =
Cohomology is computed using the standard complex A* k/I' with the
usual differential, which gives an acyclic (k, B) resolution of the trivial module 1.
Then the i-th homology group of the complex Hom(kB)(A k/I', M) is H'(k, B; M).
Homology is computed using the same standard complex. The i-th homology group
of the complex A* k/I'®(k,B)M is Hi (k, B; M).
Let b' be the orthogonal of b in k*; b = (k/b)*. Then
Hom(k,B)(Ak/I', M) ~ A'b'
0
M.
(kB)
The top degree of A* k/I* is d = dim(k/b).
Moreover as (k, B) modules:
A' k/I' ~ A'(k/b). Now we have an identification of (k, B) modules.
ty:A'(k/b)-+ A d-ibL (9 Ad(k/b)
because Ad-i bJ- is dual to Ad-i (k/b) and we can view A'(k/b) as the space of linear
maps from Adi (k/b) to Ad(k/b).
Thus we obtain Poincare duality
d
Hi (k, B; M) ~- H
i (kB;M0(k/b))
43
The left hand side is L-'HomK(Il, LM) while the right hand side is Rd-iHomK(1t, PM ®
This proves the assertion. F1
Proof of Theorem 4.4:
Xio
= Ax o Lfor
A E t, dominant.
Let M be a (Dx, K) module on the flag variety X. Since A is dominant, it suffices
to prove that:
r(x, 'K)
=
Lr(x,
m)
2CHM = a~q+p0 .M
X - H x X Xq H xKnH X + X
Let us consider
'CH.M
has and (0, H)-module first: F(H xX,p'M) = R(H)0 T(X, m)
where R(H) denotes the ring of regular functions on H. Also
P(H x Kf
Finally, r(X, a,q+p
X,q+p*M) = R(H)OKnHr(X,M)
) = R(H)
0
P(X, m). So at the level of H-modules
h,KnH
the functor CH commutes with F(X,.).
Now we examine the g-action using the functorial properties of L. Built in the
definition of L, there is a projection formula
LKnH(V ®M) = V ® LnH(M)
for any (g, H) module V and (g, K n H) module M. A (g, H) module M is just an
(h, H) module together with a map g 0 M -+ M of (h, H) modules. The projection
formula gives a map g O LH(M) = L4(g 0 M) --+ L(M). Thus L4(M) is a (g, H)
module, when M is a (g, K) module.
Z'(M) comes naturally equipped with a DA-module structure. But we can view
it in the same functorial way as above, because the t.d.o.Dx has a G-action. If V is a
(DA, H) module on X and M is a (DA, K n H) module on X, we have the projection
formula:
LHfl(V O.M) =V
44
ZeHflH(A)
In particular if we take 1 = 0 and if M is a (Dx, K) module on X, we obtain a
map
DA
L' ()
(M)
-+
which give the same structure of Dx-module on L'(M) as the one given by the
direct definition of
L.
The forgetful functor F obviously commutes with L(X,.). By parallelism, it is
easy to see that the following diagram commutes:
(DX, K) - mod
r(x')
(g,K) - mod
(DA, H) - mod
r(x,.)
(g, H) - mod
This proves that L commutes with r(X,.),
By adjointness, the same is true for L and A. Indeed for any (g, K)-module M
and any (DX, H)-module M, we have
Hom(D,,H)(AxL H(M), IM) = Homg,H)(LH(M), r(X, m))
= Hom(g,KnH)(M, r(X, FM))
=
Hom(g,KnH)(M, FP(X, .))
=
Hom(A,KnH)(AA(M), FM) = Hom(DKnH)(Ax(M), FM)
=
Hom(p,H)(ef AA(M), .).
n
II.5 H-spherical (D, K)-modules.
We continue with the complex connected reductive linear algebraic group G.
Suppose a and 0 are two commuting involutions of G with respective fixed point
sets H and K. Let P be a parabolic subgroup of G attached to H by an Iwasawa
decomposition, and let X be the flag space of type P. Put X* = H - P; it is the
unique open H-orbit in X. Let D, be a sheaf of twisted differential operators on
X, A E t*, and consider a (Dx, K)-module M. Recall the functor L from the previous
section; we will only use 'CH = LnH
KnH.
ince H/KnH is an affine variety, the
action morphism a : HXKnHX -+ X is affine. In case LHM is located in several
45
degrees we consider only its zero component. Because of theorems 1.6.3 and 11.4.4,
we will assume in this section that A E t* is dominant.
5.1 Definition: M is H-spherical if L(X,
Since H acts semisimply on r(X,
HM),
spherical if HomH(F(X, LH.M), C) : 0. But
CKM) 0
0.
it is equivalent to say that M is HH
commutes with r(X,.), so this is
still equivalent to:
HomH(LHr(X, m), C) = Hom (h,KnH)(r(X, m), C) $ 0.
Thus our definition of H spherical module is compatible with the natural one for
(g, K)-modules.
For a point x in X, and a subgroup R of G, we denote by R, the isotropy group
of x in R; R. = R n P,.
5.2 Definition: M has trivial H-isotropy if for every point x E suppM,
j : x -+ X, the isotropy group K n H,, acts trivially on the fiber j'M of M at x, and
A - pp = 0 on h,.
Let a,, = t,1 /h, n t,. Then if M has trivial H-isotropy, A
A
(Dx, K)-module
E a*
at the point x.
M is called standard if it is the maximal extension i.C of
an invertible K-sheaf C on an affinely embedded K-orbit i : Y -* X, it is called
costandard if it is the minimal extension iL, and M is simple if it is the middle
extension i4.2. Of course L corresponds to a representation r : K, -+ C' , y E Y,
such that dr coincides with A - p, on k. n t,, so we will often denote C by Oy (A, r).
We have also the sequence i,
-+ ii.L -
i.L.
5.3 Theorem: Suppose that A+ PL is B-dominant and regular. Let M be a standard
(D,, K) -module on X corresponding to the data (Y, r). If M is H-spherical then
Y n X* 0 0 and M has trivial H-isotropy.
Proof: As for (D, K)-modules, one can easily see that the open H-orbit X*
is the only H-orbit which may give rise to a standard (D, H)-module containing
the trivial representation of H.
dimHomH,(C, j
H(M))
By H-equivariance dimHom(O,H)(O,
where j : x c
KM))
=
X is any point in the open H-orbit X*.
46
jO((M) is simply the fiber of L'(M) at x.
But Y n X0
$ is equivalent to Y n X0
$
suppJ2')(.) = H -suppM = H -.
It is non-zero if and only if x E
because X* is open. When this condition is satisfied, there may exists a non-trivial
map C
-+
joC(M)
only if H, acts trivially on
jo L(M).
By the definition of
LH
this is equivalent to r being trivial on (he, K n H,). [I
The converse assertion is likely to be true but we cannot quite prove it. The
difficulty lies in the fact that in general j'C'M 0 Cizj'.M.
Let us examine the
simplest example.
5.4 Example: SL2 (R)/R*
G = SL 2
H = diagC*
a
K ={(b
b
2
2
a I a -b21
X = P1 with homogeneous coordinates [zo, zi]. K has three orbits: Y1
{[1, -1]}
=
{[1, 1]}, Y- 1
and the complement Y.. H has also three orbits: [1,0], [0,1] and the com-
plement X*. Take x = [1, 2] E Yo n X 0 , j : X C-+ X.
Take M = i.Oy for i : Yo
<-+
X and A = p, i.e. M corresponds to the princi-
pal series representation of SL 2 (R) containing the trivial representation as a submodule, and every even K-type appears once. Then it is easy to compute that
j1'C(M)
=
C3 . But, j'M = C and H. = K, = {
1}. So
LH
does not commute
with j'. On the other hand if At* is not integral, then A E t, L' (i.Oyo(A)) contains
only two H-invariant vectors. This agrees with the fact that for all A E * = C
Aso( 2 )(SL 2 (R)/R*; xx) = Px + P*
where PX is a principal series (g, K)-module.
47
Let us explain how to compute the fibers of
K(i.Oy(r)).
Consider the diagram:
E
where x E Y nX* so that H = M withP2 = MAN. H' = {(h, h-'-x) E HxX}
and E = (H x Y) n H' = {(h,h- 1 -x) E H x Y}.
All k E K n H act freely on H' by: k -(h, H-'- x) = (hk- 1 , kh-1 -x), and H'/K n H
is the quotient variety. M E M acts also on H' by m - (h, h-1 - x) = (mh, h-1 -x).
Obviously these two actions commute. M acts in the same way on E. We start with
the (DA, K)-module i.Oy (A, r) and view it as a (DA, K n H)-module. The restriction
of Oy (A, r) to Y n X0 give rise to a K
n H-equivariant
local system of rank one
that we denote by C(A, r). We will consider the tech cohomology of this locally
constant sheaf.
5.5 Theorem:
HomH(C, j*J..A) = H'(Y
n X,
C(A, r))KnH
where s = dimY + dimM - dimH n K.
Proof: All the Hom's have to be understood in derived categories. We write dx
48
for dimX, etc. Combining base change and adjointness we find:
Homm(C,j*7K
=
[dx - dH]) = Homm(Cj'aq+p M)
HomM(C,bjI'q+pM) = Hom(PH,,K
H,,M)
(b* Cq~p!.M)
q' is an equivalence between the category of (DH'/KnH,A;M)-modules on H'/K n H
and the category of (DHI,,M x K
n
H)-modules on H'; its inverse is q'*.
We
obviously have eVq+ = q't'. So we obtain:
e!p!i* Oy
= Hom(DH,KnHA;M) (q'ob*C,
(r)).
Applying base change and adjointness again, we find:
= Hom(DHIKfH,,;M) (q'ob*C, t
m~pI!Oy(r))
= Hom(DH,/KnHA;M) (q'ob*C,rs IpI0 Cy (r))
=
Hom(DE,x;MxKnH)(r*q'ob*C, (p'
0 s)!Oy(T))
Using the equalities: p o s = c 0 P2, bo[dKnH - dH] = b' and q'b* = uot*, we
obtain:
= Hom(Dx;MxKnH) (r*qoboC[-dH + dKnH], P2
1
C CY(T))
Hom(DH,,;MxKnH) (uot* C,
r*pcY (r)[dKfH
--
H]
Using base change again and adjointness again, one gets:
= Hom(DxO,x;KnH)p3 tu tC,
Wc
Oy (A) [dKnH -
dH])
Now t*C has an action of M, hence u't*C ~ p'toC. Applying ps I and the fact that
p 3 is a smooth morphism we see that p3 Iu't*C - t*C. It follows:
dKnH +
-
= Hom(DYnx,KnH)(*toC,c'OY(A)[dH - dKnH
= Hom(DYnx,),KnH)(wt 0 C[dKnH - dE
49
-
dM)
-
= HomDxO,,KnH) (toCsw*cOY(A)[dH
+ dM)
dM], c Oy(A))
Let us recall that there is a shift by [dx - dHI on the right hand term coming from
the definition of C. Moreover dE= dY + dM. Putting all the shifts on one side we
find:
Hom(H)(C,j* H iOy (A)) =
Hom(DYflxO,KnH) (Oynx*,
ynx* (A) [dx
-
dH+ dy + 2dM
-
dKnH])
Finally dx + dM = dH, and using the Riemann-Hilbert correspondence the above
space consists precisely of the K
n H-invariant
elements in:
Ho(Y n X*, C(A))
where s = dy+dM-dKnH and C(A) is the local system on Y nX* which corresponds
to the D-module Oy (A). The dimension of this cohomology space depends on the
monodromy of C(A). Note that by an obvious homotopy argument, only the action
of the group of connected components of K n H matters. E]
Let us note that K n H is transitive on the connected components of Y n X* at
n XR
O
0, cf. [Matsuki, p. 332].
For example, consider again SL 2 as in 5.4. If we take i : Y= {[1,X1]}
-
least if Y
then E consists of two points which are permuted by M = K n H = Z/2. Thus
Homm(Cj!JC(iC,(A)) = C if A E K^ is trivial on H n K = {t1}, and = 0
otherwise.
On the other hand, if we take Y the open K-orbit and x = [1,2]
then Y n X* is isomorphic to C
\
{1, 0, -1}.
The monodromy given by A acts
around the point 0. H. and K n H are both { 1} and they act trivially. Hence
HO(Y n X*, C(A)) = C
if A E Z and is 0 otherwise. Since the Euler characteristic
is insensitive to the monodromy we see that H'(Yo n X*, CA) = C3
ifA E Z and
equals C2 otherwise. This is the multiplicity of the trivial H-type in LH(iOy.).
5.6 Remark on the component group of H n K.
Sometimes, one wants to work with the symmetric space GR/Hk where Hk
is the connected component of HR = Gjy. The space of functions on GR/Hk is
11
,R (L)
hr n
(ind
ind~ (II =
(It)), wh re ind (I1) is the regular representation of
(?d
d
=
nR
the group of connected components HKc := HR/Hk. Since G is connected, this
50
is an abelian 2-group, so indy(fl) = e,,Ek CC; in particular all multiplicities in
this sum are equal to one. So the space CO (GR/Hi) is naturally divided into the
various subspaces C**(GR/HR, e) := ind RR(C,) where c runs through fHg. We
say that a representation V of GR is (HR, e)-spherical if HomHR (V, C,) : 0. This
notion can be also studied algebraically thanks to the following facts. Let G be a
reductive complex linear algebraic group defined over R. The data of a real form
GR of G is equivalent to the data of a semi-involution c of G, i.e. a conjugate linear
involution. E. Cartan showed that the conjugacy classes of involutions of G are in
bijection with the conjugacy classes of semi-involutions of G, cf. [Bien]. This gives
a bijection between the G-conjugacy classes of symmetric subgroups K of G and
the G-conjugacy classes of real forms GR of G. Moreover if K corresponds to GR
by this bijection, we have:
5.7 Lemma:
K" ~ G"
where the superscript cc denotes the component group.
Let us apply this result to the group H = G'. The involution 0 of G such that
K = G', commutes with c- and so defines an involution
0
H
of H whose fixed point
set is K n H. The real form HR = H n GR of H is associated to 0 H by Cartan's
bijection. The lemma implies:
(K n H)CC
Hg
For e E (K n H)cc, let C, be the (h, K n H)-module on which h acts trivially
and K n H acts by e. Then we say that a (g, K)-module V is (H, e)-spherical if
Hom(h,KnH)(V, Cc)
$
0- If we want to study this notion with the functor L, then
-
we must take LH+ where H+ := H* x (K n H)'c. We have Hom(h,KH)(V, Cc)
HomH+(C,, LH+ (V)) Since the sequence 1 -+ Hit -+ HR -+ Hfg -+ 1 always splits,
we can write similarly HR E Hk x Hg (although this is not canonical). So the
above definition is the same as for real groups.
The point is that theorems 5.3 and 5.5 can be formulated for (H, c)-spherical
representations: it suffices to replace the words "trivial H-isotropy" by "(H, e)isotropy", i.e. A - pp = 0 on h, and K n H, acts on the fibers by the representation
K n Hx -+ (K n H)cc4--> Cx.
51
In 5.5, one replaces KnfH-invariantsby the e-isotropic subspace of H*(YnX 0 ; C(r))
for the action of K n H.
5.8 One more example: Let us remark that if .M = i.Oy(r) is a standard H-
spherical (DX, K)-module, .R = il.Oy(r) may not be H-spherical. This happens for
example if GR/HR ~ SL 2 (R), i.e. G = SL 2
x
SL 2 ,H = diagSL2 ,K = S0 2
X
S0 2 , X = P1 x P1. H has two orbits X0 and X' in X and K has nine orbits. Let
Y be a K-orbit consisting of a line minus a point, then i.Oy is H-spherical but ii.
is not. (Compare with theorem 6.9).
11.6 D-modules and hyperfunctions.
The goal of this section is to show that a standard (D, K) modules on the flag
space X can be mapped into the sheaf BxR of hyperfunctions on X supported on
a real analytic subvariety XR.
The map we will obtain generalize the classical
imbedding
Ox c-+ BXR
First let us pass the bridge going to the analytic set-up. If Xa" is the complex
analytic variety corresponding to X, let A be the sheaf of holomorphic functions on
Xan, and set Dan = AooD. Since A is flat over 0, the functor an: D-mod -+ Da"-
mod is exact and obviously faithful. Moreover X is a projective variety, so we can
apply Serre's GAGA principle to deduce that a" is an equivalence of categories, and
it commutes with all functors of direct and inverse images.
Let a be an involutionof G with fixed point set H and let G(u, R) be a real form
of G which admits a as a Cartan involution. Let XR be the real analytic variety of
minimal parabolic subgroups in G(C, R) and let X be its complexification. Thus
X ~ G/P where P is associated to H by an Iwasawa decomposition. XR is naturally
a real analytic subvariety of X and n = dimRXR = dimcXc. The restriction of
the sheaf A to XR is the sheaf of real analytic functions on XR we write AR := j* A
where j : XR -+ X. Let PR be the functor of local sections with support in XR: it
maps sheaves on X to sheaves on X with support in XR. Let WR be the orientation
sheaf of the normal bundle of XR in X.
52
6.1 Definition: The sheaf of hyperfunctions an X supported along XR is
B = MX(A 9WR)
Sato's
theorem
says
that
0 WR)
IXR,(A
=
0
for
i
$
n.
Another way to think of this sheaf is
B = j.(j'A ® WR)[ n]
where J. and j! are functors in the category of sheaves.
6.2 Remark: The functors jI and j. used here are functors between DR-modules
on XR an Danmodules on X"". DR is simply the restriction of Dan to XR ; this is
well defined because XR is a real form of X. In the case at hand, one can give a
direct description of DR as a quotient of AROcU(g) by an ideal JAR determined
by the character A - pp, as is done in the complex case, cf. 1.4. In this way one can
develop a theory of real D-modules; in particular the shift used in the definition of
jI is [dimXR - dimR(X)] = [-n]
Suppose as before that K and H are subgroups of G defined by two commuting
involutions 0 and
-. Let KR = K n G(o, R) and HR = H n G(a, R). HR is a
compact Lie group acting transitively on XR.
6.3 Definition: K1 = Norm, (G(a,R),K)
K1 is a finite extension of KR ; K1 = KR if KR is compact.
6.4 Lemma: Let Y be a K-orbit in X.
XR. The correspondence Y
'-+
Then YR = Y n XR is a K1 -orbit in
YR is bijective between K-orbits Y in X such that
Y n XR :5 q and K1 -orbits in XR; moreover dimCY = dimRYR.
This bijection
preserves the inclusion order.
Proof: One can compare the explicit parametrizations of K-orbits in X and
KR-orbits in XR given in [Matsuki). A K-orbit is determined by a 0-stable, a-stable
Cartan subspace a and a set of positive roots R+(a), module conjugation by K, i.e.
we should take R+ (a) module WK (a) the Weyl group of a in K. For KR, we have
to take only the 0-stable a-split Cartan subspaces a; but WKR (a) may be smaller
53
than WK(a). This difference is corrected by replacing KR by K 1 . It is then clear
that Y n YR is a single K1 -orbit, possibly empty if (a, R+ (a)) is not defined over R
relatively to G(U,R), i.e. if a is not a-split.
The inverse map YR
-+
Y = K -YR is injective thanks to the above parametriza-
tion, because two a-split Cartan subspaces are conjugate by K if and only if they
are conjugate by KR. So the bijection is established. It clearly preserves the inclusion order. If Y and YR correspond to each other, then YR is a real form of Y in the
sense that it consists of the parabolic subgroups-i.e. points in Y, which are the complexifications of minimal parabolic subgroups of G(a, R), hence dimCY = dimRYR
if YR 5 0. L
6.5 Example: Take G = SL 2 (C), H = S0 2 (C), K = diagC*. Then X = CP1 and
XR = RP'. We have G(u,R) = SL 2 (R). H = S0 2 ,K = diagR* and K1 as R* n
iR*. In homogeneous coordinates, z E K acts on X or XR by [x 0 , X 1 ] -+ [zx 0 , z- 1 x]
for xi E C or R. Therefore K has three orbits: the points [1, 0], [0, 1] and their
complement Co. Similarly K1 has three orbits: the points [1, 0], [0, 1] and their
complement R. which is disconnected. Observe that R. splits into two orbits for
the action of K.
f
: Y -+ X be a map between two topological manifolds; set rdf = dimY
-
Let
dimX: the relative dimension of
f.
We shall denote by wyIx = wy/f 'wx the
orientation sheaf of Y relatively to X. This is the sheaf of sections of the line bundle
on Y whose transition functions are given by the sign of the Jacobian determinant
for the transition functions of the vector bundle Ker(df : TY -+ TX) if f is a
submersion, and Nx(Y) = the normal bundle of Y in X if
f
is an immersion. For a
general map, use the factorization into a cofibration followed by a fibration. Recall
that the functors f* inverse image, f' inverse image with proper support are defined
between the derived categories of complexes of C-sheaves on X and Y, having a
constructible cohomology. The next result can be seen as a generalization of the
Thom isomorphism.
6.6 Property: There is a canonical morphism of functors:
f* -+ fo
wYlX[-rdf]
54
which is an isomorphism when f is smooth.
Proof: We will only need this property for closed imbeddings. To do a little
better, we prove it for smooth maps and closed imbeddings. The general case follows
by showing that the above morphism is independent of the factorization of
f.
Let S be a C-sheaf on X and assume f is smooth. Then f behaves locally like
the projection of a vector bundle p : E -+ X. The Thom ismorphism
H'v(E, p* S) ~ HI-~d'(X, S ®9 ptwElX)
is given by integration along the fibers, and its inverse is the multiplication with
the Thom class of E, cf. [Bott and Tu p. 88]. In the derived category we have:
fif*S
~
fl(f1S 0 wEIX)[-rdf]
S 9 fiwEIx[-rdf]
and there is a unique isomorphism 0 : f* S -+ f'1S
the identity on S
®
wEIx[-rdf] such that f!(o) is
f*wEIX[-rdf].
If f is a closed imbedding, then
f*S
is quasi-isomorphic to the cohomology of
S restricted to Y, while f'S is quasi-isomorphic to the local cohomology ofS along
Y. Let us first take S = C. Then
H (X; C) ~e H.'(Nx Y,; C)
This is seen by working in a slice of X transversal to Y and by using the fact
that, for a point y E X,
H (X, C) = Hs(ByIBy; C) = H,(TyX; C)
where By is a small ball around y. The Thom isomorphism for the vector bundle
p: NxY --+ Y yields
HI(Y,wyix) ~ HC,(N Xy, C)
Since tensorization by wYIx is clearly involutive, we get:
f*C ~> f'C 0 wyx[-rdf]
55
Now for an arbitrary sheaf S, we use Godement's resolution of C by flasque
sheaves:
0
injective. 0
-+
-+
C
-+
S
-+
.9
-+
S
-+
- - - Since we work over a field, the g9
are even
9 0 -+ S ®91 -+ - - - is a flasque resolution of S. Let Fy
be the functor of local sections with support in Y, then there is a natural map:
s
D rygi
-+
ry(s
D gi).
S
Dy(9.)
is quasi-isomorphic to
S ® Iy (C), so we get
a natural map.
f*S D f'c
f's.
Using the result established for C, we obtain the desired morphism.
f*S = f*S ® f*C ~+ f*S ® f'C 0 wylx[-rdf|
6.7 Proposition: Let i : Y -+
X,j : Z -+
f'S ®wylx[-rdf.
l
X be two maps between topologial
manifolds. Let S be a complex of C-sheaves an X having constructible cohomology.
Then the above morphism yields a canonical map of sheaves:
iii'S -+ j*(j'S 0wzlx)[-rdj]
Proof: By adjointness we have a map iii'S -- S. Apply j* to get j*i S
-+
j* S.
On the other hand, the previous property gives as a map j*S - j'S ® wzlx[-rdf].
By composition we get an element of Hom(j*iii'S,j'S ® wyx[-rdf|). Using adjointness of j* and j, we obtain the desired map.I
The Riemann-Hilbert correspondence translates these results into identical statements for D-modules and hyperfunctions on X.
The group K1 acts on B an let us consider the subsheaf BK on which K1 acts
algebraically, i.e. the local sections of B which transforms under K1 according
to the restriction of an algebraic representation of K. The elements of BK are
automatically distributions. Let Y be a K-orbit in X such that YR = Y n XR 0 0,
and i : Y -+ X is affine. Let A E t* be integral and consider the standard (DA, K)module .M(Y, A) = ifi'A(A). We can define similarly BK 1 (A) = subsheaf of B whose
global sections are KI-algebraic.
6.8 Theorem: There is natural morphism: M (Y, A) -+ BK (A) which respects the
action of (DR, K1 ), and is injective if Y is closed.
56
Proof: This follows from 6.6 and 6.7. Indeed the global sections of M (Y, A) are
-
K-algebraic, hence they are mapped to K-algebraic elements. rd(f) = dimRXR
dimRX = -n. Note that with the shift, both modules live in degree zero. If
Y is closed, then .M(Y, A) is irreducible over (Dx, K). But by K-equivariance the
elements of M (Y, A) are determined by their restriction to YR. Hence this map must
be injective. LI
In fact, if Y is closed the image of the (DA, K) module .M(Y, A) consists precisely
of the hyperfunctions in BK1 (A) which are supported in YR. Using this result we
can refine the classification of H-spherical representations. Assume A is dominant
integral.
6.9 Theorem: Let M = i Oy(r) be a (DX, K)-module with Y a closed K-orbit,
YnX
0, (A, r) is trivial on (h,, K n H,) and A + PL is B-dominant regular, then
M is H-spherical.
Proof: One can prove that Y n X*
letter].
$
0 is equivalent to Y n XR, cf. [Matsuki,
By the above result, there is a non-trivial map
f
: i y(A) -+
BK(A).
Moreover il. Oy (A) is the unique irreducible sub quotient of if Oy (A) whose restriction
to Y is non-zero. Hence by the flabbiness of B, f(i Oy (A)) is a submodule of BK (A)1
Now using Helgason's and Flensted-Jensen's isomorphisms, the (g, K)-module M
corresponding to iOy (A) appears as a submodule of AKR (GR/HR). Thus
Hom(h,KflH)(M, 1)
$
0
i.e. HomH(L H(M), 1) 0 0 or also HomH (1, LH(M)) = Hom(o)(O, CH) $ 0.
In general if we require the morphism f : i Oy (A) -+
E]
BK. (A) to be only KR-
equivariant, then there may exist several maps including some injective ones.
11.7 Closed orbits and discrete series.
It is known that when KR is a maximal to compact subgroup of GR, the closed
K-orbits on the full flag variety of G support the fundamental series of representations in C* (G). When rank G = rank Kx., this fundamental series fills in the
eigenspaces of the discrete spectrum of Z acting on L2(G), and is called the discrete
series of G, (for a proof see [Mili~in]). Oshima and Matsuki have shown that similar
57
statements hold for symmetric spaces when one consider real flag varieties and this
result can be translated to complex flag varieties where one should consider only
some particular closed K-orbits, see [Oshima].
In this section we want to recover Oshima-Matsuki's theorem from our formalism, so as to explain the choice of these particular K-orbits. We will also prove that
the discrete series of a symmetric space has multiplicity one.
As before take G with two involutions a an 0 whose fixed point sets are H and
K. Let A be a maximally a-split torus in G, i.e. a torus on which a acts by -id
and which is maximal for this property. Let L = Cent(A; G), then L = MA where
M = Cent(A; H). A may be much smaller than the center C of L, but the adjoint
action of A on g yields a genuine root system R(a) while in general R(c) is not a
root system. Take a set of positive roots R+ (a) and let N be the nilpotent subgroup
of G whose Lie algebra is spanned by the root spaces corresponding to R+ (a). Then
P = LN is a parabolic subgroup of G and the flag variety X of parabolic subgroups
of G conjugate to P is said to be associated to H by the Iwasawa decomposition.
7.1 Definition:
rank GH = dim A = t
We say that we are in the equal rank case if rank GH = rank K/H n K, i.e. if we
can choose A C K.
Let GR be a real form of G such that KR = K n GR is compact and put
HR = H n GR. For A E a*, let L' (G/H ; X) be the space of real analytic KR-finite
functions on GR/HR which are square integrable and eigenfunctions of D(GR/HR)
for the eigencharacter X = xA. We want to describe this space as a Harish-Chandra
module. Let us first recall a result due to Flensted-Jensen, Oshima and Matsuki.
Theorem: LI(G/H; X) # 0 for some X if and only if rank GH = rank K/HnK.
If A is singular, L'(G/H; X) = 0
Therefore we will focus on the equal rank case.
Since X0 = H-P is the open H-orbit in X and HnP = M is the stabilizer of P =
x E X. Since A C K, Y = K.x is a closed K-orbit in X. WG(A) = WH(A). Choose
58
representatives w,..., WM for the cosets in W(H/K n H) := WH(A)/WKnH(A).
Then all the closed K-orbit in X whose intersection with X* is not empty are of
the form Y = KwjP, as j = 1,..., m.
Thanks to theorem 5.2, we can describe all the H-spherical standard (D, K)modules supported closed K-orbits in X.
They must be supported on some Y
and have trivial H-isotropy. It suffices to do the construction for Y = K - x. Let
t, = p/p be the Cartan factor of p. Then a ~ a, = p/pi + m and we have a
canonical inclusion a*
c
,.
The isotropy group of x in K is connected because Y
is a flag space for K; K n H. contains the group A n M of elements of order 2 in A.
Let i : Y -+
X. Let r : K, -- CX be such that dr coincides with A - pp on
kx n t,. Let Oy(r) = indK (r) and set .M(Y, r) = i.Oy(r).
7.3 Proposition:
* M(Y,r) is an H-spherical (DX,K)-module if and only if A
trivial on K
E a* and r is
n Hx.
" When j runs through 1 to m and r through all possibilities, these modules
M(Y,,r)
constitute all the irreducible H-spherical (DP, K)-modules on X,
supported on closed K-orbits.
This is clear by theorem 5.2 and the fact that il = i..
Put M(Y, A) =
I'(X, m)(Y, A)).
Now we can formulate Oshima-Matsuki's result.
7.4 Theorem: In the equal rank case for A dominant and regular in a* there is an
isomorphism of (g, K)-modules:
0@M(Y, A)~ L2 (G/H; x).
j=1
Proof: A - p, is integral in t* since it is a character of A and T, surjects on
A. Hence there is a line bundle L and X such that i'L = Oy(r). Let BK1 (LC) be
the sheaf of K1 -algebraic hyperfunctions along XR with values in the line bundle
L, i.e. BK(1
)==(,n(L WR), cf.
6. Let YR, = Y n XR and put BKI (yj,) be
59
the space of hyperfunctions in BK1 (,C) supported in YR,,. By theorem 6.8, there is
a bijective morphism of (DA,K1)-modules:
.M(Yj,,r) -+ BK 1 (Yj,T)Since A is dominant, the global section functor is exact and this bijective morphism carries over to global sections. It is not difficult to see that YRJ is a single KR-orbit. Indeed A C K and by Matsuki's parametrization of KR-orbits on
GR/PR, there are m = #W(H/K n H) closed KR-orbits on G/PR. X* is a complex neighborhood of XR, hence if a closed K-orbit does not intersect X*, it cannot
intersect XR. We have found that there are m closed K-orbit which intersect X*.
Thus it suffices to prove that if a closed K-orbit intersects X*, it also intersects XR.
This was done in [Matsuki, letter]. As a result , we do not need to use K 1, but only
KR.
Again by the dominance of A, the Poisson transform is an isomorphism. Combined with Flensted-Jensen's isomorphism this gives an imbedding of (g, KR)modules
m
eM(Y,,A)
j=1
c-+
AK(G/H;xA)
where AK (G/H; x\) is the space of real analytic KR-finite functions on GR/HR with
eigencharacter Xx for D(G/H).
Now the deep result proved in [Oshima-Matsuki is that a function in AK(G/H; x,\)
is square integrable if and only if it is the image of a hyperfunction in BKR (L) supported on some Y n XR.
Thus the image of the above imbedding is precisely
L2K(G/H; XA). El
7.5 Corollary: Suppose that A + PL is B-dominant. Let V be an irreducible Hilbert
space representationof G with infinitesimal characterA+PL. Then dim Homc(V, L 2 (GH))
1.
Proof: Let us denote by the same letter V, the Harish-Chandra module of V.
By theorem 7.4, V imbeds into some I'(X, i.Oy (A) where A E a* is P-dominant and
i is the inclusion of a closed K-orbit Y = K -p into X. If A + pt E t* is B-dominant,
60
then the enveloping algebra U of g generates r (X, Dj), cf. 111.2.2.2.
Therefore
r (X, i. Oy (A)) is an irreducible (U,K)-module and for distinct orbits Y's,they are
inequivalent because the
(DA,
K)-modules i.Oy(A) are inequivalent.
Hence V =
P(X, i.0(A)) and occurs with multiplicity one in L' (G/H, X). This gives the result
for almost all infinitesimal characters, in particular the regular ones.Fj
Even if A is P-dominant, it may happen when A is small enough that A + pt is
not B-dominant. D.Vogan has checked by coherent continuation that the modules
r(X, i.oy(A)) - which can also be described as some Ap(A), cf. [Zuckerman] and
111.2.3.2 - are irreducible (U,K)-modules. as long as A E a* is P-dominant. By
theorem 7.4, to prove multiplicity one, it suffices to prove that these modules are
inequivalent for distinct orbits.
In chapter III, we will prove this property for
classical groups and G 2 . We have not found a way to handle the case of groups of
type E, and F in this limit range of A. The point is that the following description
of r (X, i. 0 (A)) as K-module is not explicit enough.
7.6 Proposition (Blattner Formula): For A E t* dominant and Y a closed
K-orbit, there is an isomorphism of K-modules:
F(Xi*i'O(A)) = EY2.O(-1)Hj(Y, O(S(n n s) ®CN_, 9 A d(n n s)))
where n is the nilpotent radical of some p E Y, g = k + s and d = dim (n n s).
Proof: There is a K-invariant gradation on i. Oy (A) given by the Euler operator, radial to Y in X and for which
gri.0 (A) ~ S(Tx/Ty) 9 0(A) 0 fl-'(A)
The shift by f1Xj-Iy (A) is there because to define i.Oy (A), one has to transform
Oy (A) into the right D-module of top degree differential forms
apply i4 and finally multiply back by U'(A).
Oy (A)
on Y, then
As a K-sheaf on Y, fl-1(A)
~
0 (Ad(n n s)).
To prove the identity 7.6 it suffice to apply the functor of global sections in the
derived category to the above equality of graded modules and to compute the Euler
61
characteristics. Hi(X, i. Oy(A)) = 0 for j > 0 because A E a* is P-dominant. So
the left hand side is simply r(X,i.Oy(A)). L
From the Borel-Weil-Bott theorem it follows that the representations of K which
may occur in r(X, i*i'O(A)) have an extremal weight of the form
V = A + Pn - Pc + EaER(nns)nara E
tk
Here t* is the dual of a Cartan subalgebra of k such that a* C t*; pn := p(nns), p,
p(n n k) and Pn + P = p. In particular if A + pn - P, is K-dominant, then A + pn - pc
will be the smallest K-type of r(x,ii'o(A)).
Given integral weight A E a*, it defines a character Xx of D(G/H) , there is a
(g, K) isomorphism
L._(G/H;
where ij
x)
~WEW(H|KnH)P(X, ii
o(A))
is the inclusion of the closed K-orbit 'Y into X. Restricting to K, we
have by Blattner's formula:
top
r(x, itiw!O(A)) = Ei;>o(-1))Hi(wY, Oy(S(wn n s) o Cw(A Thus the set of possible K-types is w(A+pn-PCEaER(-nns)n
P) 9 A(wn
n s))
-a), where w E WH(a)
represents a coset in W(H/K n H) and w makes A + p, - pn is K-dominant. Since
Wp is not conjugate to p by K, those sets
of K-type are different. Taking some
large K-dominant K-types U, of highest weight 1i = A
+
pn - Pc + EaER(nfl)naa
which occurs in
r(x, i~i'Ox(A)),
we have,
Hi(Y,0y(p))
=
U,
j=0
=
0
otherwise
Since yi and wpt are not K-conjugate, U, and Uwvp are inequivalent.
Hence if
we know that both these K-types occur, the (g,K-modules F(X,i*i'Ox(A)) and
F(X,iiw!Ox(A)) will be inequivalent. and all the constituents of L'(G/H; XA)
62
will be inequivalent. Unfortunately there can many cancellations in Blattner's formula, and in general it is very difficult to say if a possible K-type occurs or not.
11.8 An algebraic Poisson transform.
The purpose of this section is to describe a functor which transforms a (D, K)
module on X into a G sheaf SK(M) on G/K. This functor is analogous to the Poisson transform in its effect on modules. Its definition could be formulated in terms
of real flag varieties and real symmetric spaces, and the definitions are related by a
restriction morphism, as in section 6. As for the real case, the inverse functor should
be a kind of boundary value map using the compactification of G/K described in
[Springer] and due to Oshima, De Concini and Procesi.
First we review what is the Poisson transform. Consider a real form GR of
G, and a spherical principal series representation of GR induced from a minimal
parabolic subgroup PR
Ind (A) ~ {f : GR -+ C
I. f(gp)
= eA-(p)f(g) g E GR, p E PR.
f E C** (GR)
}
where A E a* and AR is a maximal R-split terms of PR. GR acts on Ind(A) by
left translation, denoted
e.
Let KR be a maximal compact subgroup of GR. Then
we can decompose GR = KR - AR -NR and PR
=
MR ' AR - NR. This representation
contains exactly one K-invariant vector, say vA, because by Fr5benius reciprocity,
we have:
HomKR (1, Ind(A)) = HomKR
(1, C(KR
/MR)) = HomMR(11
C.
vx can be explicitely written.
v, : BR = KRARNR -+ C: kan
i-+
e ~(a)
There is a G-invariant pairing between Indp (A) and its contragredient IndG(-A)
given by integration on K.
(u,v) =
u(k)v(k)dh for u E Ind
63
(A), v E Ind
(-G)
Hence every vector v E Indp (-A) defines a GR equivariant map.
S, : Indp(A) -- C* (GR)
Sv(u)(g) = (t(g-')uv) g E GRU E IndpG(A)
Since vx is K-equivariant and equal to 1 in KR, we obtain an intertwining operator
S := SV_'
S: Indp(A) -- C*O(GR/KR)
S(u)(g) = ((g- 1 ))u, v-) =
which is called the Poisson transform.
u(gk)dk
When A is dominant (recall that the roots
of n are negative). Indp(A) has a unique irreducible submodule J(A). This is true
also when A is singular because the unitary spherical principal series is irreducible
[Kostant]. J(A) is the socle of Indp(A) and it consists of the functions which have
the smaller growth at infinity.
From Langlands-Militie study of exponents, we
obtain that v,\ E J(A). Dually Ind (-A) has a unique irreducible quotient J'(-A).
v-, can be viewed as a nonzero vector in J'(-A). Moreover v\ is cyclic for Indp(A)
in the sense that G - vA is dense. It follows that when A is dominant, the Poisson
transform S is injective.
8.1 Observation: A representation of G may occur as a subquotient in C**(G/K)
although it has no K-invariant.
For example, take GR = SL 2 (R), KR = SO(2), P = B,
A = p = 1. IndG(p) e C(RP 1 ) vA(kan) = 1.
The G-module structure of C*(RP1 ) is one irreducible submodule C = the constant functions and two irreducible quotients D 2 and D2 which are square-integrable
representations.
For
f
E C*(RP), (Sf)(gK)
= fSO( 2 )f(g-'k)dk. The image
by S of a finite Fourier series E2=,cnein on RP1 is the harmonic polynominal
= 0 cnz"
+
n'-n
n on Dr
SL 2 (R)/SO(2). For K-finite vectors, we have
S(C)
=C
S(D2)
=
S(D2)
=2C[2
zC[z]
64
(C)K
C
(D2)K
0
(D 2 )K
0
In fact not only Ho(k, D2 ) = 0, but also H,(k, D2 ) = 0 for all i, since H1 (k, D2 )
Ho(k, D2 )*.
=
Note that the obvious K-invariant functional on C* (D2), namely
6. =evalution at the origin, is zero on S(D 2 ) and S(D2).
In conclusion, we observe that the KR-spherical irreducible representations of
GR give only a small part of C*(GR/KR), but the Poisson transforms gives a
whole eigenspace of the ring of differential operators on GR/KR. This is the general
situation as was proved by Helgason for K-finite vectors and by Kashiwara, Oshima
et al. for representations of GR, see [Schlichtkrull].
8.2 Theorem: S : B(GR/PR;LA)Z4A(GR/KR;xX)
is a GR-isomorphism when A
is dominant, with inverse a boundary value map 3.
Oshima has constructed a compactification of GR/KR for which GR/PR appears as a piece of the boundary (dimGR/KR = dim GR/PR + dimAR).
We pass to complex algebraic groups G, K, A, P, M. We already know how to
go from (D, K)-modules on G/P to hyperfunctions on GR/PR. Now we want to
transform a (D, K) module on G/P into a (D, K)-module on G/K. By analogy
with the real situation, we should work with the diagram:
G/P+'-G/M-F+G/ K.
n is affine but not proper: its fibers are isomorphic to the open K-orbit in G/P.
Since we want to integrate along the fibers of x, this may look annoying, but it is
rather difficult to ask a smooth map to be at the same time affine and proper (and
different from the identity). The advantage of an affine map is that the direct image
functor is exact.
T M is a (DG/K, K)-module
Let M be a (DA, K)-module, then .c7r
on G/K with
eigencharacter XX with respect to the image of Z in D(G/K): the algebra of Ginvariant differential operators on G/K.
8.3 Definition: Kv7r' is the algebraic Poisson transform of G associated to K and
P.
De Concini and Procesi have constructed a compactification C of G/K' where
K' is the normalizer of K in G, which exhibits G/P as a piece of the boundary,
65
cf.
[De Concini-Procesil and [Springer].
spherical vector in a representation of G.
It is the closure of the orbit of a KIt is a smooth projective variety on
which G acts with a single open orbit: G/K and a single closed orbit: G/P. An
analog for D-modules of the boundary value map is the nearby cycle functor whose
monodromy invariant part is: i*j. where i and j are the inclusions of G/P and
G/K in the compactification of G/K. It involves a choice of A corresponding to the
infinitesimal character Xx. Since C is constructed using only a maximally 0-split
Cartan subgroup A, it is natural to focus on the case A E a* . By theorem 2.1,
this is even necessary to deal with K-spherical representations. Then for a close
of A dominant, one would expect the nearby cycle functor to be the inverse of the
algebraic Poisson transform 1cgr'; I hope to be able to prove it soon.
GR/KR is real analytic submanifold of G/K. We can restrict K.j'.M to GR/KR.
For K-spherical modules, we should obtain a subsheaf of the sheaf of real analytic
K-finite functions on GR/KR which are eigenfunctions of D(G/K). So that this
algebraic Poisson transform and the real Poisson transform. An application of these
considerations would be to define globalizations of (D, K)-modules as W. Schmid
does it for Harish-Chandra modules.
Finally, let us observe that in the same way, we can transport (D, K) modules on
G/P to (D, K) modules on G/H. This is useful for G/H may be the complexification
of an indefinite symmetric space, and the real Poisson transform is not defined in
general when HR is not compact. Since the fibers of G/MH -+ G/H are isomorphic
to the open H-orbit X* in G/P, this explains in another way the condition in 5.3
on the support of an H-spherical (D, K)-module.
66
Chapter III. Microlocalization and Singularities
III.1 Basic Microlocal Notions
1.1 Constructions and Definitions
Given a smooth complex algebraic variety X, one defines on the cotangent
bundle T*X of X the sheaf of Ex microdifferential operators, see [Kashiwara], or
[Schapira]. The sheaf ex is in some sense the localization of Dx, just as the sheaf of
holomorphic functions is a localization of the sheaf of polynomials. More precisely,
a microdifferential operator is invertible wherever its principal symbol does not
vanish, [Schapira, 1.1.3.4]. The sheaf ex is a coherent, noetherian sheaf of rings
with has a Zariskian filtration.
If p= (x, 0) E T*X, then
gr(ex,p)
Oxx[
1,.-
, en]
where n = dim X. If p = (n, e) E T*X, e 5 0, then
gr(ex,p) ~_O,.x,p[ T -1, T
where P*X is the projective cotangent bundle of X, and P is the image of p in
P*X. Let ex-mod denote the category of sheaves of &x-modules on T*X which
are quasi-coherent over OT*x. The coherent ex-modules form an abelian category.
Let 7r : T*X -
X be the projection ; then &X is flat over r-lDx. Therefore the
microlocalization functor mic is exact:
mic : Dx - mod -+ ex - mod
M '-4 ex
r,-1D,
7r~ 1 m
Moreover the functor mic preserves coherency and its image is the subcategory of
x-modules defined on all of T*X. Since ex
I T*X
~ Dx, its inverse functor on this
subcategory is simply the restriction to the zero section.
The support of an Ex-module M is also called its characteristic variety and is
denoted by charM. If M = mic N, char M is simply the characteristic variety of N
in the sense of D-modules. The variety char M is a closed analytic subset of T*X,
67
stable for the action of C' on T*X and involutive, i.e. (char M)' C charA. One
can also define the characteristic cycle of M, denoted [charM].
An x-module M is called holonomic if it is coherent over Ex and char M is
a lagrangean subvariety of T*X, i.e. (char M)' = char M. The subcategory of
holonomic ex-mod is denoted by ex-modh. There is a natural duality on ex-modh
which exchanges left and right Cx-modules:
* : X-+ ext" (M, ex[n])
The property exte (M, ex) = 0 for j # n = dim X characterizes the holonomic
x-modules among all coherent ones.
The sheaf &x operates on the sheaf C, of micro-functions on X. One can also
defines similar sheaves Am, BM, Cm, DM, em for a real C'-manifold M. Then CM is a
sheaf on T*M whose corresponding presheaf associates to an open set V C T*M the
quotient of BM(M) by the space of hyperfunctions on M which are micro-analytic
at every point of M, [Kashiwara, p. 14].
As an example, take X = C, Dx = C[z, a], then ex contains the element
defined away from the zero secton of T*C and (a - 1)-' = (a-' + 8-2 + a-3 +-
a-'
).
The infinite order differential operators do not belong to ex.
1.2 Microlocalization of a non-commulative ring.
Let A = UjA; be a filtered unitary ring such that A- 1 = 0 - this implies that
A is complete with respect to the topology induced by this increasing filtration.
Suppose that grA is a commutative noetherian ring. Then 0. Gabber has defined
the localization of A with respect to a multiplicatively closed subset S C grA. We
follow the presentation of [Ginzburg]. Consider the category of homomorphisms
f : A -- B such a that
(i)
B is a complete Z-filtered unitary ring.
(ii) f preserves the filtration.
(iii) the elements of (grf)(S) are invertible in grB.
68
There is a universal object is : A -+ As in this category; i.e. for any morphism
f : A -+ B in this category, there is a commutative diagram.
A
B
As
In particular, As is a complete and separated Z-filtered ring. The map is : A -+ As
is strongly compatible with the filtrations and
S-1 grA.
gr(As)
If OVS, then As k 0. The map a : A
-
grA is called the principal symbol map.
We have defined As to have the following property: if the principal symbol of a E A
belongs to S then is(a) is invertible in As. Finally, As is a flat A-module.
1.2.1 Remark: For any conic open subset V C SpecgrA, let S = S(V) be the
multiplicative set of elements of grA invertible on V, i.e. which do not vanish
at any point of V.
Then V -+
As(v) is a presheaf defined on the conic open
subsets of SpecgrA. We denote the corresponding sheaf by A = mic(A), and call
it the formal microlocalization of A. This terminology agrees with 1 in the sense
that 6x consists of the elements in mic(Dx) which satisfy a certain convergence
property. Also if M is a coherent noetherian A-module, then the sheaf associated
to V
'-+
As(,)
OA
M is a coherent sheaf of micA-modules denoted micM and is
called the formal microlocalization of M. The functor
mic: A - modc
M
-
A - modc
A®AM
is exact. The support of mic M is the characteristic variety of M.
1.2.2. Construction As: Let t be a transcendental over A. Consider the graded
ring FA = Et'Ai where A = UiA, is the Z-filtration on A.
It is a subring of
A[t, t 1 ]. When the ground field is C, FA should be regarded as a bundle of rings
over C* whose fiber at z E C* is A ~ FA/(t - z)FA and which specializes to
grA = FA/ + FA when t -+ 0. If b E A is a locally ad-nilpotent element, i.e. for
69
every a E A, there is an n E N such that (adb)"(a) = 0 where adb(a) = ba - ab,
then the localization Ab of A with respect to the multiplicative set {bn I n E N}
is well-defined. For example, the localization of the ring FA at t is isomorphic to
A[t, t-1].
Now consider the multiplicatively closed subset S of grA we had at the beginning.
Let FS C FA be the set of homogenous elements in tkAI - for some
~ At/At_1 c grA belong to S. Set
k- such that their images in tkAk/t k-AkI
A(k) = FA/tkFA and S(M) = FS/tkFA n FS. Using the commutativity of grA, it
is easy to check that every u E AMk) is ad-nilpotent, (in fact (ada)k+l I AM') = 0).
This guarantees the existence of the non-commutative localization (S(k))-1A(). The
projective system
-+ FA/t3 FA --+ FA/t2 FA --+ FA/tFA ~ grA
combined with the universal property of localization gives rise to the projective
system.
-+
(S( 3 )) -'A( 3) _ (S( 2 ))-'A(
2
)
-+
(S( 1))-'A(1 ) ~ S-1grA.
Set B = lim(S')-1 A('). This ring is the localized version of FA. The equality
A = FA/(t - 1)FA suggests the definition.
As =
B/(t - 1)B
=lim lim(S')~1A(').
mult. by t
This presentation of As is not practical for applications, but the important
things to remember are the properties of As.
1.3 Microlocalization of DA
We shall need twisted sheaves of formal micro-differential operators in the same
way as DA is a twisted version of Dx. Let X be the complex flag space of type P of
the group G and let A + t* be as in chapter 1. Set ?r : T*X -+ X. We have defined
the twisted sheaf of differential operators DA on X, cf. 1.4.3. The sheaf DA has a
natural filtration by degree, and there is a symbol isomorphism o
Applying 7r-
1
gr
~+ 'r.OTx.
we get an injection & gr(7r-DA) -+ OT.X which maps gr(7r-lDA) onto
70
the sheaf of germs of regular functions on T*X which are polynominal in the fiber
variables. Fix an affine open subset U in X, then T*U C T*X can be identified
with Spec(grDA(U)). For a conic open subset V of T*U such that 7rV = U, let S(V)
be the multiplicatively closed set of elements of gr(7r-DA)(V) = (grDx)(U) which
are invertible on V. Then as in section 2, taking (grDx(U) for A, we can define
x(V) :
(gr7r-DA)(V)s(v). It is easily seen that the open sets V of T*X which
project onto affine open sets of X form a base of the Zariski topology of T*X. Hence
we obtain a presheaf V
ex(V) defined on the set of conic open subsets of T*X.
'-f
By definition, the corresponding sheaf ex is the formal microlocalization of Dx.
1.3.1 Definition: The sheaf ex on T*X constructed above is called a twisted sheaf
of formal micro-differential operators - tmo for short.
The sheaf 7r-'Dx is subsheaf of ex. If D E Dx, 7r-'D is invertible in ex wherever its symbol does not vanish. We also have a symbol isomorphism a : gre
~>
EjEZOT'X(j) where OTrX(i) denotes the sheaf of germs of homogenous functions of
degree j in the fiber variables. Moreover OT-X is faithfully flat over eEZ 0 T-x (j),
[Schapira p. 77], so we will often identify a graded gr(ex)-module M with the module OrT*X,grrM. From the general properties of microlocalization in section 2, we
have the following result.
1.3.2.
The functor mic
Property:
Dx - modc
x - modc :
-e
e4 0,-i1, M is exact and faithful. Its image consists of the ex modules which are
defined on all of T*X; on this subcategory, the left inverse functor is the restriction
to the zero section of T*X.
1.3.3. Remark: Strictly speaking, microdifferential operators are formal microdifferential operators which satisfy a certain convergence property, [Schapira, p.
11]. If E-ooijsrMP, (x, ) is the expression in local coordinates of a microdifferential
operator D defined over an open set U c T*X and pj is homogenous of degree j
in the variable
= (, ....
,). Then D E ex, i.e. D is a microdifferential op-
erator over U if for every compact subset K c U, there exists c > 0 such that
63p
EO
) (,8EK
(P-
( j is finite. So we want the terms
su
p. ,(x, () j to
be the coefficients of a Taylor series on C which converges in some disk of radius e.
71
In this approach, it would be natural to define a sheaf of twisted micro-differential
operators as a pair (, i :
7r~'x
-- ex) locally isomorphic to the standard pair
(ex, ix).
1.4 Microlocalization of Ux
We adopt again the notation of chapter I. Let Z be the center of the enveloping
algebra U = U(g) and pick X E Max Z. Define
Ix = {z - X(z)
Iz E
Z}U
Let p be a parabolic subalgebra of g and define:
J, = Annu(U ®[p,p] C).
I and Jp are both two sided ideals in U. Set
UX := U/(Ix + Jp)
The natural filtration on U induces a filtration on Ux.
1.4.1. Lemma: C := Spec grUx = closure of the Richardson nilpotent conjugacy
class of p in g*.
Proof: [Borho-Brylinski I, p. 455 & 4561E)
C is a closed cone in g* and taking Ux for A in
2, we can define the formal
microlocalization UX of Ux on C. We regard it as a sheaf on g* supported on C.
For any conic open subset V of C, UL(V) consists of Ux together with the inverses
of the elements of Ux whose principal symbol do not vanish of any point of V
n C.
In particular, if V contains 0 E g*, then C C V and Ux(V) = Ux.
111.2 Microlocal study of the moment map.
2.1 I and A as i. and i.
Let G be a complex connected reductive algebraic group and P a parabolic
subgroup of G. Let X be the flag space of G of type P. The cotangent bundle T*X
is isomorphic to the set of pairs (p, x) such that p is a parabolic subalgebra of g
72
conjugate by G to Lie P and x E p C g*. G acts on T*X by conjugating the pairs
(p, x) and it preserves the canonical symplectic structure of T*X. This action gives
rise to a so-called moment map u which in our case is just the projection
u : T*X -+ g* : (p,X) -+ x
it is G-equivariant with respect to the coadjoint action on g*. The image of tz is
C = Gpi- = the closure of the Richardson nilpotent conjugacy class of P in g*. Our
reference for the basic properties of the moment map and their implications for the
relations between D-modules and g-modules is [Borho-Brylinsky I & III].
Now consider an admissible subgroup K of G, that is a subgroup having only
finitely many orbits on the full flag variety of G, and let k be its Lie algebra. Let
M be a coherent (DA, K) module on X for any A E t*, cf. Chapter 1; and let
M = I'(X, M) be the corresponding coherent (g, K) module. If A + PL is dominant
and regular, then M generates M over DA, cf. 1.6. Let Char M C T*X be the
characteristic variety of M. The support of M - denoted Supp M, also called the
associated variety of M - is the subvariety of g* defined by the annihilator of grM,
for a good filtration on M. Finally let W be the set of K-orbits on X.
2.1.1. Lemma:
a) Char M C UYEwTjX
b) p(TjX) = K(pJL n k-I) for all p E Y
c) If A + p(u) is dominant and regular, then pL(Char M) = Supp M.
Proof: These results are proved for A = pe in [Borho-Brylinski III] see 2.5 for
a), 2.4 for b), 1.9 for c). a) and b) are independent of A. The translation functors
give equivalence of categories inside the dominant chamber; this implies c). E
Let X = Xx be the infinitesimal character defined by A E t and the HarishChandra morphism p : t --* Spec Z. Recall the algebra Ux = U/Ix+J, introduced
in 1.3. Then M is naturally a (Ux, K)-module. We also have the sheaf of algebras
Ux = mic Ux on C c g*. micM is a sheaf of (Ux, K)-modules whose support is
contained in C C g*. Let i. and it* denote the direct and inverse images between
calEA-modules on T*X and Ux-modules on g*.
73
2.1.2 Proposition: The following diagram commutes.
(&Ax, K)
-modc on T* X
.
mic T
(Dx, K)
(Ux, K)
-modc on g* -_T
T mic
-mode on X+ 2P
(Ux, K)
-mode on e E G
2.1.3. Remark: For M E Dx-mod, F.M is first a module over TDP. Since DA is a
t.d.o with a G-action, we have a map U -- FDA which factors through Ux. Hence
by "restriction of scalars" we may view FM as a Ux-module.
Proof: If we call v : X -+ e the map of X onto the point e, then r is v, and AX
is v*. By naturality of the construction of the functor mic, the following diagram
also commutes.
T*X4
X
g*
4e
We get the projection formula for (Dx, K)-module F
.(tI*UX
-Dyr - 1,7)
=
Ux (& u(X,
7)
2.2 Associated variety of a submodule of I'.m
First let us prove a simple lemma. Let D be a t.d.o. on X.
2.2.1. Lemma: Let D =P(X, D). if r: D - modc -+ D - modc is exact, then it
sends simple objects to simple ones, or to zero.
Proof: HomD (M, TN) = Homp(AM, N) where A is the localization functor
with respect to D, i.e. AM = D OD M. Let N be an irreducible D-module, and
suppose M is a D-submodule of TN. By adjointness, the nonzero map M -- TN
gives a nonzero map AM -+ N which therefore must be epi because N is irreducible.
Applying r we get a map rAM --+ N which is still epi or zero by the exactness
of r. But there is a natural map M --+ AM, adjoint to the identity on AM. The
composite map M -- TN is our original map, hence it is epi. This implies the
irreducibility of TN. L]
74
Recall that by restriction of scalars, any rDx module on X is a Ux module
for X = xA E Max Z. We are interested in the following problems: let X be an
irreducible DA-module. When is 1 M irreducible over Ux ? When is
.M completely
reducible over Ux?
2.2.2 Proposition: a) If the moment map A : T*X -+ C c g* is birational and
has a normal image, then rDA = Ux, for every A E t*; and the filtrations coincides.
b) If A + pt is dominant then Ux
+FDx is surjective.
Proof: Part a) is due to Beilinson, Bernstein. The result follows from the fact
that the map S(g)/grix+grjp-F(X,grDA) is an isomorphism, which is independent
of A.
Part b) is due to Borho-Brylinski. It follows by applying the translation principle to the isomorphism Ux, ~+ (X, Dx). Note that when C is not normal, or y4 not
birational, the natural filtration on Ux, may be different from the operator filtration
on r(X, Dx).
1
This proposition answers our problem in almost all cases. Unfortunately, some
examples which escape are discrete series of symmetric spaces. So it would be a
shame to disregard them. Applying the technique of microlocalization, we can go
one step further.
Let us consider a closed conic Lagrangean subvariety
is the set of K-orbits in X. Put Z = u(A)
g
A of UYEW Ty X
where W
C. Recall that the generic point Z*
of a scheme Z is any point whose closure is Z. One can think of Z* as being an
open subset of Z whose closure is Z. We say that the moment map pL : T*X is an isomorphism in a neighborhood of
A'
if there is a neighborhood U of
A*
C
in
T*X and a neighborhood V of Z* in C such that it : U -+ V is an isomorphism.
Let M be an 6x (resp Ux) -module on T*X (resp. C) with support
We say that M is irreducible at its generic point if M
over x
I A*
I A*
A
(resp. Z).
(resp. Z*) is irreducible
(resp Ux | Z*). It is clear that if M is irreducible then it is irreducible
at its generic point.
Let M be an irreducible ex-module whose support is
75
A and
suppose p.M
#
0.
0
2.2.3 Lemma: If i is the isomorphism in a neighborhood of A and if A is Pdominant, then 1z,,{ is irreducible at its generic point.
Proof: The first hypothesis means that it is birational and that Z contains an
open dense subset of normal points of C. We can take Z to be this open set and
let
A
S
A'
00
=
-+
p- 1 (Z). Let
A'
and
A'
be their neighborhoods in T*X and C such that
Z' is an isomorphism. Then 4(A') ~ Ux(Z'). Moreover since A is P-
dominant, A. is exact with inverse p*. Hence (p*.M) I Z ~
over (it*S)
IZ
~ Ux
,(.M
I A)
is irreducible
| Z.
The associated variety Ass(M) of a Harish-Chandra module M is the zero set
of the ideal Ann(grM) in S(g) for a good filtration on M. It is clear that when M
has an infinitesimal character Ass(M) = supp(mic M).
2.2.4 Theorem: Let M be an irreducible DX-module on X A > 0. Let N -+ P.M be
a g submodule of P.M. Then:
Ass(N) = Ass(P.m)
Proof: Hom(g,K)(N, r, m) = Hom(D,K)(AN, M).
Thus we get a nonzero map: AN --+ M, which must be surjective since M is
irreducible. By hypothesis P is exact, hence PAN -+ PM is still surjective and the
composite map N -+ PAN --+ .M is the original inclusion. Looking at associated
varieties we get: A,,(N) 9 A,,(Pm) 9 A,,(PAN). (1)
Now A.,N = supp(mic N) and A,.(JTAN) = supp(ji.pi*mic N). But supp(u*mic N)
p~'(suppmic N)and supp(pj) 9 j(supp7) for any ex-module for T*X. Hence
supp(po~*mic N) C ppc1 (suppmic N) = supp(mic N). Comparing this inclusion
with (1), we get
A.,(N) = A..(P.m) = A.,.PAN.L
Remark: CharM D char(AP.M) and they may be different.
The following consequence of the above theorem is clear in light of 2.3.2 and
2.3.4.b.
Corollary: Suppose that A E t* is P-dominant. If a (g, K)-module of
76
the type Ap(X) is reducible, then all its composition factors have same associated
variety.
Now let M be a simple (DA, K)-module on X with characteristic variety
A.
Suppose A is dominant and P(X, m) $ 0.
0
2.2.5 Proposition: If i is an isomorphism in a neighborhood of A, then every
proper quotient of p.(mic M) is supported on a subset of Z = /(A)
of smaller
dimension.
Proof: By 2.2.3. we know that p.(mic M) is irreducible at its genenic point.
Let L be a (Ux,K)-submodule of tii(micM). By 2.2.4 and the above remark, the
support of L is Z. Thus if we take L simple, the L is unique and (it*.(mic
))/L)
IZ
is zero. This implies that supp (p.*(mic M)/L) is contained in the closed subset
0
Z - Z which is of dimension smaller than Z. l
2.2.6 Corollary: If It is an isomorphism in a neighborhood of A and if P.M is a
semisimple (g, K)-module, then P.M is irreducible.
Proof: Given an exact sequence 0 -+ L -*
M -+
N -+ 0 of (Ux, K)-modules,
we have suppM = suppL U suppN. Since P m is semi-simple, every submodule L of
P.M is also a quotient. By 2.2.4 L should have support equal to Z and by 2.2.5, L
should have support strictly smaller than Z. Hence, either L
=
LM or L = 0. L
If P.M is a unitary Harish-Chandra module, then it is semi-simple. In the next
section we give a criterion for unitarity.
The examples suggest that the number of possible submodules of p.(mic M)
should be bounded by the number of connected components of pt' (z) n A, for
0
z E Z.
By an argument on characteristic varieties, this can be proved in very
special cases, see 3.2 step 3.
2.3. Vogan's complete reducibility criterion.
It is known and it can be seen with the functor L introduced in section II.4 that
the global sections of standard (D, K)-modules on the full flag variety are canonically
the K-finite duals of the standard (g, K)-modules constructed in Vogan's book
by cohomological parabolic induction.
We recall Vogan's result which says that
77
cohomological induction from a 8-stable parabolic subgroup preserves unitarity;
hence, the (g, K)-modules induced in this fashion from a unitary representation are
completely reducible. The 0-stable assumption can be relaxed to the hypothesis that
the 'K-orbit of the parabolic subgroup is closed, but we won't need this generality
here. For the study of D-modules, it suffices to know the result for induction from
1-dimensional representations.
Let Z be a closed K-orbit in the flag space X of type P. We have a smooth
fibration r : B -+ X. 7r- 1 Z is a K-stable closed subvariety of B; thus it is the closure
of one K-orbit, say Y. Observe that F =
ir'Z
is smooth by construction. If 0 is
the involution of G defining K, then the K-orbit of P E X is closed if and only if
P contained a 0-stable Borel subgroup. If P itself is 0-stable, then P contains a
0-stable Borel subgroup. For A E g* a linear form fixed by 0 and purely imaginary
with respect to the real form g(0, R) (such that kng(0, R) is a compact subalgebra),
let L(A) = Cent(A; G) and U(A) be a 0-stable unipotent subgroup of G such that
P(A) = L(A) - N(A) is parabolic. Then P(A) is clearly 0-stable. Assume P is of
this form and let p(u) be the half sum of the roots in the unipotent radical U of
P. Suppose A + p(u) is the differential of a character of P. Consider the associated
line bundle 0A on P and pull it back to 0A on B. If i : Y c-* B is the inclusion
and c = codimBY, then the global sections of the standard (DA, K)-module i. 6y(A)
form the space H (B, 6A). We view it as a (g, K)-module.
2.3.1. Lemma: H (B, A) ~ Ap(A)
For the definition of Ap(A) see [Vogan 1984). This lemma appears in [Zuckerman] without the bar over Y. If j : Z c
P is the inclusion of Z = K.P, then we
have of course H (P, 0A) = Hp'(B, 6A), by base change.
2.3.2. Lemma: Hz(P, 0A) ~ AP(A) as (g, K) modules.
This lemma is not difficult to prove using the functor L.
AP(A
=
AAPA(A) =AALA,Ln
I', L
gpro
Ln)(C
K(C
P(u)) KLnK AA
A
C
~ Lk, flind
LnK) (CA
78
C p(u))
(C
0 C,))
Consider (D, L n K)-module CA+p(U) over the point P and let i, : P
<-+
P be the
inclusion. Then
AX(CA+,(U)) = AA(U(g) OP CX+O(U))
ip*CA+,P U'
K
AAAp(A) = LnKP*(CA+p(u))
Since in our case Ox is a well defined G-homogeneous line bundle on all of P, then
LnfKip*(IX+P(u)) =
a*q+prip*(11x+o(u))
iz*(OA
Z)
qKXLnKP
a
=
where
K x P
, pr
P
Thus AxAp(A) = izA(OA
I Z)
p
T i,
C
4-P
= M1(P,
K-P=Z
x). Applying the global section functor we
obtain the result. L
Now we will state Vogan's unitarity result in the particular case of Ap(A).
Let
h be a Cartan subalgebra of p such that A E h*. Let R(u,h) be the roots of h in
g/p. With our normalization the infinitesimal character of Ap(A) is A + p(t).
2.3.3.
Theorem:
If CX+p(u) is a unitary representation of (f, L n K) and if
Re(a, A) > o for all a E R(u,h), then Ap(A) is a unitary (g,K)-module.
This is 7.1 e) together with 8.5 in [Vogan 19841. In fact the final remark in
Vogan's article shows that in some cases unitarity is still true in some strip "below
0". For instance, if P = P'
and G = SLn+1 (C), t* ~ C, and the unique root a
in R(u, t,) is n + 1, so that 2p = n(n + 1). Then for A E C such that ReA > -n,
the above unitarity result is still true, and in fact the vanishing theorem 1.6.2. also
holds; compare with [Vogan 7.1.dl.
2.3.4.
Corollary: a) If Re(a, A + p(t)) > 0 for all a E R(u,h), and if CA+p(u) is
a unitary (t, L n K)-module, then Ap(A) is an irreducible unitary (g, K)-module.
79
b) If Re(a, A) ;> 0 for all a E R(u,h), and if CA+p(u) is a unitary representation
of (t, L n K), then '(P,iz* C) is a completely reducible (g, K)-module.
Proof: b) follows from 2.3.2 and 2.3.3.
a) follows from 2.2.2.b), 2.3.2 and 2.3.3., using the fact that i,.OA is an irreducible (DA-lp(e), K) module.
111.3 Nilpotent varieties
In paragraph 2.2 we saw that the properties of the moment map lead to results
about the irreducibility of DA-modules considered as g-modules. In this section we
pursue this approach and we describe some known results about the singularities of
the image of A. Then we deduce some consequences for the reducibility of standard
H-spherical (g, K)-modules. We have seen in 2.2.2 that if the moment map It is
birational and has normal image, then the global section functor r sends an irreducible D-modules M to an irreducible g-modules. The study of examples suggests
that it should suffice to assume that it is birational and that an open dense subset of
pu(CharM) is unibranch. We have not been able to prove this assertion, but we can
at least give a criterion for unibranchness in terms of multiplicities of Weyl group
representations.
111.3.1 Normality and unibranchness
First we recall a proposition due to Springer for a) and Richardson for b).
3.1.1. Proposition: Let X be a complex flag space of type P for the connected
reductive algebraic group G and let i : T*X -+ g* be the moment map
a) i is a proper map and Impz is the closure of the Richardson nilpotent conjugacy class C, of P in g*
b) i is finite over C, of degree b,= Cent(x,G)/Cent(x,P) I for any x E C,.
See [Steinberg, 4.2].
In particular, since T*X is smooth, when b, = 1, g is a resolution of the singularities of its image, and if all fibers of 1L are connected, then the image of A is
a normal variety. This is the case when B is a Borel subgroup, then X is the full
80
flag variety, and Imp is the whole nilpotent cone )* in g*. Indeed Kostant proved
directly that )*
is normal by showing that )*
is a local complete intersection,
smooth in codimension 1. In fact .1* is even a complete intersection whose defining
ideal is the set of G-invariant polynomials on g* without constant term and the
only singular point of )*
is the origin.
To study the nilpotent element of g*, we may assume that of g* is semisimple
and also we can identify g with g* via the Killing form. It should be said also all
results that we shall review could be formulated for the unipotent elements of the
grou itself, since the exponential maps - over C - is a diffeomorphism between the
set of nilpotent elements in g and the set of unipotent elements. When X is the full
flag variety, we denote it B; otherwise P. So in general the moment map pt sends
T*P into M. Let us put P. =
- 1 (x) for x E M. The fiber can be described easily:
Pz = {p E P I x E n(p)}
where n(p) is the nilpotent radical of p. P are called Spaltenstein varieties. Unless
P = B, the varieties P, are different from P. the fixed point set of the flow generated
by x on P:
P' = {p E P Ix E p}
Both are projective varieties, but the following result is false for P, in general.
3.1.2. Lemma: The varieties P. are connected.
In fact, as we see from 3.1.1. b), when x is a Richardson element for p. P
consists of bp points.
Now we will investigate the normality of a nilpotent variety, i.e. the closure of
a nilpotent conjugacy class. Since a product variety is normal if and only if every
factor is normal, we may assume that g is a simple Lie algebra.
3.1.3.
Theorem: (Kraft-Procesi) If g = stn, then every nilpotent variety is
normal. If g = spa, on, then every nilpotent variety is semi-normal and it is
normal if and only if it is normal in codimension 2.
By g = On is meant g = so, but G = On.
The fact that conjugation is
taken under On affects only the 'very even' classes which appear only when n is a
81
multiple of 4. A variety X is semi-normal if every homeomorphism Z -+ X is an
isomorphism; so roughly it means that the nilpotent variety in the classical groups
have no cusp singularities. In fact Kraft and Procesi have determined that the only
type of abnormal singularity which can appear is the product of an affine space with
two singularities of type A, glued together at one point. A point x of a variety X
is called unibranch if the preimage of x in the normalization of X is connected. An
example of singular unibranch point is
; the typical example of non-unibranch
point is
. A unibranch point is also called topologically normal. If an
algebraic
variety X is seminormal and all its points are unibranch, then X is normal.
Normality (and seminormality) is a local, open condition. Hence the set of
abnormal points is always closed. But the set of multibranch points need not be
closed. In the following picture, all points on the z-axis are abnormal, but 0 is
unibranch.
Let us denote by IH,"(X) the n-th local intersection cohomology group of X at
x with coefficients in Z. An equivalent way to express the unibranchness of a point
x in a complex variety X is to ask that IH2(X) = C. Assume that the moment
map i : T*P -+ U, is birational, where C,, is the nilpotent variety determined
by the Richardson class of P. From the theory of Borho and MacPherson, the
dimension of IH2(Z,, should be expressible in terms of multiplicities of Weyl group
representations. We describe the answer which is quite elegant but unfortunately
still hard to compute. It generalizes to arbitrary G a formula in [Borho-MacPherson,
p. 710] proved for SLn.
All conjugacy classes in g have even dimensions (over C). For z E R, let Cx
82
be the conjugacy class of x and 2d. = codim (C,,: X). One has d. = dimB2 . Let
d, be the codimension in R of C,. Then dimP.
d2 - d,. Let p, be the special
representation of the Weyl group W of G associated to the Richardson class C, by
Springer's correspondence. For any x E R, W acts on H'(B.). Let C, = R,.
3.1.5.
dimIH.,,)
Proposition:
When it is birational, z E NP, we have dimH*(P,) =
= mtp(pp, HzP(Bz))
Proof: Let A* = R*A*Qi'.p.
If t is an irreducible representation of 7r- 1 C.,x E i,, let V(,, be a Q-vector
space of dimension equal to the multiplicity of p in the monodromy representation
of 7rC2 on the top cohomology group of P. Let i* : C. c-+ N, be the inclusion,
and L, the local system on C2 with monodromy representation p. We shall say
that the pair (C., p) is relevant to u if dimP = d2 - d, and V(,,)
is a semismall map - i.e. dim(fiber) 5
$
0. Since y
dim(stratum) - we can apply a result of
Borho-MacPherson to deduce for z E A,
H'(Pz) = e(c2,X,)IH:2(dz-dp)(Cz; L~,) ®V(2 ,,)
where the sum runs over the pairs (C2 , P) relevant to yi and such taht z E Ox g R,.
A similar formula holds on the full flag variety
H'(Bz) = e(.z,)IHi-2d (Ox; L,) ® Vx
But here the Weyl group W acts on both sides.
Borho and MacPherson have
shown that the right hand side is the decomposition of H'(B,) into irreducible
representations: W acts on V(x,,,) by an irreducible representation p(,,) and
(1) dimIHi-2d (0., L,) = mtp(p(3 ,,), HII(Bz)).
In this way, one obtains a bijection called the Springer correspondence:
(2)
{(x, p) I x E R/G,
P E ( 7riCz)^, p appears in H2d.(B2 )}
+-+
{P(,,)} = W^
If we plug formula (2) into (1), we obtain:
E
(3) dim H'(Pz) =
z
2
mtp(p(,
2 1 ,), H+ dp(B,)) - dim V(,,)
EC.9)/p
83
®
(4) H*(Pz) =
IH;2 (dz-p)(O, LV)
® V(Zj)
2EOn CXp
Since C C C, we have d, ;> d,. Hence for -2(d. - d) to be non-negative, we
must have d. = d, and so x E C,. On the other hand for relevant (x, P):
dimV(,w,) = mtp(p , H2(dz.dp)(PZ))
When x E C,, P. is a point by the birationality of it, therefore:
dimV(
=
1
if
=
0
otherwise
=
L
Collecting this information, and since p, = p(z,), (3) and (4) become respectively
dimH*(Pz) = mtp(p,, H 2dp (B))
H*(Pz) = IH,(-A,,) F1
Note two particular cases of this proposition:
-if z E C, , then z is a normal point of n,, hence H*(Pz) = C. H 2dp(B,) is the
top cohomology group of Bz, and the identity 3.15 reduces to Springer's result: the
special Weyl group representation pp occurs with multiplicity one in H2, (B,).
-if z = 0, then 0 is unibranch because i-'(0) = P. B0 = B, H*(B) is isomorphic
to the space of W-harmonic polynomials on t*, so H* (B) is equivalent to the regular
representation of W. The identity 3.1.5. reduces to Joseph's result: p, occurs with
multiplicity one in the space of W-harmonic polynomials on t* of degree d,.
III 3.2. Spherical Representations and Characteristic Cycles.
We will concentrate our attention to the flag spaces X of type P where P is a
parabolic subgroup of G associated by an Iwasawa decomposition to the centralizer
H of an involution o- of G.. These are the flag spaces which appear in chapter II.
There exists a Richardson nilpotent element x for P which is regular in a real form
G(o-, R) of G such that H n G(o-, R) is compact. Therefore, the centralizer of x is
84
contained in some parabolic subgroup conjugate to P, cf. [Kostant-Rallis], and by
3.1.1.b. the moment map t : T*P --
p, is birational.
To decide whether the global sections of an irreducible (DA, K)-module M on P
form an irreducible (g, K)-module or not, we shall do the following:
1. If .A, is normal, then 2.2.2. a) applies and gives a positive answer.
2. If )p is not normal, we look at the characteristic variety of M. If its image
by 1 contains some normal points of M,, then 2.2.4. applies and gives a
positive answer.
3. If g(CharM) does ot contain any normal point of n,, then we look at the
characteristic cycle of M and Pm. If the coefficients of the cycle of largest
dimension are equal, then we can again conclude that P.M is irreducible.
The application we have in mind is a direct proof of the irreducibility of discrete
series representations of symmetric spaces.
Hence, after step 1.
, we will only
consider (DX, K)-modules which are supported on closed K-orbits in P. Also because
very little seems to be known about the normality of nilpotent varieties in the
(Note: For G 2
,
exceptional groups, we will only consider the classical groups.
irreducibility is preserved because the only nilpotent variety which occur are either
the whole cone or the point zero.)
Step 1: The weighted Dynkin diagrams of the nilpotent orbits which are regular
in the real form of G corresponding to H, are obtained as follows. Consider the
Satake diagram of this real group. Put labels 0 on the black dots and labels 2 on
the white dots. Note that these Richardson classes are always even. To translate
these Dynkin diagrams into partitions we apply the inverse of the rule described in
[Springer-Steinberg Ch. IV]. The answer is, with the obvious restrictions on p and q.
85
h
sin
s(gl,
sin
son
sln (R)
sl2n
'P2n
s2n
s02n+1
sop X s02n+l-,
sop, 2n,+ 1-
s02n
Sop X SO2n-p
sop,2n-p
S 0 4n
g12n
son
s 0 4n+2
gl 2n+1
sn+2
--
-
spn
SPq X SPnq
sPq,n-q
-
4
spn
gln
spn(R)
g(R)
x gln_,)
so, xso-,
Satake diagram
sup,n-p
-
--.
Partition
e-
e
-O
s
(2p + 1, 1"-
p-
1
e(n)
(n, n)
.
e-o
2
)
g
e=~0
O-
-
(2p + 1, 12(n-p))
(2p + 1,
K
12(n-p)-1)
(2n, 2n)
-
o-.
(2n + 1, 2n + 1)
((2q + 1)2,
1 2(n-2q-1))
(2n)
N.B.: When 2p = n in case su,,, or so,,, or q = n/2 in case sp,,, the corresponding partitions are simply (n) or (n, n).
In light of [Kraft-Procesi], we can draw the following conclusions, which were
first observed by D. Vogan:
1. if g = stn all nilpotent varieties are normal.
2. if g
=
son, all nilpotent varieties considered here are normal.
3. if g = spn, the nilpotent varieties considered here are normal if and only if
h = gen or sp, x sp,(n = 2q) or sPq,q+l(n = 2q + 1)
The conjugacy class (2n,2n) in s04n is very even. This partition describes one
04n-conjugacy class which splits into two disjoint SO 4 n-conjugacy classes.
The
image of the moment map is the closure of one component. Kraft and Procesi have
proved [Prop. 1.5.4] that this SO 4 n-nilpotent variety is normal. For the other cases,
it suffices to apply 3.1.3 and the fact that these partitions have no degenerations of
type (2n - 2n) -
(2n - 1, 2n - 1,1, 1), cf. [Kraft-Procesi].
However, for the symptectic groups, the partition ((2q + 1)2,12(n-2q-1)) show
that the corresponding conjugacy class is not normal. In codimension 2, it has a
singularity which looks like two singularities of type Aq glued at one point , times
an affine line. So for q = 1, the picture 3.1.4. give a heuristical idea of the kind of
singularity we have in hands.
86
Using 2.2 a) one obtains:
3.2.1. Proposition: For a flag variety X of type P with P minimal in some real
form of G, U surjects onto P(X, D\) for all A E t*, if g =.s4 1 , son, G2 , and if g = sp,
with gR = spn(R), SPq,, or SPq,,+.
Step 2: Take g = sp, h = sp, x sp,,. The group K will determine the real
form of g and h. First we consider K = GL, so that GR = SP,(R) is the split
real form and HR = SPt(R) x SPe
(R). The rank of the symmetric space GR/HR
is min(e, n - f). After a few calculations, one finds that there is a unique closed
K-orbit Y which interests the open H-orbit. Moreover the image p(TjX) C g*
contains nilpotent elements associated to the partition ((2 + 1)2,j2(n-2t-2)); these
are generic in C, = pi(T*X), hence normal points. Using 2.2.5. we conclude that
if M is a standard (Dl, K)-module supported by the closed K-orbit Y with A E t*
then T(X, M) is an irreducible (g, K)-module. However U(g) does not surject onto
P(X, DA) for certain dominant A such that A + pt E t* is not B-dominant.
Now take K = SPk x SPnk. Then GR = SP(k,n-k) and HR = SP(r,e-r) x
SP(n-e+k+r,k-r) for some r < k. (We suppose t < n-t). In the equal rank case,
one must have t < r and 2t < k. It suffices to consider the situation r = k > 2.
After a few calculations, we find again taht there is a unique closed K-orbit Y which
intersects the open H-orbit. But the most regular elements of p(T.X) correspond
to the partition (3 2t, 12(n-3t)). Comparing it with the partition for the Richardson
class in C, we see that if the rank t of G/HR is 1, then we can conclude that
irreducibility is preserved for dominant A and DA, K)-modules supported on Y. In
higher rank we cannot conclude anything. However, observe that the centralizer of
a generic element x of pz(T X) is always connected. It would be interesting to know
if this implies that x is unibranch, i.e. u-'(x) is connected. However using 11.7.5
and D. Vogan's result asserting that the (g, K)-modules P(X, i. oy (A)) which give
rise to square integrable representations of G on GR/HR are irreducible, we obtain
the following multiplicity one property:
3.2.2.
Property:If G is a classical group or G 2 , then the discrete series of
L 2 (CR/ HR) is multiplicity free, for any real form G
87
of G and any symmetric
subgroup H of G.
Step 3: Or last resort is to compute the coefficients of the characteristic cycle
of F.M (also called the associated cycle of P.M). For Y a closed K-orbit and M a
standard (Dx, K) module on M. The characteristic cycle of M is simply [A] = [Tj X],
because M is irreducible. The characteristic cycle of Fm is the direct image in Ktheory of [A]. So we can write:
[Char P.M] =
tz. (A) =
X(Pz,
0(A))[i(A)] +
lower dimensional terms
Here X(Ps, O(A)) is the Euler characteristic of the variety, x is generic in p(TjX)
and 0(A) is the restriction to P of the sheaf 0(A) that we have extended from Y
to X.
If x(P, 0(A)) equals 1 as it is the case if P, is a normal point (and it is birational), then P.M is irreducible at its generic point. Indeed, the coefficient of the
variety Z of largest dimension in Char (rPM) bounds the number of consitutents of
P.M whose associated variety is Z.
88
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