The unramified principal series of p-adic groups:
the Bessel function
by
Mario A. DeFranco
A.B., Princeton University (2009)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
INS?'P E
WUE
MAA -TTs
~F
O TECHNOILOGY
Doctor of Philosophy
JUN I
2 01
at the
BARIES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
A uthor .................
Signature redacted
Department of Mathematics
May 2, 2014
redacted
Signature
..
......
C ertified b y .......... ...... .. .......
.......
(}..
. ... ... ... . ... ...
..e
Benjamin Brubaker
Associate Professor
Thesis Supervisor
Accepted by
Signature redacted
Alexei Borodin
Co-Chairman, Department Committee on Graduate Students
The unramified principal series of p-adic groups: the Bessel
function
by
Mario A. DeFranco
Submitted to the Department of Mathematics
on May 2, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Let G be a connected reductive group with a split maximal torus defined over a nonarchimedean local field. I evaluate a matrix coefficient of the unramified principal
series of G known as the "Bessel function" at torus elements of dominant coweight. I
show that the Bessel function shares many properties with the Macdonald spherical
function of G, in particular the properties described in Casselman's 1980 evaluation
of that function. The analogy I demonstrate between the Bessel and Macdonald
spherical functions extends to an analogy between the spherical Whittaker function,
evaluated by Casselman and Shalika in 1980, and a previously unstudied matrix
coefficient.
Thesis Supervisor: Benjamin Brubaker
Title: Associate Professor
3
4
Acknowledgments
I would like to thank my advisor, Benjamin Brubaker, who introduced me to this
field of study and supplied invaluable guidance and support throughout my time at
MIT. I would like to acknowledge Professor Ju-Lee Kim for a helpful conversation
about Iwahori subgroups and Professor Arun Ram for a helpful conversation about
unipotent subgroups. I would like to thank everyone at MIT for providing me plenty
of time and space to write this thesis. Finally I would like thank my parents, Toni
and Mario, and the rest of my family for giving me so much encouragement in too
many ways that I can count.
5
6
Contents
1
Introduction
2
The Subgroup Iht and Vectors
3
Defining Integrals
27
4
Computing W(p(a) - 0U,;.) via a Quotient Space
31
5
Intertwining Operators Applied to
37
6
Evaluation of the Bessel Function
51
7
Computing WH(X, w)
55
8
9
#
21
7.1
Statement of Theorem .......
..........................
7.2
Explicit Bruhat Decomposition ......
7.3
Paths in a Root System .......
7.4
Nesting Coordinates .......
7.5
Final Integration
7.6
The Support Conditions Are Independent of z . . . . . . . . . . . . .
74
)
95
.....................
.........................
...........................
7
57
65
68
. 71
............................
The Matrix Coefficient S(p(g) - #
55
8
Chapter 1
Introduction
Let G be a connected reductive group over a non-archimedean local field F with split
maximal torus T. Assume that G is defined over o, the ring of integers of F. Let
B = TU be a Borel subgroup for G. The simplest class of representations of this
group are the unramified principal series - those arising by parabolic induction of a
character x of T(F)/T(o). We denote the resulting representation by Ind (X). In
this thesis, we examine two matrix coefficients of the representation IndB(x); one is
known as the "Bessel function" while the other has been previously unstudied.
Recall that a matrix coefficient associated to a representation (p, V) is a function
on the group G, built from a linear functional L : V
-
C and a vector v E V as
follows:
g '-
L(p(g) -v).
(1.1)
The linear functional is often assumed smooth when working in the category of
smooth representations of G(F). We do not assume that here.
Matrix coefficients are key objects in the study of p-adic harmonic analysis and
automorphic forms and representations. An important step in gaining a deeper un-
9
derstanding of matrix coefficients is to determine their values at specific elements
of G. That is, we seek explicit formulas or closed expressions that are more illuminating than the initial definition (1.1).
Two prominent matrix coefficients are the
"Macdonald spherical function"and the "spherical Whittaker function"; the terms
and constructions are explained below. The Macdonald spherical function (which we
abbreviate to "spherical function") was first investigated by Macdonald [Mac] and
later Casselman [Cas]. The spherical Whittaker function was studied by Casselman
and Shalika [CS], who proved a conjecture of Langlands about the precise formula at
torus elements. These authors succeeded in establishing explicit formulas for their
matrix coefficients in terms of certain polynomials significant in combinatorics and
representation theory.
This thesis continues the pursuit of such explicit formulas. The main result is the
evaluation of the Bessel function at torus elements of dominant coweight in terms of
representation-theoretic polynomials. To our knowledge, it had not been recognized
in the literature that this function should even have an explicit evaluation. This thesis
also makes apparent an intimate connection, or analogy, between the Bessel function
and the spherical function; this connection had also been unknown. Specifically, the
Bessel function shares important properties with the spherical function that were
described in [Cas]. We subsequently consider another matrix coefficient, previously
unstudied, and show that it is connected to the spherical Whittaker function in such
a way that extends the analogy we have demonstrated between the Bessel function
and spherical function.
The matrix coefficients mentioned above are constructed by mixing and matching
certain linear functionals and vectors of Ind (X) that are invariant under the action
of certain subgroups of G. These subgroups are K = G(o), a maximal compact
subgroup of G, and a unipotent radical U of a Borel subgroup of G.
10
First, there is the K-invariant functional S, known as the "spherical" functional in
comparison with the archimedean setting, where K = SO,(R) is a maximal compact
subgroup of G = SL,(R). It satisfies
S(p(k) -v) = S(v) for all k E K and v E IndB(x).
Second, there is the (U, 0)-invariant functional W = W(U, V)), where 0 is a nondegenerate character of U. This functional is called the "Whittaker" functional,
again in comparison with archimedean case where matrix coefficients made from this
functional are associated to a differential equation due to Whittaker. It satisfies
W(p(u) - v) = 0(u)W(v) for all u E U and v E IndB(x).
For an arbitrary irreducible smooth representation, it is not clear that non-zero
functionals with these properties exist. However, it is true for unramified principle
series. We note that (K, 1) and (U, 4) are examples of generalized Gelfand pairs:
given a a subgroup H of G and a character 0 of H, recall that (H, 4) is a generalized
Gelfand pair for G if the induced representation IndG(0) is multiplicity free. This
ensures that any G-module homomorphism from V to IndG(,0) is unique up to scalar
and this fact may be exploited in applications.
The invariant vectors in Indi (x) used in the construction of the matrix coefficients
are the K-fixed vector #K, known as the "spherical vector", and the (U, 4)-invariant
vector
#e,
known as the "Whittaker vector". They satisfy
p(k) -$K =OK
for all kE K=G(o)
11
and
p(u) - bU,, = P(u)#U,,
for all u E U.
For an arbitrary smooth representation of G(F), it is of course possible that such
invariant vectors do not exist in the representation.
However, they do exist for
unramified principle series.
Thus we arrive at four possible matrix coefficients for Ind (X) formed from these
choices of functionals and vectors. The spherical function is the matrix coefficient
constructed from S and
#K,
and the Whittaker spherical function from W and
#K-
As stated above, this thesis considers the Bessel function, constructed from W and
#gO;
it gets its name function in the archimedean case which satisfies a differential
equation due to Bessel. We also evaluate the matrix coefficient from S and OUO
which has been unstudied. In the remainder of this introduction, we discuss known
results and methods of proof.
We summarize some appearances of the Bessel function in the literature. The
asymptotics of Bessel function on p-adic groups are studied in the thesis of Averbuch
[Ave]. Cogdell, Kim, Piatetski-Shapiro, and Shahidi use the Bessel function to obtain
stability of local
-factors in [CKP-SS] and [CP-SS.
They also use "approximate
Whittaker vectors" as a way to define the Bessel function. In addition, Gelbart and
Piatetski-Shapiro use the Bessel function to study e-factors. Baruch [Bar] and Lapid
and Mao [LM] study general properties of the Bessel function integrals. The matrix
coefficient construction using a vector in Ind)BG(X) appears in Baruch and Mao
[BM] and Soudry [Sou] but they do not name the vector they use as the "Whittaker
vector". The authors of [BM] restrict their attention to rank 1 groups and their
double covers; we note that the the limiting process used there to define the integral
of the matrix coefficient is generalized in this thesis to arbitrary G. These authors
12
do not seek explicit formulas for these functions on p-adic groups. Finally, in [P-S,
Piatetski-Shapiro employs a vector similar to 4
in the finite field setting, but there
it is called the "Bessel vector".
I first encountered the notion of the Whittaker vector while examining [GKMMMO]
which deals with constructions of archimedean Whittaker functions. After computing the corresponding matrix coefficient in the p-adic setting for certain groups, I
conjectured the explicit formula for arbitrary G. Upon learning the similarity with
the formula for the spherical function, I examined Casselman's paper [Cas] to see if
there was a connection. This led me to seek analogues of the objects in [Cas]. We
briefly compare [Cas] and this thesis below.
First we describe the setup of the main result, Theorem 1.0.2. The root datum
(X, R, Xv , Rv) of the pair (G, T) determines G as an abstract group whose presentation is given by chapters 9 and 10 of [Spr]. Here X and Xv are lattices in duality
by the pairing (,), and R C X and Rv C Xv are dual root systems. Let o denote
the ring of integers in F and p the maximal ideal of o. Let 7r be a fixed generator
of p and q be the order of the residue field o/p. Let K be the subgroup G(o) of G;
K is maximally compact. Fix a Borel subgroup B of G with unipotent radical U.
Let R+ is the choice of positive roots in R that corresponds to this choice of Borel
of group with R- denoting -R+.
In general, for any positive system S of R, we
write the corresponding unipotent group as U(S) and set U- = U(R-). Then U(S)
is generated by unipotent elements u.,(t), where -y E S and t E F. Let D be the
simple roots in R+ with D- = -D.
The Haar measure m on G is normalized so
that u 7 (o) has measure 1 for each y E R. Let 0 denote a C-valued character of U
that is trivial on ua(o) but nontrivial on u,(p- 1 ) for each a E D. We also use
4
to
denote a character of U~ that is trivial on ua(o) but nontrivial on ua(p 1 ) for each
a E D-; this should not cause confusion.
13
Let T be the maximal torus in B. T is generated by elements p(t) for p E Xv
and t E F. Say that p E Xv is dominant if (a, p) < 0 for all a E D. We denote
A(7r) by 7rA. Now, we let C(XV) denote the vector space of rational functions in the
indeterminates 7rcv", a E D. In C (Xv), we denote 7r& r O" by -rcv+Ov for a, / E R.
We give C(Xv) a topology by completing with respect to the ideal generated by the
7r-av, a E D (that is, geometric series in 7r-&' are convergent). Let X : B -+
be the tautological character on B defined by X(7rA)
=
7rA
C(Xv)
(caution: we take this
notation for C(Xv) from [HKP] where it is used slightly differently). Let p be the halfsum of roots in R+. Let 6B the modular character of B. We have
j1(2r") =
q-(P).
Let W be the Weyl group NG(T)/T with we the long element of W with respect
to R+.
The Weyl group acts on X and Xv, and for w E W we let Xw be the
unramified character of B defined by Xw(7r")) = 7rw'"). W also acts on C(Xv) by
w(7rP) =
irw(p)
and we extend this action by linearity. In addition, for a positive
system, S set w(U(S)) = U(w(S)). We let we denote the long element of W with
respect to R+.
Let IndG(X) denote the principal series representation of G constructed by normalized induction:
IndG(X) = {f : G -+ C(Xv)if(bg) = 6/ 2 X b)f(g) Vb E B,g E G}.
The action p of G on IndG(X) is right-translation. Note that we do not require the
functions
f to be
locally constant, for we will need to consider functions that are not
locally constant everywhere on G.
The Whittaker vector <5, is defined via the Bruhat decomposition
G= U BnwU
wEW
14
where the element n
is defined as follows. For a E D, the element n0 is defined by
nc,= u(1)u-a(-1)ua(1).
Given a reduced decomposition w
=
sa...sam with ac E D possibly non-distinct, nw
is the element
nw = n, ...
nam
where n, is independent of the reduced decomposition.
Definition 1.0.1 In Ind (X,) there is a vector called the "Whittaker vector" which
we denote by
#vo.
We define it as a C(Xv)-valued function on G by
0",(bnwu)
=
/2Xv
X
(b)#v ,(nw)k(u)
(1.2)
where
6
OvOn)=
and 6 is the Kronecker delta. We let
#
tV,wt
denote
The notation is meant to convey that the Whittaker vector transforms under righttranslation by u E U as
OuP(g)
= #u,"P(g)(u).
We define the Whittaker functional W = W(U-,
W(f) =
When
#u,Vp)
f
=
k
4)
by integration over U-:
f(u)4-'(u) du.
, the above integral is not absolutely convergent. We define W(p(g)-
to be a limit of absolutely convergent integrals as explained in chapter 3. We
15
now can state the main theorem.
Theorem 1.0.2 Let ao E T(o) and p E X'
be dominant. Then
-1/2
~
W(p(aoir") -
) =
(7r")
w
WEW
(p_1 1q--R
17
)
.
(-ER+
This document contains the proof of this theorem when R is a direct sum of root
systems of types A, B, C, and D with some discussion of type E. The proof for
exceptional types is forthcoming. The type dependence is solely contained in chapter
7.6.
As mentioned above, this document will describe the similarities between the
Bessel function and spherical function. We now summarize the proof in [Cas} of the
formula for the spherical function. Recall that K is the subgroup G(o) of G.
Definition 1.0.3 For v E W, let 0'
E IndG(X') denote the normalized K-fixed
(spherical) vector defined by
#v (bk) = 6/2 v(b).
We let OK denote #1.
The Iwasawa decomposition
G=BK
shows that
#K
is well-defined. The spherical function F(g) is then defined by
I (g) =
K(kg) dk.
'
16
Compare our Theorem 1.0.2 to Theorem 4.2 of [Cas].
To establish this formula,
Casselman makes use of the Iwahori subgroup I of K.
Definition 1.0.4 Let S C R be a choice of positive roots. Define I(S) to be the
subgroup of G generated by T(o) and the elements u,,(x) subject to the following
conditions:
x EP
if 7 E -S
x Eo
if YE S.
(1.3)
We call such u,,(x) the unipotent generators of I(S).
For our fixed positive system R+, we let I denote I(R+).
The space of Iwahori-fixed vectors Ind3(X)'
c IndG(x) has a basis (the "standard
Iwahori-fixed basis") given by the set {#K-w}wDefinition 1.0.5 For v, w E W, define
#vg.,
to be the vector in Indi (Xv) determined
by
#K;w(g)
Let OKw denote
= OK(g)1BnflI(g)-
. Define the vector space VK(Xv) C Ind (Xv) to be the space
spanned by the set {K;w}w; thus VK(Xv)
=
Ind((Xv)I.
Casselman then introduces another basis of VK(X) which he constructs using the
intertwining maps Tw : IndG(Xw) -+ Ind (X) (see chapter 5). These new basis vectors
are {Tw(#
)}w. Casselman shows that
#K =
OK
Zw(TJ
wEW
in this basis is
7
yER+
17
)TW('K l)
(here we use the formulation of [HKP3). It remains to determine
1K TW (OK)(kir'") dk.
Casselman proves that this integral evaluates to the simple monomial term, and the
formula is proved.
The formula in Theorem 1.0.2 was conjectured by the author after computations
for certain G. Motivated by the similarity to the spherical function, we seek a proof
of Theorem 1.0.2 analogous to that found in [Cas] for the spherical function. At each
step of the proof in [Cas}, this thesis constructs corresponding objects in our proof
for the Bessel function that play the same role. These objects include a previously
unstudied subgroup of G which we call Iht and new vectors in the principal series
representation we call Ou,,p;w (chapter 2), which take the place of the Iwahori subgroup
and the Iwahori-fixed vectors, respectively. Due to the similarities surrounding these
matrix coefficients, we call these two frameworks the K-fixed side and the (U, 4) side.
We do encounter difficulties not present in the K-fixed side; to overcome them we
develop new techniques, including novel ways to compute the Whittaker functional
(chapter 4) and combinatorial objects we call paths in root systems (chapter 7). At
this point, we do not have a foundational reason why the analogy between these
matrix coefficients exists; searching for such a reason is one of many avenues of
future work.
Now we state relations in G used throughout the thesis. These are taken from
the presentation of G given in chapter 9 of [Spr]. Fix a total order on R. Then, for
linearly-independent a and
#
E R, there exist structure constants c7,;i,j
18
E F such
that
U.(X)uo(y)
=
uQ(y)
(
uia+jO(cC,;ijXziyi)
u0(x)
(1.4)
\ia+j,3ER;i,j>O
where the product is taken according to our fixed order on R. We call this the
commutator relation. We will make a slight abuse of notation where we allow the
integers i or
j
to be negative and set
Ca,O;i,_j = Ca,-/;i,j-
There are also structure constants da,, E F satisfying
nflu(t)na = us,.()(da,/t).
Lemma 1.0.6 The structure constants may be chosen so that cQ,f;i,j E Z and d,
E
{±1}.
Proof 9.5.3 in [Spr].
As da,O E {±1}, we will often suppress this constant during integration as it can be
removed by a re-scaling of variables.
We use the presentation of G given in chapter 9 of [Spr] where the field is F. For
p EX,
we have the relations
p(x) u(t) p(
1 =W
,
and
yv(x)n
=
uy(x)u
19
(-x'-)u^(x).
(1.5)
We will often make use of the following implications of (1.5). For xy
y
u_,(x)u,(y) = u7( +
1+ xy
1
1 + xy
x
1+xy
#
-1,
(1.6)
and, for x # 0,
u-7(x) = U,(-x~')-v'_-)nyu,(-x-1).
20
(1.7)
Chapter 2
The Subgroup Iht and Vectors
OU,b;w
We define a subgroup of G called Iht.
This subgroup will play the same role in
evaluating the Bessel function as that of the Iwahori subgroup I in evaluating the
spherical function.
Definition 2.0.7 Let S C R be a choice of positive roots and let ht
=
ht(S) be the
height function with respect to S. Recall that if 7 E S is
7
=mia1 +... + mrcr
where ac are the simple roots of S and mi positive integers, then
ht(7) = M1 + ... + M.r
Define Iht(S) to be the subgroup of G generated by T(o) and the elements u-,(x)
21
subject to the following conditions:
x
E
if Y E
Pht (--y)
-S
(2.1)
if Y E S.
x E Plht (^)
We call such u,(x) the unipotent generators of Iht(S).
For our specific choice R+ of positive roots, we let Iht denote Iht(R+) and ht = ht(R+)
when there is no possibility for confusion.
Iht shares many properties of I. Indeed, I and Iht are isomorphic as groups
(Lemma 2.0.12). A key property we will often use is the following.
Lemma 2.0.8 Suppose ua(x) and ua(y) are unipotent generators of Iht- Then in
the commutator relation
Ua(x)uO(y) = U(y)
Uia+Uj(cij;aOQXiY)
Ua(x)
each element in the product is also a unipotent generator of IhtProof Follows from a direct check of the cases according to the signs of a,3 and
ia + j/3 along with the assumption that cij,. ,, are integers. El
Lemmas 2.0.9, 2.0.10, and 2.0.11 are proved in exactly the same manner as they are
proved for I. They may be proved by writing an element g E G as
g = bnu
for some b E B, w E W, and u E w- 1 (U-) n U. Write u as a product of unipotents,
ordered by a convex ordering of w- 1 (R-)
n R+. Starting from the right end of this
22
product, if u^, E I (or Iht), then apply a step of the explicit Bruhat decomposition
(chapter 7.2) if u, V I (or Iht) and proceed to the next unipotent. Otherwise, if
U-1 E I (or Iht), then move to the next unipotent. Then end result is an expression
in BnYI (or BnyIht) for some y E W. We call this the explicit Iwahori (or Iht)
decomposition.
Lemma 2.0.9 Let S C R be a choice of positive roots. Suppose u is an element of
the form
JJ nyz
u=
'YES
where the product over S is taken in any order. If u
E Iht(R+), then each u,(x,) is
a unipotent generator of Iht(R+).
Lemma 2.0.10 Let w E W. Then an element i e Iht has a unique factorization of
the form
i = i -(w)ai+(w)
where i_(w) E (iht n w(U-)), a E T(o), and i+(w) E (iht n w(U)).
Lemma 2.0.11 G is the disjoint union
Bn,,Iht-
G =U
wEW
More precisely,
G=
U
Bn,,,(Iht
n w-'(U-))
wEW
and this decomposition is unique. Moreover, if g is of the form
g = bnwu
23
with b E B, w E W, and u E w-'(U-), then g E Bnw(Iht n w-(U~)) if and only if
u E
'ht-
Let pV denote the half-sum of roots in RV+.
Lemma 2.0.12 If pV E X', then
Iht
~
nw,
I
(nwelrPv)1.
Sometimes pV V RV+. In that case we extend the field F by adjoining 7ri/ 2 to obtain
2pv (71/2)
which serves the same purpose as 7rP in the above lemma. Either way, Iht is isomorphic
to I via an automorphism of G.
Lemma 2.0.12 follows from fact (ii) of the following lemma. Fact (iii) will be
needed in the proof of Theorem 5.0.31.
Lemma 2.0.13 Let a be a simple root and -y any root in R.
(i). (p, av)
1
(ii). (p, -yv) = ht(yv)
(iii).
-
(3, -,v) = ht(yv) + ht(w(yv))
,3EW-'(R+)flR+
Proof We make use of the invariance of (,)
24
under the Weyl group W.
(i). Because sa(R+\a) = R+\a, we have
,P_
, v) =(s.P
a, saV))
= (p -
2'
-av
~aV
= Therefore (p - 2, a')
= 0 and (p, av)
=
-2
(c,
av)
(ii). This follows immediately from (i).
(iii). The statement is equivalent to
(-p-+
+
0
t
=
(P, W(-Yv)).-
/3Ew-'(R+)nR+
But we check that
-P+
E(
= WR()p),
,3EW-1(R+)nR+
so the identity follows from the invariance of (,)
under W. E
Now we can define the "Iht-standard basis".
Definition 2.0.14 For v, w E W, define 0",,.
to be the vector in IndG(Xv) deter-
mined by
0'pw(g) = #Ov, (9) 1 B.. I, (g)Define the vector space Vu,lp(Xv) C Ind (Xv) to be the space spanned by the
for all w.
Just as
kK;w
OK
wGW
25
#,
it is clear
=UV
>
u,,O;w
wEW
by the Iht-decomposition for G (Lemma 2.0.11).
26
Chapter 3
Defining Integrals
This section explains how we interpret the integral
IU- Ou,,;w( ua)
W 1(u) du
that is not absolutely convergent.
Recall that F has a topology induced by the non-archimdeadean metric and that
FN has the product topology, for a positive integer N.
Also recall that U- is a
topological group whose topology is generated by the base of open sets
MY (p")
about the identity element, where y E R- and n > 0. Furthermore, U- is homeomorphic to FN as topological spaces, where N
Definition 3.0.15 If
f
=
IR+.
is homeomorphism from some open subset 0 C FN into U-
such that
U- \ f(0)
27
f
has Haar measure 0, then we call
a parametrization of U-.
An element of FN is described by an N-tuple
(4i,
... , tN)
of elements tj E F. Recall that a non-zero element t of F is of the form v7r", where
v E o', n E Z, and ir is a fixed generator of p. The integer n is called the valuation
of t and we call v the unit of t.
Definition 3.0.16 Let
f
U- be a parametrization of U- . Given an
: (FX)N ->
N-tuple of integers
7 = (ni, ... , nN),
define the subset
f(ii) C U- to be the image
w,
f(oX7r"1 ... ,)o
)7rN
Let . denote a finite set of N-tuples of integers. Define the subset
f(9) C U- to
be
U
f ()=
f(x)
W1EJ
Each set
f(T)
parametrizations
f
(ox)N.
We require the
to possess the following property.
Definition 3.0.17 Let
respect to Bni,
and homeomorphic to
is compact in U-
f
U-. We say that
: (Fx )N
U if whenever f(t
1
,...,
tN)
is of the form
aulnwIu2
28
f
is torus-monomial with
for some U1 , u 2 E U and a E T, then a is of the form
a
=
a'
Ma)
aED
where ma is some monomial in Q[ti, tl 1 , ...
If a parametrization
on
f(..),
f
, tN,
tN1
is torus-monomial, then the Whittaker vector
#
for a fixed -/, in the sense that the unramified character
only finitely many values. As
is bounded
J / 2 x takes on
f(.0) is compact, it follows that
/f P)
U4(ua)@
-1(u) du
is absolutely convergent for for any a E T. Likewise the corresponding integrals for
p(a) - #v,0-
are also absolutely convergent.
Definition 3.0.18 Consider a sequence sets of N-tuples {fA}k. We say that
-
oo if for any positive integer M, there exists some ko such that for all k > ko,
_fk
contains each N-tuple (n 1 , ... , nN) satisfying nil < M for 1 < i < N.
For a torus-monomial parametrization
f
W(p(a) - #U,;w) = lim
k--+oo
We will present, for each
and a sequence
f,
k -* oo, define
.. (ua)O-1(u) du.
k
#U,;w, torus-monomial
parametrizations
f
that depend on
a reduced word for we. We show that the above limit exists and is independent of
the choice of reduced word. The parametrizations are constructed in chapters 4 and
7.2.
29
We then define the Bessel function Wup(a) to be
Wu,1p(a) = E W(p(a) -#vp-.)WEW
30
Chapter 4
Computing W(p(a) - <p
) via a
Quotient Space
The main result of this section is Lemma 4.0.23 which partially constructs the torusmonomial parametrization used to define W(p(a)- Ou,';); the remaining construction
is in chapter 7. In this parametrization, Lemma 4.0.23 expresses W(p(a) - OU,.,) as
a product of two factors. This expression is central to the proof of Theorem 5.0.31.
Recall that W is defined as an integral over U-.
We break up this integral
according to a subgroup H of U- and its quotient space. We state the following
theorem.
Theorem 4.0.19 Let G be a locally compact group and H a closed subgroup. Then
there exists a G-quasi-invariantmeasure IG/H on the quotient space GH and
j
f(g)dp(g)
=
J
(IH f(gh)dPH) dPGH(gH).
Proof Chapter 1 of [OV].
31
1
For a w E W, we apply this theorem with G = U- and H = w- (U-)
n U-. The
following result characterizes certain coset representatives of U-/(w-'(U-) n U-)
we need for Lemma 4.0.23.
Lemma 4.0.20
UT(w 1(U-) n U- n Iht)
U~ n (BnwIht) =
where the disjoint union is indexed over the elements Ti E U-
n (Bnw(w-1 (U-) n U n
Iht)). That is, we may assume -9 are coset representatives of the form
Ti = bnwi E Ufor b E B and i E w- 1 (U~) n U n Iht.
Proof Follows from the Lemmas 2.0.9 and 2.0.11.
Lemma 4.0.21 Let sn.. .s, be a reduced word for w E W, where si is the simple
reflection for ai E D. The representative7U of U-/(w-1(U-)
nU-) may be written as
n
Ti
u-ai (ti).
=
i=1
Furthermore, when U is of the form auni with a E T, u
then a is of the form
H aV(M)
acrD
where ma is a monomial in ti,tj1 , ... ,tn,tj.
32
E U, and i E w- 1 (U-) n U,
Proof For a fixed w E W, almost every element u of U- may be written as
HU-ai)
\i=1
(4.1)
H
/
(-Ew~l(R~)nR~
This may be proved by writing unrI nwe as
ulnwU2
for some u 1 , u 2 E U; this can be done for almost every u E U-. Then we may write
U =
'iinw
t'2/Ui-(ai)(ti)
for some u'E U- n w- 1 (U-) and where
wi = si...s1 .
This in turn may be written as
U1'
naiUai(ti)
n2-
Now, starting with i = n and proceeding to i = 1, apply the relation
naua,(t) = uac (t-))a(t)u~(t-1)
and move ua1 (t-1) to the left as in Lemma 7.2.1. The resulting expression for u is of
33
the form
u = bu'
with b E B and u' E U- of the form (4.1). Thus b = 1, proving the statement.
To see that a is a monomial in the ti, t;-1 , proceed from i = n to i = 1, applying
the relation
u-C,(t) = uoj(t-)av(t-1 )n-uc(t
1)
to U and move ua,(t~') to the left using the relation
ua(x)ua(y) = U-a(y(1 +
xy)~1)aV(1 + xy))ua(x(1 + xy)- 1 ).
Then by induction it follows that the parameters of a are monomials. I
We now define two terms that arise from the quotient integral of Lemma 4.0.23.
Definition 4.0.22 Let a = ao7r" with ao E T and p G X' dominant.
WH(XV,
Define
w) and WU-/H(Xvwa) by
WH(XIW) = w-'(U-)nU- ov~,v1 ;w(nwu) du
and
WU-/H(X, w, a) = 6-1/2XV(a)
1/2
Ih
v(b)V-'(a-ua-1)_I n.rs
)dU.
where T! = bnwi according to Lemma 4.0.20 and du- is an abbreviation for the
quotient space measure dp -/w-1(U-)nU-(U(w- 1 (U) n U-)). The torus-monomial
parametrization defining Wu-/H(xv, w, a) is constructed in the proof of Lemma 4.0.23
and in chapter 7.2.
34
Lemma 4.0.23 Suppose that a = a01r" with ao E T(o)/T and [ E X' dominant.
Then
W(p(a)
=
WH (X, w)WU-1H(X, w, a)
(4.2)
Proof We have
1
J/
U -U#j,;.(a)W
(u) du =
J-1/
2
X (a)
j
u-U
1
.. (u)W- 1 (aua- ) du.
conjugating out from a change of variable in u. Apply Theorem 4.0.19 with H
=
w-1(U-) n U- to get
j
(a)
i/2Xv
fU-1H
By the support of #
JH
#OU,0w(Uh)0-1 (a-a-')O-~ (aha-1) dh du.
U
we may assume that u = -Uh E BnwIht. Lemma 4.0.20 then
says we may assume h E HnlIht and U = bni with i E w- 1 (U~)nUnIht. Therefore
h lies in the kernel of
4,
and the dominance of a implies that aha- 1 does as well.
We obtain
6112 Xv(a)
6x(b>#"j,,(nihp1(aua1)dh di.
H
JU-1H JH
Next we move the unipotent generators that comprise i to the right over h. We
refer to Lemma 7.2.1 about moving unipotent elements. Make changes in h to cancel
the effect of these moves; these changes take the form of the operator B. which is
discussed in chapter 7.2. Since each new unipotent must be a unipotent generator
(Lemma 2.0.8), these changes do not alter the fact that h ranges over H n Iht or that
aha-1 lies in the kernel of
4.
35
We finally obtain
1/2Xv
(a) f-H
J H,
1 )1IBfwIh
H/X(b)4,;w (nh)1-'(a7a-
(U)dh du
which factors into the two integrals WH(Xv, w) WU-/H(Xvw, a). L
Theorem 7.1.1 gives the following important result needed in chapter 5:
WH(X , W)
=
-7r-w(-')qht(w(y))-ht(y)-1
I-
1
7yEw- (R+)nR+
36
(4.3)
Chapter 5
Intertwining Operators Applied to
oU,0;W
There exists a unique intertwining map T,, : Ind (X.a5v)
-
IndB(X,). It may be
E IndB(Xs""). For the purposes of this
expressed via an integral for appropriate
document, we will let T, denote the integral itself.
Definition 5.0.24 For a E D and
#
a function on G, define T,, by
f
(noau(x)g) dx.
lim
Ts.(#)(g) = m-oo
J,-m
Remark 5.0.25 In the K-fixed side, the intertwining maps T, are used to construct
the vectors T,(q
) for each w E W. These vectors are in fact a basis of VK(X) and
are instrumental in the proof of the formula for the spherical function. Therefore,
we would like to consider the vectors T,(#w"73)) in hopes that they constitute a
basis of Vu,,p(X).
Upon computation, however, it turns out that these vectors may
not even lie in Vu,,(X).
Nevertheless, we can still obtain a critical result "modulo
37
the kernel of W: we consider a multiple integral determined by W o T,, over the
IR++I
1-dimensional set nruaU~. The Ts, thus aid in establishing relations among
the Wu(,#;). To explain this result (Theorem 5.0.31), we first define actual maps
between the Vu,,,(X).
Definition 5.0.26 For -y E R, define cw(-y) E C(Xv) to be
1
- q1r7V
1- 7r--tv
Definition 5.0.27 For w E W and a E D such that e(saw) = 1 + f(w), define the
map TI', : VuO (xso)
T.'a(#,
S.)
T'0
=
=-r
-+ VU, (X') by
(cw(v- 1 (a)) + 7v1(av)) kV,;w - q
"vU)#,p;W
+ (cW(v-
1
(-Y))
7r
#V(V)O$
(5.1)
+ q(5.2)
5
define T', by
For a reduced word of w = s ... s1,
T' = T,', 0 ...
a T'.
Lemma 5.0.28 The maps T' are independent of the reduced word of w used to
define them.
Proof The T' are essentially multiples of the maps T,,
fixed case; see Theorem 3.4 of [Cas].
applied to VK in the K-
These maps on VK are independent of the
reduced word for w because the intertwining integrals are themselves independent of
the reduced word. The same independence follows for the T'. The definitions (5.1),
(5.2) then determine T' on all of Vu,,(xv).
38
Recall that Gindikin-Karpelevich formula for the K-fixed vector states that the
takes the K-fixed vector 0'a" to a multiple of
intertwining map T,
#'
; there is
analogous result for the Whittaker vector.
Theorem 5.0.29 For a E D,
TS(#"gV) = T
,'
=W(V-(a))#,.
Proof Recall that the Whittaker vector
#U
E IndG(xv) is supported on the double
coset Bn,,,U where it is evaluated as
#jp(bnvu) _ e12XV(b)b(u).
We may assume that g
bnrmu for some b E B, w E W, and u E U. After a change
=
in x, we get
lim 6
2
xv (b)
f
q c" (nau,(x)niwu) dx
for some c, d E F. Consider three cases for w.
(i). w
=
we. We have
#v
(bwu) = 6112 xv(b)(u). In the argument of Os', apply
the relation
nu,(x)
=
u (x-1 )av (X-')u
-(a
1
)
to the left over u-,(x- 1 ) by conjugation to obtain
and move the n,
6
2XV
(b)
j
#OL(ua(x-
1
)av (x- 1 )nwu-w(o)(x-1 )u).
39
Since -wt(a) is a positive simple root, the integral reduces to
i xv(b)V)(u)
IF 612 xsV
v (-1))V(x-1) dx.
Letting m -- oo, we get cW(v-1(y))J1/2 Xv(b)*(u), verifying the theorem in this case.
(ii). w = sawj. Then #rY,(bnu) = 0, and
jp-m
$s (no,(x)bna nw,u) dx =
00' (riwt uwe(a) (da,we x) u) dx
IF
=
1/2 Xv(b)
&u)
j
0(da,,Wx) dx
=0
for sufficiently large m.
(iii). w 5 we, scwj. Then 0yo(nw) = 0. The integral in this case is also 0 because
nau-a(x)nw
is not in BnwU for any value of x, and the final case is proved.
The second equality easily follows from the definition of T,', and by pairing up w
with saw for each w E W.0
This formula will be used to construct the normalized intertwining maps Tb'. Ultimately we show that the set of vectors Tw(# -1 1) is the desired basis of Vu,,P(x).
Definition 5.0.30 Define TZ: Indo(xSaV) -- IndG(Xv) by
=
T
40
For a reduced word of sa...si of w E W, define Tw : Ind (Xw'") - IndG(xv) by
Tw = Tw o ...
o Tw.
We check
T8Vi(#O'(q)
T
o Tw(#vg.)
=O#",
(5.3)
=".
for all w E W. Equation (5.3) implies
Two
Tw
= Tw
W1
W2
W1W2
for all w 1 , w 2 E W. These facts will help us express
(5.4)
#
in terms of the T,(Ow"'.).
We emphasize that the purpose of the intertwining maps in this paper is to establish relations among W(p(a) -
.V). The intertwining maps serve the same purpose
in [Cas] and [HKP] in the K-fixed side. Casselman first describes the intertwining
maps on VK(x) by first directly computing the intertwining integral for T8,,(#K;1)He then obtains the result for arbitrary w by applying the action of the IwahoriHecke algebra (see his proof of Theorem 3.4 in [Cas]). At this point, we do not have
recourse to such a tool for the (U, 4) side. Therefore we prove the analogous result
outright for general w. This is the content of the the next theorem.
Theorem 5.0.31 Let a = a0 irA where ao E T(o) and A E X'
is dominant. Let
a E D and v E W such that f(sav) = 1 + £(v) and let w be any element of W. Then
cwPV(V-
(-y))W(p(a) - 0'",w) = W(p(a) - Ts,("Q;))
41
Proof Consider the Whittaker functional applied to the intertwining integral W o
Ts, (p(a) -
J
J4
w(naua(x)u)-(u)
dx du.
We will construct a parametrization of this domain of integration that is torusmonomial to evaluate the integral as a convergent limit. Then we will evaluate this
integral in two different ways which will give the two sides of the equation in the
theorem.
To construct the parametrization, we begin with the parametrization u 0 (x)7h,
where TUh is the parametrization of U- from chapter 4. We then perform changes of
variable to this expression to create a parametrization that is torus-monomial.
First conjugating a to the left and then making a change in the u and x gives
B l/ 2
(a)Xv(a)
J
# ,;w (nou.(x)u)-'(aua-1 ) dx du.
Recall
U
U=
n Bn,Iht
yEW
and that the set
ncaUa(x)BnyIht c Bn.Iht only if y
= saw or y =
w, depending on
x. We may therefore restrict the variable u to Bna.Iht and BnriIht. We divide the
proof according to whether f(saw) = 1 + f(w) or f(saw)
=
-1+
f(w).
f(saw) = 1+ e(w) :
u E Bn.
,,Iht
-/2()"
: Consider
j
j.
IF
1
U,(nuo,(x)u)P 1 (aua- ) dx
42
du.
Since f(saw) = 1
+ i(w), we have (saw)-
1
(a) E R-. Write u as Tih as in chapter 4,
with h E H = (sw)'1(U-) n U- and give -th the torus-monomial parametrization.
Express Uh = bnrni. Move b to the left over ncua(x) and make a change in x to
reverse the effect of this move. As in the proof of Lemma 4.0.23, we get
(a)X (a)
SB
J UnBn ,,w Iht
6 1/2 Xv (b)
B
j
F
Re-arranging, we get in the argument of
ky-(1)aua(x)nan h)V
U~
n
Iht-
dx du.
#"y"
(5.5)
nwuw-l(-a)(x)h.
That w
1 (aa
O;noaxcwhO1(fu
(-a)E R~ implies x E Pht(w'(-a)). Now uw-1(_,)(x)h ranges over w- 1 (U-)n
Make the change in x so that (5.5) so that has the torus-monomial
parametrization. Integrating with respect to x and h gives
WH (XsV w)
and with respect to U gives
WU-1H (X,
sow, a).
Now (4.3) implies
WH (X S,
WH (XV
w)
r
qM1
sw)
q
sow(-Yv) +
E
where
'7ER=
'YER+
_1
43
sc~w (-yV)
1
al
and
(htsaw(7v) -ht 7).
-
(ht w(yv) -ht-y)
M1 =
'YERt
'yER_
We simplify A, and M 1 . The fact R'
,
1
=
R _w, \w- 1 (a) shows that A, =av
We claim M 1 = 0. First we collect the terms involving roots and immediately see they
reduce to -ht w- 1 (a). Re-expressing the sum of coroots using w(Ri
)
-R
gives
-htav+
E
ht-yv-hts.(yv).
'fERtwt
This expression reduces to
(a,V)
-1+
7YERWjive
which is equal to ht w- 1 (a) by Lemma 2.0.13. Thus M 1 = 0.
Putting together these results with Lemma 4.0.23, we see the contribution from
integrating over these regions is
vi
W(p(a)-
vsc-
U
q
u E Bn.Iht
We now express u as 7h, with h E H = w- 1 (U~) n U-. By reasoning similar to that
of the previous case, we consider
6l/
2
(a)xv(a)
B
~
JU-nBn,,Iht
6 1/2xv(b)
Sev
j
B
F
(nuo (x)nwh)-
U
Apply the relation
U-I(x)nc = Ua(x
1 )av (x-1 )u
44
-(x 1 )
1(aUa- 1 )dxdu.
and re-arrange to obtain
6
61/2 Xv(b)
B 1/2(a)XV(a)
5 1/2 Xscv(av (X-1))(NU,,Ow(nwu-w-l(c,)(x-
SU -nBn,, Iht
That -w- 1 (a) E R- implies x E pht(w 1(a)). Make a change in h
n wg-is o h
w-(e)h.
Integrating with respect to x and h gives
ht(w-1(C,))
-raV
(
(1 - q-1)V-1
WH (Xs v)
IF
1 IF--
while with respect to I gives
WPU- 1H(x,
w, a).
Now
H (Xav, W))
WH(Xv, W)
v-'(A)
where
A2 =
-sa(w(yv))
1
YER+W'tI_
Re-express this sum as before to get
(a, V)aV
-YERZ.t
45
+ W(.Yv).
)h).
which is (1 + ht w~ (a))av by Lemma 2.0.13. Putting together these results yields
cw(v-'(y)) + 7r" (C)W(p(a) - #vUP;w)
The two cases together show the total contribution is
(ca(Xv) + 7rv-1(a))W(p(a) - Ov;.)
- q- 17r -(av)W(p(a) - #,;)
which is W(p(a) -T,'/(#7";.)).
e(saw)
=
-1+
£(w)
u E Bn,,..Iht
By the reasoning above, we consider
6J 1 / 2 x(a) JUflBn.scwIht
That -w-
1
2
x(b)
j
#yP# ,(naua(x
(a) E R+ implies x (E p1-ht(w-'(a)).
newh)O(aia-1) dx du.
(5.6)
The parametrization of h is of the
form
h = UW-1(a)(t)h'
(5.7)
with t E phtw-(a) and h' a parametrization of w-'(U-) n U-. Apply to
ncaUa(x)u-a(t)nriwh'
the relation
ua(x)u-.(t) = U-(
t
x
+ f)av ((1 + xt)-)u( 1 x
I+ xt
1+xt
46
(5.8)
and the fact that 1 + xtw-l(,) E o'
allows us to write the argument as
U,(
using the left U-invariance of
in the limit f
#P
x
X )nwh'.
1 + xt
The integrand is now independent of t which,
-+ oo, ranges over pht(w-1(a)). Applying the explicit Bruhat decompo-
sition to h' yields
__a I7r Ariwtu
U-a(
for some 7rA E X'
X
1 + xt )0orawu
From chapter 7, we may assume that each
and a' E T(o).
parameter t^, of h' has valuation ht(-y). This, along with x E p1-ht(w 1 (a)), implies
that
ir-Au0(
X
1 + xt
E U-(o).
)irA
is then independent of x as
This implies, after one more re-arranging, that
well. Thus the integral is absolutely convergent. Integrating over x, t, h' and -9and
letting m,
-+ oo yields, respectively,
m(p1-htw-1(a)M(phtw1(a))WH (Xsv,
)
WU-/H (Xv , sw , a).
Combining
1
1
m(1-htw (a))M (phtw (a)) _
and the previous calculation
WH (X
",CW))
qirv-l(av)
WH (XV, SaW)
47
-1
gives us the contribution
'1'v)W(p(a) - #Ov.)
-7r"V
u E BnwIht
By the reasoning above, we consider
J1/ 2 x(a) flBnwIht l/ 2 x(b)
j
U;w(ncua()nwh))(auia-1 ) dx du.
(5.9)
The relation
E pl-ht(w-l(a)), as -w-'(0) E R+, yielding
implies x-
j-1/2
-1)
-
nnur(X) = ua(x-')av
()x
(a
2
1/2 X/
/U-nBn.,.It
.B
v ()
F BX
(-
Xsav
(
1
1
)-(
(x1)uwu-(x'nh.
The same argument of the previous case shows that
#"
is independent of x.
Integrating as before gives
(1 - q-1)V-1
1-7r-cV
S
WH (X sav,7 )WU- /H (Xv, w, a),
as in the case f(saw) = 1 + f(w), u E BnwIht. This expression is then
(cw(v-1(y)) + q~ r"v(av)) W(p(a) - #"y,,)
48
The two cases together show
W(p(a)#u,.p;w) + (c', +
W(p(a)Ts. (#u,,;s.w)) = -7r'
)W(p(a)#u,,;sw)
which is W(p(a) - T',(0'-'.w)).
We now perform the same integration in a different order to obtain the right side
of the equation in the theorem statement.
For
#
E IndG (X), write W(T,,
j j
(0)) as the double integral
1
du.
#(u-a(x)nau)V- (u)dx
The relation
1
u-a(x)na = uc,(x-)av (x-1 )u-(x
)
and the change of variable
u -
U_a(-X-
U
yields
(F/ 61/
2
xV(av (X-1))-1 (u,(x-1) dx)
(I -
q#(u)
1
(u) du).
The integral with respect to x equal to cw(v- 1 (a)) and with respect to u is W(O).
Corollary 5.0.32 For w 1 , w 2 E W and a as in the lemma,
-1
-1
W(p(a) - Tj(#w))
=
W(p(a) -
Proof This follows from straightforward induction.
49
'U -2-
50
Chapter 6
Evaluation of the Bessel Function
This chapter evaluates the Bessel function Wu,,p(a) at torus elements a of dominant
coweight.
Lemma 6.0.33 expresses the Whittaker in the basis T,'1,(O'7
1 ),
w E W.
We use Corollary 7.1.2 from chapter 7. The arguments in this section closely follow
those in [HKP].
It follows from the definition of T.' that T,(#5~. 1) is a linear combination
1=
v<w
such that the coefficient a,
a
is nonzero. In particular,
17 -cw(Y)-
=
=
q- 7r.
(6.1)
YEw(R-)nR+
Therefore, there is some triangular, invertible matrix which expresses the change
between the basis the
Let's express
#g,,
#
1
.. and the T (
in this basis.
51
Lemma 6.0.33
W(
qER+-1
'YER+
wEW
Proof Set
.)
d.
#g=
(6.2)
wEW
for some constants dw. Recall
=U1
Z
U4,;w.
wEW
Equating the coefficients of
#U,
.;w, in both expressions yields
dwaww, = 1.
Therefore
I-cw(y)q7r~v
dwe =
-
'ER+
'YER+
S1-q7r~,v
1 7r-,Yv
Now the dw satisfy
dw1W2 = W1(dW2)-
To see this, apply T , to (6.2) and then apply the Weyl group action w, to recover
vectors in Vu (X). From the expression for dwe, it follows
1-
dw = w(f
1
YER+
52
qiryV
7
V_ ).
by taking w 2 = W, W1 = wfW-w.
n
Theorem 6.0.34 Let a = ao7r with a E T/T(o) and
W(p(a) -
Ub) = 61/2 (")
E X' be dominant. Then
ZW(17J
-JER+
WEW
Proof Apply p(a) to the expression for
7V )Y
#
in Lemma 6.0.33.
Wo T', = W implies
w(
WV(p(a) - #7,3)
wEW
l
1-
1-
q7rYv
1r7y
)W(p(a)
-YER+
By Theorem 7.1.1
WY(p(a) - #',11)
=
61/2(ir-')W(irj
_1_yv7
YER+
These results together provide the formula. L
53
1
The fact that
54
Chapter 7
Computing WH(X, w)
7.1
Statement of Theorem
Recall that WH(X, w) is defined by
WH(X,W)
=
Jw-'(U-)nU-
U,;
du.
(7.1)
This chapter is devoted to proving the following theorem, which was used in chapter
5, and its corollary, which was used in chapter 6.
Theorem 7.1.1
-rV
WH (XW)=
ht(w(yv))-ht(y)-
-Yw(R+)nR+
Corollary 7.1.2
W(qdUp;1
=vJ~
1 7
PER+
Proof Apply the theorem with w = 1 and use the support of
55
.1,p~
plus the fact
that
ht (7v) =
ht(-y)
YER+
'YER+
for any root system R.
Our approach to proving Theorem 7.1.1 consists of the following steps. We first
parametrize w
1
(U-) n U~ using unipotent root elements ordered by a convex or-
dering of the roots in w
1
(R-) n R-. After an "explicit Bruhat decomposition" in
section 7.2, we express elements of n,(w
1 (U)
n U- as the product
n4
u+,
with u+, u' E U and a E T. We make this decomposition because the definition of
the Whittaker vector requires its argument to be in this factored form. This Bruhat
decomposition then determines a sequence of changes of variable that simplify the
torus element a. Under this change, X(a) depends on only the valuations of the integration variables. These changes determine the torus-monomial parametrization.
We thus partition the domain of integration into cells for which each integration variable has a fixed valuation (each cell is of the form f(iT) in the notation of chapter 3).
Integration over such a cell then becomes an integral over a product of unit groups
0
'. We show that this integral is zero for every cell except one.
To perform the integration over each cell, we introduce in section 7.4 coordinates
in the unit variables which we call "nesting coordinates" that make the integrand
easier to handle. In particular, they isolate sub-integrals of the following forms:
/
0, n > 1
=
=(Ur-n)du
j/
(z7r-1) dz = 0
(7.2)
(7.3)
These are are two basic ways to prove an integral of V) vanishes. Section 7.6 identifies
key terms in the integrand with combinatorial objects in the root system R. After
56
analyzing these objects, we can finally carry out the integration in section 7.5.
7.2
Explicit Bruhat Decomposition
Let u be an element of U-. Let N be the the number of roots in R-. After a sequence
of N steps, we express u as u'NaNntWIUN with UN, u' E U and aN
c
T, provided that
u is sufficiently generic. Any element of U- can be written as
U
=
7
UY(ty)
(7.4)
yER-
for some t- E F. In general, the roots 'y may be taken in any order. For our purposes,
we always choose convex orderings; the convex ordering ensures that, among other
things, the group operations we apply are well-defined and will eventually terminate.
Recall that a convex ordering of the roots in R- is equivalent to a reduced decomposition we by the following correspondence. Let N be the number of roots in
R- and let {aN, ---,
2,
a
I
be a sequence of (possibly non-distinct) simple roots in
D+ that corresponds to a reduced decomposition of we:
We =
SQN'.Sa 2 S'1'
Define wj by
w= sa ...so,
with wo = 1. For i > 1, define -yj by
Wi=
i(ai)
57
Accordingly, we say that
'yi
maps to the simple root ac.
The ordering determined
by yi+1 -< -yj is a convex ordering of R-. We will achieve the Bruhat decomposition
U=
via a sequence of steps. Our initial expression for u is (7.4).
NaNnW-1guN
N
After the jth step the expression for u appears as
U =uajnwi1
Y)(fyj)
Uj
(7.5)
where
j =
+1s,(
J7),
i=j
aj = Aj(-1) J7yv (fI),
i=1
uj is some element of U, A3 is some element of Xv, and
fy,j
rational expressions. The expression (7.5) reduces to (7.4) when
the final form when
j
E F(tyl,
j
...
, ty)
are
= 0, and attains
= N.
We show what each step of the Bruhat decomposition entails. Assume that we
have expression (7.5) after the jth step. As -wj(yj+1)
is the simple root aj, apply
relation (1.7) to u-,g, in the product over roots 7 -< 7j to obtain
11 um(-Y)(f-Y,i)
The rightmost factor uj+
1
(fN+)
uaj+(f .Ql~)a,
(+)n
a
(
)
(.)
in (7.6) will remain in this position and contribute
to uj+1. The other unipotent factor ucj
1
(fj+ 1 ), however, will contribute to u
.
Therefore we have to make sure that we can "move" this unipotent to the left;
Corollary (7.2.2) provides this result. We postpone the proof and assume for now
58
that
uC,+ (fyi+,)aY+1 (f-i7,ij)"
H uwj (Y)(fY-,j)
747j+1
is equal to
u'ca,+(fj+i)na3
for some rational functions
(I
W(
1yj+
-<~Y
(7.7)
1 (y
j+1 (f
i+
fy,j+1
and u' E wj(U). Therefore this u' may join the
unipotent element u and we set u+1 to be
Uil= ujajnw,3 u ri, a3
in addition to setting
aY1(f
aj+1 =
=+Wa-n Wi±
)n
and
Uj+1 = uajy (f -~~)uj.
This completes the j + 1-th of the Bruhat decomposition.
The following technical lemma about moving unipotents in general is needed to
prove the result assumed above.
Lemma 7.2.1 Let
'yN
-<
...
--< y1 be a convex ordering of S C R. Suppose we have
a product consisting of u-, in this order with other unipotent elements uO = udim
interspersed between them:
U-7N
'YM-1
...
7-n+1
U-Y
)On,1U)3,2''...
59
n
-1,1
00n-1,2
... 'U,1,1
O1,2-'''
71
(7.8)
The 1,m may be any roots in R subject to the following conditions:
(i) If #i,m E S, then
(ii) If
E S,
m E,
#i,m = yj for some -yj -< -yj.
then 3i,m = -7j for some -y >- -y.
Then we may re-write the above product as
1
u'
1 u,
(7.9)
i=N
for some u' E U(S~) and with parameters of the u, possibly altered.
Proof We obtain the expression (7.9) by sequentially "moving" the uf3 "to the left",
"passing over" the other elements uy in the product. We show each move must end
in one of two ways: either we reach the far left end of the product, where the uo
contributes to the u' factor; or we reach a u^, with -yj = 3, in which case the uo is
absorbed into uny by the relation
u, ()u0(y)
= U, (X + y),
(7.10)
for whatever parameters x, y that these unipotent elements have. It is in this manner
that the parameters of the u, may be altered, as referred to in the lemma. The
element we choose to move will always be the "leftmost" of the u,,
u,3,
1
in particular
in (7.8). Of course, during this move we may create "new" unipotents by the
commutator relations, which we label as uI3 ,, appropriately. We will show that these
new unipotents also satisfy conditions (i) and (ii). Thus, we show that conditions
(i) and (ii) on products in the same form as (7.8) are invariant under moving the
60
leftmost u3 to the left.
First, suppose
3
,
, G S. Then by condition (i),
to the left and absorb it into uYi as in (7.10).
3
n, =
-yj -< -y,.
We move u,
Now, if we have 7i between -yj and
y, such that 7j and 7 span an irreducible rank 2 root system within R-, then u,,
creates new unipotents as it moves to the left over uy, with roots in this rank 2
system, appearing in a block immediately adjacent to u- on its left. The convex
property of the ordering implies that each of these roots must be equal to some
with 7y -< 7y
-< -y.
'Yk
Thus new unipotents of this move satisfy condition (i) (in this
case, any new unipotent must lie in S so condition (ii) does not apply).
On the other hand, suppose
3
n,,1 E S-, so -1,
= yj >- -7 . Then we move up,3r
to the left end of the product. Condition (ii) ensures that this move does not create
torus elements by relation (1.6). Now, if we have -yj -< y with -- y and 7fj spanning
an irreducible rank 2 root system of R, then u3,
creates new unipotents as it moves
to the left over u . To see that the roots in this rank 2 system satisfy conditions
(i) and (ii), consider the sequence {-Y5,
... ,
-72,
-71,
7N, 7N-1, - ,7j+1} which is a
convexly-ordered positive system. This proves the invariance of (i) and (ii) under
moving uo, 1 to the left.
Now, we claim that iteratively moving the leftmost uo to the left will eventually
yield a product in which no uf3 appear, that is, a product of the form (7.9). This is
proved by induction on the position of the leftmost unipotent u3 1 . with n
the base case.
=
N as
l
Corollary 7.2.2 In the previous notation for the explicit Bruhat decomposition, the
expression
Ua
S(f-Y)
61
(f-)
(7.11)
may be re-written as
(7.12)
H U-(f-y,j+1)
'
for some u' E wj(U) and fyi,j+1 E F(tY..., ti)
Proof Apply the lemma with S = wj(R-). E
The explicit Bruhat decomposition shows how to express u E U- as u'NaNWfUN.
Recall, however, the argument of the Whittaker vector in (7.1) is n.u for u
w-
1
j
=
(U-). We can view nu
£(w)
and
fy,j
E
U-
n
as a j-th step in an explicit Bruhat decomposition, with
= ty. Thus we can use the same procedure to express neu in
the form u'NaNnwe UN. We therefore assume w = 1 from now on, as the general case
translates easily from this one.
Using
0
uaj+1
~
UN
1
1(f
,j),
(7.13)
j=N-1
we write that integral for W(Ou,+;i) as
fU-
j#,.;i(u)
du =
fU-
X(aN)
(f\<j<N
1BIht (u) du.
/y~~
(7.14)
We now describe changes of variable in the t, that render the rational expressions
fY,,j into monomials.
Changes of Variable From Moving u-. Given -y E R- and a convex order -< on
R- , consider the product
(
Y,(tY,)
\s'-<y/
62
U_7(t-1).
Move the unipotent u-, and then all new unipotents u3 to the left as described in
Lemma 7.2.1. The u, may get absorbed by a u, via
usi(t 7 )uO(x) = uY (t, + x),
where
#3 =
-yj and x is some polynomial in the parameters. In this case, we make the
change of variable
t, ++ ty
- X.
The order of these changes is determined by when they occur in moving the new
unipotents to the left by always moving the leftmost u,3 . We call the set of all changes
of variable that occur in this manner the changes of variable from moving un.
We denote the application of these changes by the operator By. We also let B>. 7
and B-.<y denote the changes of variable from moving uy, for -y'
- y and -y' -<
-y, respectively. The order of these changes is determined by the convex ordering,
starting from the most succeeding root. The B signifies that the changes are derived
from the explicit Bruhat decomposition.
The B-<,
simplify the elements
aN
and
UN
so that there parameters are mono-
mials in t,. The next lemma describes these monomials.
Lemma 7.2.3 In the notation of the explicit Bruhat decomposition, let m,, denote
B>-y,
(f,,j) and my,
In addition, Bsy 1(fyj+,1)
to denote m,,,.
= By,(fy,
Then
).
Proof Follows from the explicit Bruhat decomposition.
63
On the other hand, the changes B-,y, complicate the support condition 1 BIht (u) in
the integrand of (7.14). Namely, in the original coordinates (7.4) of u, the statement
u E BIht is equivalent to
(7.15)
t^ E phty
for all -y E R-. After the changes of variable from moving uy, the conditions (7.15)
appear as
E pht"
We now partition the domain of integration FN into cells on which B y1 (x(aN))
is constant. Recall FN is parametrized by
t
=
t71 I
(t7N7I'
7y
E F.
An element t E F is of the form
t
=
v7r-
where u E oX and n E Z is v(t), the valuation of t.
Definition 7.2.4 Let T denote an N-tuple of integers
7 =
(nN, ---, n1).
Call the subset of FN such that v(t.,) = ni for each i the cell C(ff).
ty,
=
On C(),
v,7rni with ni fixed and v., free to range over o'.
Since x is unramified, X(B-<),l (aN)) depends only on 7T: let x(7T) denote the value
of X(B,
(aN)) on the cell C(T). Clearly FN is the disjoint union of C(T) as E ranges
64
over all N-tuples. Therefore we consider the integral I(T) over C(7T):
I(T) = X(7)
j
(( m-1)
Jcn)y<O
/
(7.16)
dt,.
1pht,(Byl (t-))
Y<O
)
-Y<O
We will show that this integral is 0 for every 7 except one, namely 7E
=
(ht(-YN), -. , ht(-yi)).
Paths in a Root System
7.3
We first examine the structure of these functions by defining combinatorial objects
which we call "paths in a root system". This definition is derived from moving the
unipotent element u-, to the left and thus will be used to describe the functions
BO(t,)
Definition 7.3.1 Let S C R be a positive system. Let P denote the 3-tuple of the
following sequences. For an integer f(P) > 0, let {ae(p), ..., a,} and {be(p),
... ,
b} be
sequences of integers in {±1, +2, ±3} and {P(p), ... , P 1, PO} a sequence of roots in S.
Define P,i to be the subsequence {P, ..., P}. Define E(P 1,O) to be
a 1P 1 + biPo
and, for 1 < k < f(P), define E(Pk,o) to be
E(P,o) = akPk + bkY(Pk-1,o).
If E(P,o) E R for all 1 < k < £(P), and if b1 < 0 with all other ai, bi > 0, then say
P is a path in S. Furthermore, given a convex ordering -< of S, if E(P) -< Pt(p) and
Pai
-< P
for 0 < i < e(P), then say P respects the convex ordering -< of S.
65
Lemma 7.3.2 Let P be a path in R- that respects the convex ordering. Then for
k > 0,
E(P,o)
-<
E(P+1,o)-
Proof Follows from convex ordering.
Definition 7.3.3 For a path P in R, define c(P) E Z to be
f(p)-1
c(P)
=
Jl
CessiE(Po);ajba
.
i=O
Furthermore, define t(P1,o) to be
t(p1,o) = taltbi
and t(P,o) to be
t(Pk,o)
= takt(Pk_,o)k
with t(P) = t(Pe(p),O).
Remark 7.3.4 With these definitions, we can characterize the elements in each
change of variable from moving u-po. Each change is of the form
tE(p)
H-+
tZ(p) - c(P)t(P)
where P is some path that respects -< of R- beginning at P and which sums to
E(P).
Definition 7.3.5 Let P and Q be two paths in R. We say that
off P if Qo = Po and E(Q) = P for some i
66
$
Q E T branches
0. We call such a pair of paths (P,Q)
a branching pair, and that P is the absorbing root for this branching (as this
root corresponds to a unipotent absorbing another unipotent). We give the following
rooted tree structure on paths in R. Let T be a rooted tree and label the vertices
of T by paths in R-, not necessarily distinct. We call T a rooted tree of paths
in R- if the following property holds for each vertex v of T: the path of a child
vertex of v branches off the path P of v such that the paths of the child vertices have
distinct absorbing roots in P.
Definition 7.3.6 Let (P,Q) be a branching pair of paths in R. Let k be the greatest
index such that P =
Qj for
all i < k. We say that
P -<Q
if
Pk+1 - Qk+1,
Q -<P
if
Qk+1
<
P+1-
Definition 7.3.7 For a path P in R-, define the operator Bp inductively by
Bp(ty)
=
>7
t, -
t(P -< Q)c(Q)BQ(t(Q))
E(Q)=Y, Qo=P
and extend this action to Q(t,, ... , tYN*
Lemma 7.3.8
BO(t,) = t, -
Z
E(P)=y,
67
c(P)Bp(t(P)).
Po=0
7.4
Nesting Coordinates
The integral (7.16) is actually an integral over (ox)N.
(=X(i)
(
N
Irt"
i=1
f
J(ox)N
)
()i(m
Y<o
(\y
1,htey(Bay1 (ty))
<
O
dv,
(7.17)
with t-, restricted to C(T).
We will make a change of variables among the units
vy that is suited to the structure of the m,, and B-<. (ty). This change will reveal
"nesting phenomena" (7.19) among the my which will make the integral over (ox)N
easier to evaluate.
Changes of Unit Variable for Nesting Coordinates. Recall that m(-y, a) denotes the multiplicity of the simple root a as a summand of y. For a root 7y E Rand another root -y -< yj, call the set of changes of variable
for all -y -< -y the changes of unit variable for -y.
of these changes by the operator Ny,. We let N>.
We denote the application
denote the changes for -y' >- -y,
starting from -yi. The N signifies that these changes create the nesting coordinates.
Now N>.+1 (m.,) is a power of the uniformizer 7r times a product of unit variables.
We describe exactly which unit variables appear in this coefficient.
Lemma 7.4.1 Let v(y, a) be
J
v(-y, a) =
0>-y,3#maps to a
68
vo,
with an empty product defined to be 1. Then the unit variable coefficient of N",, (m,,j )
is
V-Y H v(7Y3+1, a)nt(,j(^),C,).
aEDProof The proof is by induction. We note that the lemma is true for all mY,O in
which case
inYO =
t 7 . Fix 0 > j < N and assume it is true for all N yj,(m,j) for
- -<_7j+1. Now apply N,,
to see that the unit variable coefficient of N>. y,r(my,j)
1
for -y -< 7 3+ is
v,[v 7y3 ±v(7Yj+,
JJ
aj+1)](w(7),a+1)
v(Y+ 1, a)m(wJ(7)'))
(7.18)
aED\aj+1
The
j +
1-th step in the Bruhat decomposition is to conjugate uW,(
by aY+1 (N".,, (m-,,+)).
N>73,j (rmj 3
7
) (N,+
1
(m,j))
Using the induction hypothesis on the unit coefficients of
), this conjugation has the effect (in terms of units) of multiplying (7.18)
by
aj+1)](Wi(7),a+1).
[v y+v(7+1,
7
Now
rn(wj(-y), %+1) - rn(wj+1(-7), aj+1 ) = (wj(y), aY+i)
and
v(7yj+, a)m(w(^),a)
for all a 54 aj+1. (In the case
j
=
-
v(yj+2, a)m(w+l(Y)
, a)
N - 1, we let
J-
v(YN+1,a)=1
I3ER-,3mapstoa
These facts combined with (7.18) proves the induction step. El
69
Corollary 7.4.2 Let -y E R- map to a E D-. Then the unit variable coefficient of
N>,(m,) is
3 maps to Q
13y,
Now we may apply N.< 1 to the integrand of (7.17) to get
P(N-<(m1171
1 )) =
Jl[
(7.19)
'P(v~(ir." + V-1( rn2 + ... + v,-1r"Q-m)...))
aED-
Y<O
where we have re-indexed the vI, as va,i such that y is the i-th most succeeding root
that maps to a, and na,i are integers depending on ii. This is the nesting phenomenon
referred to above.
The rational functions B,,
(-y) are also suited to N-.y<; Lemma ?? describes how.
Lemma 7.4.3 Suppose -yj E R- maps to a simple root a. Let 3 be the next root
succeeding -yj that also maps to a, if such a root exists. Then for any yy
-<
,
NyjB-O(t,) = v"(w1(7)'")Ba(t,)'
where B p(t,)' denotes a function independent of the unit variable vj (if 3 does not
exist, then -< 3 is interpreted as E R~). In other words, v , completely factors out
of NyB- (ty).
Proof For 7 -<
/, v',, completely factors out of Nyj(ty) to the power m(wj1((y), a);
here we have used that m(wj_(Y), -a) = 0 for any -yj -< -y -< 3. Furthermore, for
any path P in R that respects -< with PO -<3 , then v-, also factors out of t(P) to
the power m(wj-1(E(P), a). The lemma follows from these facts and the definition
of B.<-3 . 0
70
7.5
Final Integration
This section proves Theorem 7.1.1 using the main result of section 7.6. The proof
motivates a "subvariable" z of a unit variable vY which is treated in that section.
For an N-tuple f, consider the integral I(T) given by
N
X(F)
fj
1|7r ni
dv,.
11 lht(,) (NR- B-.,y (ty))
O(NR-m,~)
/ER(
i(.x)N
YER
ER-
v
(7.20)
Lemma 7.5.1 Let 71 =
ht(7y)
for some j.
(nN, ..- , n 1 )
be an N-tuple of integers. Suppose that nj=
Then
H
ht()(Byj(ty))
IP
7ER-
1pht)(ty)
=
YER-
for t, restricted to C(Ti).
Proof The operator B'Y, is a sequence of changes of the form
t
-+
t, - c(P)t(P)
where P is a path in R- with E(P) = -y and Po = -yj. We claim each such change
leaves invariant the product
H
I pht(-y) (tY)
.
YER~
Indeed, apply the change for one such P to get
IIPht(y)(t
7
-
c(P)t(P))
H
y'ER~,#y
71
Iht(-')(ty,).
The conditions on the ty along with the assumption that the valuation of t-, is ht(Y3 )
imply
t(P) E pht(Y)
as E(P) = y. Since c(P) is an integer, we get
mod pht()
t1 - c(P)t(P) = t.
and the desired invariance is proved. L
Definition 7.5.2 Given an N-tuple i, let j = j(W), 1 < j < N, be the greatest
index such that for all k < 1, nk = ht(-Yk).
Successively applying Lemma 7.5.1 gives
JJ
tht(
7
)(B
y (t-y))
71
rj 1pt-) (Bayj(ty))
=
yER~
YER-
The condition 1 ,ht(-)(NR-Byj (tyj)) is simply
t£f E Pht(%)
so we may assume nj > ht(%y). We claim that (7.20) is 0 unless nj = ht(-yj).
Assume nj > ht(-y). Recall the re-labeling of 7j as (a, i), where -yj is the i-th
most succeeding root that maps to a. We integrate va,4 over o'. If no root preceding
-yj maps to a, then from (7.19) we need to consider
V-1
Vir 7rn
72
;
otherwise, we have
V- (rn.,j + v-+i1
va~i""
+v,i+lv7r"
n)
where no,, is the negative of the valuation of M , and v and n are independent of
both va,i and va,i+1. We claim
naj = ht(yj) - nj -
1
given our condition on j. This may be proved using the form of the m., (Lemma 7.2.3)
and induction similar to that in the proof of Lemma 7.4.1. By Lemma 7.4.3, each
support conditions 1,ht(-) (NR-B,
(ty)) is independent of vcj. Thus the integral of
va,i over o' in the first case is 0 by (7.2) because nj > ht(-yj); and likewise in the
1 +v E Pn-ht(fj). In the second case,
second case is 0 unless m = nci and 1 + v a,i±1
1 n 3 hly)
vc,i+1 must be of the form
Vai+1 =
-1
+ zirnj -ht(-yj)
(7.21)
for any z E o. This is how the subvariable z of va,i+1 is determined. We would like
to integrate z over o to get 0 by (7.3), but the support conditions may depend on
z. We assume that the support conditions are independent of z which is proved in
Section 7.6. Thus we apply (7.3) to see that the integral is 0 in the second case as
well. This proves that l(n) is 0 unless
nf = (ht(yN), - ,ht(-y1)).
73
7.6
The Support Conditions Are Independent of
z
In the notation of section 7.5, we show that the product
(7.22)
1 IP t(t)y(NR-B-<7j(t7))
-YER-
restricted to C(T), where j = j(f), is independent of the subvariable z of v,,i+1. In
the following, a primed function
f' will
denote the result of applying NR- to
f
and
factoring out the maximal power of v,,i+1 (which may be 0); in particular, primed
functions will be independent of v,i+i.
Our analysis will depend on the type of the root system R. We first examine the
possible levels of paths in R-.
Lemma 7.6.1 Let P be a path in R- with P0 =y,. Consider a collection of paths of
any level from P such that no two branchingpaths share the same junction. Let m be
the number of all junctions. Then m is less than the multiplicity of ac in w 3 (E (P)).
Proof Apply wj to the roots in paths in this set to express wj(E(P))
c
R- as a
sum of other roots in R- with aj appearing at least m + 1 times in the sum, one
from each junction and one from P itself. E
Type A In type A, simple roots occur with multiplicity at most one. By Lemma
7.6.1, there are only level 0 paths and thus Lemma 7.3.8 gives
B
= B-,(t) =,(t1)
c(P)B<-,j(t(P)) E pht()
E
r-(P)=-YP0='Y
74
(7.23)
Apply NR- to (7.23).
The unit variable v,,i+1 factors out of NR- B,
power 1, and out of NR-B,,(t(P))
(ty) to the
to the power 0 by Lemma 7.4.3. Also for such
P,
(t(P)) = Bay (t(P))
B,
because there are no level 1 paths. Dividing NR- B,
S
B-<,(ty)' - va-i1
(tx) by v,,i+l gives the condition
c(P)B ,y,(t(P))' E pht(y)
(7.24)
E(P)=-Y,PO=-yj
By (7.22),
B--<yj(t(P))' E pht')h(7)n
Then substituting (7.21) into (7.24) shows that (7.24) is independent of z, using
c(P) E Z.
Type D The roots in D, r > 4, are of the form
±ej t Ej,
where i
$ j
and where {EI}1<i<;r is a set of r orthonormal vectors. We let R- consist
of the of the roots
ei t ej
where i >
j, with simple roots DD- ={ji+1 - Ei1<i<,
U
E2+E1.
The roots in D- that may appear with multiplicity 2 are {Ei+1 - Ei}1<i<,-1 while
75
all other appear with multiplicity at most 1. When aj is a root of the latter type,
the analysis done in type A suffices to prove the z independence; when it is of the
former, then any path has at most one junction. We analyze the terms arising from
such paths and show that, while those terms containing z dependence may not be in
Pht( 7 ) individually, the sum of them is. This sum is equation (7.31).
Suppose that P is a path in R and Q branches off of P. We write the root
sequence for P as
{Pprec, EQ, Psucc}
where Ppr
and Pucc denote the subsequences of P's sequence that precede and
succeed EQ, respectively. Not that Psucc is a path in R but Pprec is not necessarily
one. We make an abuse of notation and define t(Pprec) to be
t(Prec) =
t(P)
tr(Q)t(PSUCC)
As for type A, we now determine how each condition
NR-B- I(t-y) E P ht(-y)
depends on v,
and thus on z. From Lemma 7.3.8,
(7.25)
B-,,(ty) = B-yj(ty) - Ec(P)By,(t(Pprec))X
P
B-<yj(tr(Q)) -
Apply NR- to (7.25).
1(P -<Q)c(Q)Ba,j(t(Q)) B-<yj(t(Psucc)).
Now v,in factors out of NR-By(t.)
to the power 2; out
of NR-Ba,(tE(Q)) to the power 1; and out of NR-B.<yj(t(Pprec)), NR-B.<y(t(PUcc)),
76
and NR-Bayj(t(Q)) to the power 0. Then
v2+NR- B-. yj(ty) = B- y (t7)'
v- ,Vic(P)B-.
yj(t(Pprec))x
-
(7.26)
P
(B.<,(tE(Q)
E(P
)' - vai+Z
Q
Now
B<73j (tr,(Q))' - va +1 E c(Q)B y,(t(Q))' = vjNR- B-.(ts(Q)).
(7.27)
Q
Therefore we re-write the right side of (7.26) as
V3 +1 c(P)B.<y,(t(Pprec))' (v- NR-B.<-yj(tE(Q))) B.<, (t(Psucc))'
B<y7 (ty)' P
(7.28)
+ Va+lc(P)B-yj(t(Pprec))'B.y%(t(Psucc))' E
1(Q -< P)c(Q)Bayj(t(Q))' .
Q
By (7.22), we may assume
B.V
B-<yj(tF,(Q)))
-(pe)'gNR-
B <y(t(Psucc))' E Pht(-y)+ht(-yj)-nj
using
B<^j(t(Pprec)) = B j(t(Prec)),
B.<-yjt(Psucc) = B <yt(Psucc).
77
Applying (7.21) to the first v-j in (7.28), this equation simplifies modulo pht(-y) to
B-.<y,(ty)'
(V~ iNR- B-.,j(tr,(Q)) B-.<^tj(t(Psucc))' (7.29)
V -c(P)B-.<yj(t(Pprec))'
P
1(Q -< P)c(Q)B.<, (t(Q))'
+ v-2c(P)B<
7 , (t(Pprec))'B-yj (t(Psucc))'
Q
Now, using (7.27), we collect the z-dependent terms in (7.29) modulo pht(-t) to obtain
v-z7r"n-ht(7j)B<7j
(
13
E(P, Q)c(P)c(Q)t(Prec)t(Q)t(Psucc))
(7.30)
(P,Q) respects -<
where we have the following definitions. We say that (P,Q) is a branching pair if
P and
Q
are paths in R with
Q
branching off P, and that (P,Q) respects the convex
ordering if both P and Q respect the convex ordering. Define
-1
E(T)
c(P, Q)
: T 1 ) -< T('
:{1:
=
-<
c(P)c(Q)
and
t(P, Q) = t(Pprec)t(Q)t( Psucc).
The sum in (7.30) is then
E(T)c(T)t(T).
(7.31)
T respects -<
We show that this sum is 0 by constructing an involution among the terms in the
sum, pairing up each term with one that is its negative. This involution is deter-
78
mined roughly as follows. Each term in the sum arises from a branching pair (P,Q).
Essentially, we "transfer" a certain root from one path in this pair to the other, thus
constructing a branching pair (P, Q).
We first describe a result for the structure constants.
For simply-laced root
systems R with a, 3 E R, we set cc,, = ca,,0;1,1 following notation of [Spr]. As usual,
if either a, # or a+3 V R, then we set c0 ,3 = 0. The next lemma states two standard
relations among the structure constants.
Lemma 7.6.2 Let R be a simply-laced root system and let a,/3, y be linearly independent roots in R. Then
(i) cO =
-COa
(anti-commutativity)
(ii) ca,Oc,+O,y + cOy,cO+y,a + c-y,acyc+,
= 0 (Jacobi relation)
Proof [Spr]. L
We apply relation (ii) to structure constants of the branching pair (P,Q).
Lemma 7.6.3 Given a branching pair (P,Q), let
a = min(Q)
# = E(Q) - min(Q)
-y = E(PSUCC).
Then exactly one of the sums 3 + -y, -y + a is in R, and
caO Ca+,,yy =
ca,+,-y
cc=
:3 + - E R
- ca,, co,-Y+a : -y + a E R
79
Proof That exactly one of the sums ,+7, -y+a is in R follows from a straightforward
check using the characterization of roots in type D. Therefore, exactly one of the
three terms in the Jacobi relation is 0, depending on which sum is not in R, leaving
the stated equation between two remaining terms. E
Operations on branching pairs. We construct a branching pair (P',Q') out of
P and
Q
by re-arranging the roots in the paths; t(P',Q')
=
t(P,Q) because all the
relevant roots remain unchanged, and c(P', Q') is equal to c(P,Q) up to sign. There
are four operations we consider.
Commute. Suppose that min(PSUCC)
$
P0 . If E(Q) + min(Psucc) V R, then set
(P,Q)COrnn = ({Pprec, min(Psucc), E(Q), Psucc\ min(Psucc)},
Q).
Clearly c(P,Q)comm is equal to c(P,Q).
Transfer to the preceding sequence. In the case E(Q) - min(Q)+ E(Pucc) E R,
transfer min(Q) to Pprc; set
(p, Q)Prec =
({Pprec,min(Q), E(Q)
-
min(Q),Psucc}, Q\min(Q)).
Then it follows from Lemma 7.6.3 that c(P, Q)prec
-
c(P, Q).
Transfer to the succeeding sequence. In the case min(Q)+E(Ppre) E R, transfer
min(Q) to Psucc; set
(P,Q)succ
=
({Pucc,
(Q)
-
min(Q), min(Q), Pucc}, Q\ min(Q))
80
Again from Lemma 7.6.3, c(P, Q)sUcc = -c(P, Q).
Switch. We set
(P,Q)switch
Pprec,
(Psucc) Q} , Psucc)
Then c(P,Q)switch = c(P,Q); indeed, we get one -1
by changing
cE(Q),(P-c) to cE(Pmc),E(Q)
and another -1
from the E factor.
The initial input of R is a branching pair (P,Q) that respects the convex ordering.
It generates a sequence of intermediate branching pairs that do not respect the
ordering until a pair that does respect the ordering is obtained. This pair (P,Q)R is
the final output and satisfies c(P,Q) = -c(P,
Q)'
and t(P,Q) = t(P,Q)R.
Root Transfer Algorithm R for Simply-Laced Root Systems
Step 1. Given a branching pair (P,Q),
la. If min(Q) -< min(Psucc),
lai. If EQ - min(Q) + EPu~c E R, then transfer min(Q) to Pprec. With the
resulting pair, go to Step 1.
laii. If min(Q)+EP
E R, then transfer min(Q) to Pc. With the resulting
pair, go to to Step 2.
1b. If min(Pucc) -< min(Q),
ibi. If EQ + EPsucc - min(Psucc) E R, then apply the commute operation to
min(Psucc) and EQ. With the resulting pair, go to Step 1.
81
1bii. If min(Psucc) + EQ E R, then transfer min(Psucc) to
Q,
i.e., apply the
inverse of the transfer to the succeeding sequence operation to min(PSUCC).
With the resulting pair, go to to Step 2.
Step 2. Given a branching pair (P,Q),
2a. If either (P,Q) or (P,Q)switch respects the convex ordering, then output
that pair.
2b. If max(Pprec)+ EPsucc E R, then apply the inverse of the commute operation to the roots max(Ppre) and EQ in P. With the resulting pair, go to
Step 2.
2c. If max(Ppre) +EQ E R, then apply the inverse of the transfer to preceding
sequence operation to max(Pprec). With the resulting pair, go to Step 2.
Lemma 7.6.4 The root transfer algorithm R for simply-laced R is well-defined and
is an involution on branchingpairs (P,Q). Furthermore, t(P,Q) = t(P,Q)R.
Proof First observe that during the steps of the algorithm no roots are added to
or removed from the union of the root sequences of
Q, Pprec,
and Psu
1 c. Therefore
t(P,Q) = t(P,Q)R.
We claim each intermediate pair (P,Q) possesses the following properties:
(i)
Q respects
the convex ordering and P does not, but P absent E(Q) does respect
the convex ordering, so that
max(Ppr,)
-<
min(Psucc),
max(Ppre)
-<
min(Q).
82
(ii)
E(Psucc) -< min(Psucc),
E(Q) -< min(Psucc),
E(Psucc) -<min(Q),
E(Q) -<min(Q).
(7.32)
Property (ii) also holds for pairs that do respect the convex ordering.
We will use the fact that in simply-laced root systems R, for any a,3 E R, the
quantity (a, ,v)
lies in {0, ±1, ±2} and satisfies the following conditions:
(a,fl
a+ f E R <
a = # <--
-1
(7.33)
(af3v) = 2.
First we prove that min(Q) -, min(Psucc) assuming that Psucc and Q respect the
convex ordering. We may assume that a3 = Ej+1 - Ej for a fixed 0 < i < r - 1,
where Po = -yj. Recall that the integer sequences for any path in D, are of the form
a = {, ..., 1, 1}, b = {1, ..., 1, ,1, -1}.
Then wj_ 1 (E(Q)) and wj_1(E(Pucc)) must be
of one of the following forms:
{Cm
-
Ei+i,
Ei - Em', Em
-
Em', Ek + Ekt,
-Ei+i
+ Ei
for some m > i + Ii > m', k > i + 1, k > k'. In addition, EQ + EPucc E Dr since
P is a path. We directly check the possibilities listed to see that min(Q) cannot
equal min(Pucc); thus la and lb are exhaustive. Lemma 7.3.2 applied to the initial
input implies that steps lai and 1bi always generate intermediate pairs that possess
properties (i) and (ii). Likewise we check that steps laii and 1bii preserve property
(ii).
83
We claim that the algorithm must eventually enter step 2. For if it did not, then
at some point in the algorithm, we have a pair (P,Q) at step 1 such that either Psuc
or
Q
consists of the single root PO while the other has root sequence
{0n, ..., 1, PO}
with n > 0 such that /i + Po V R and 31 -< Po,13 +1 -<
f3i for
all i. By construction,
(W, + .. +#1 - PO) + (- PO) E R
and thus
(On + .. + 01 - PO, -PO)=-.
The assumption that the algorithm never enters step 2 implies that
(#i, -PO)
for each i > 0.
=0
Then (PO, PO) = 2 gives a contradiction.
This proves that the
algorithm must eventually enter step 2.
Now, let (P,Q) be an intermediate pair at step 2. If Ppr
EQ + EPsuc.
is empty, then EP
=
Therefore either
EPsucc -< EP --< EQ
or
EQ -< EP -< EPsucc
by the convex property. We get from these two possibilities that either (P,Q) or
(P,Q)switch, respectively, respects the ordering by (ii).
If Pprec is not empty, then max(Prec) succeeds at least one of EQ, EPsuc,; oth84
erwise, the convex property would imply the initial pair did not respect the convex
ordering. If max(Pprec) succeeds only one of the two roots, then by (ii) either (P,Q)
or (P,Q)switch respects the convex ordering. If max(Pprec) succeeds both, then neither
(P,Q) nor (P,Q)switch respects the convex ordering and the pair enters step 2b or 2c,
which we check are exhaustive. As before, we also see that 2b and 2c preserve (i)
and (ii).
The algorithm terminates because 2a and 2b decrease the number of roots in Pprec,
and we proved that the algorithm terminates when Ppr is empty (though Pprec being
empty is not necessary for termination). We must show that (P,Q)
#
(P,Q)'. The
root transferred at step laii or 1bii, call it 'y, ensures that the pair output at step
2a is distinct from the intitial output pair, except possibly in the case that P and Q
have identical roots at the places succeeding -y. But in that case, we get that
- + 2/3 E R
for some root
3, a contradiction in simply-laced R (For non-simply laced R, there is
a modification of the algorithm at this point).
To check that the algorithm is an involution, see that step 2b and 2c or inverse to
step lai and 1bi, and the root transferred at laii or 1bii is the same root transferred
there by applying R to (P,Q)".
Type E In E 6 , the simple root
64 -
E3
may appear with multiplicity three.
Therefore for there are two types of rooted trees of paths in R- that begin at
64 - E3:
those with two children of the root, and those with one child of the child of the
root. In the support conditions, we add and subtract terms as we did in type D and
collect the z dependent terms. The powers of z that appear are z and z2. There is a
85
different vertex-relation coefficient each power of z. There is an equivalence relation
on the set of rooted trees that respect the convex ordering. We say T1 ~ T 2 if T
can be obtained by applying a sequence of root transfer operators Re, where e is an
edge of a rooted tree. The sum of these terms weighted by the structure constants
and vertex-relation coefficients is 0. In contrast to Type D, where the number of
trees in an equivalence class is always 2, this number for Type E can vary, with
possibilities including 4, 5, 8, 9, or 16. The strategy is to characterize all possible
equivalence classes and show the sum over each class is 0. The equivalence class for
a tree with m edges is described by an m-regular graph. Below we list the vetex
relation coefficients vr 1 and vr 2 in E 6 .
vri(T) =
-2
:
T(1)
1
:
(
1
:T(1,2)
-PT1,1)
-<
and TM
T(1,1) and V(2)
T-<
()
and T(1,1) -<T
and
<
T()
:(1)
1
: V(,') -< V() and
otherwise
86
-(1,2)
and T(1,1)
-2
0
(1,2)
-<
(1,1)
T(M
<T(1,1,1)
T(1,1,1)
-1
: T(1 ) - T('1' and TM --< P,2)
1
: TOl) -<T(',) and
_- 7l)
T(1 ) -< T(1 ,2) and T 1'1 -< TI)
1
1
r2(T)
rT1,2)
-< T 112'
and T 1' 1)
-<T)
1) -< T(1 1)and T 1 1 )-<rl)
1,1
0
otherwise
Root Transfer Algorithm for Type B
The root transfer algorithm for Type B extends that for Type D with some
modifications. As in type D, we use the characterization of roots in Br. Recall that
the positive roots of B, consists of the vectors ej ± ej and Ej, j > i
roots Ei+1 - Ej, i
>
>
0 with simple
0, and Eo. The coefficient sequences for paths may now include
±2. We describe the points in the type D algorithm that are modified.
The first modification deals with inputs (P,Q) for which the algorithm outputs
two pairs instead of one. That is, in type D we could partition the sum into sets
of two terms such that the two terms add to 0. But in type B, we partition the
sum into sets of two or three terms, such that the terms in a triple add to 0 as well.
The following conditions are to be placed at the indicated spots in the algorithm
for type D. Here, (P 1 , Qi) always denotes a pair where min(Qi) if of the form ei
and has coefficient 2 in Qi; (P 2 , Q2) a pair where min(P 2 sc)
coefficient 2 in P 2 succ; and (P 3, Q3) a pair where min(P 3 suc)
form ej and has coefficient 1 in both paths.
87
of the form ej and has
= min(Q 3 ) is of the
laiii. Given a branching pair (P, Qi), if min(Q1) -< min(P succ) and min(Qi) has
coefficient 2 in Q1, then transfer min(Q1 ) to Pic to make the pair (P 2 , Q2),
where Q2 = Q1\ min(Qi) and P 2 'ucc = {min(Q 1 ), P1
suc}
but with min(Qi)
having coefficient 2 in P 2 succ. In addition, transfer one copy of min(Q1) from Q,
to Pi succ to make (P 3 , Q3), where Q3 = Q, but with min(Q1) having coefficient
1, and P 3 = P 2 but with min(Qi) having coefficient 1 also. With these two
pairs, go to Step 2.
lbiii. Given a branching pair (P2 , Q2), if min(P 2 suc)
-<
min(Q 2) and min(P 2 succ) has
coefficient 2 in P 2 ucc, then transfer min(P 2 SUCC) to Q2 to make the pair (P 1 , Q1),
where Q, = {min(P 2 succ), Q2} but with min(P 2 succ) having coefficient 2 in Qi;
and where P'succ = P 2 succ\ min(P 2 succ)}.
In addition, transfer one copy of
min(P2succ) from min(P2succ) to Q2 to make (P 3 , Q3), where
Q3
= Q, but with
min(P 2 succ) having coefficient 1, and P 3 = P 2 but with min(P 2 succ) having
coefficient 1 also. With these two pairs, go to Step 2.
1c. Given a branching pair (P 3 , Q3), if min(P3 succ) = min(Q 3 ) = 7, then -Y must
have coefficient 1 in both paths. Construct two pairs (P, Qi) and (P 2 , Q2)
out of (P,Q).
Here Q, =
P1 succ = P 3 succ\7
Q3
but with -y having coefficient 2 in Q1, and
Furthermore Q2 = Q 3\7 and P 2 SUM = P3 succ but with y
having coefficient 2. With these two pairs, go to step 2.
The second modification deals with terms in the sum that do not correspond to
pairs of paths, but to paths with a 2 in the b sequence. Specifically, let S be a path
with root sequence
{..., Sn+1 , Sn., }
and coefficient b, = 2 such that ESO,n is of the form Ej and ESo,n is of the form
88
ej + Ej. #,+1 is of the form Esj - ej. Let (P,Q) be a branching pair with
Pprec =
{..., Sn+3, Sn+2}
Q = {Sn+1, Sn, ---}
Psucc
=
{Sn, Sn- 1 , ... }
2c(s) = c(P,Q). Apply Step 2 of the algorithm for type D to (P,Q). The output
corresponds to the term that cancels the term that corresponds to S.
The third modification deals with a path S such that So = E0, b1 = -2,
and
S1 = Ej + 6o. Let (P,Q) be the branching pair
Pprec =
{..., S3,32}
Q = {S1, So}
Psucc
=
{So}
Apply Step 2 of the algorithm for type D to (P,Q). The output corresponds to the
term that cancels the term that corresponds to S.
Type C The root transfer algorithm for Type C also extends that for Type D
with some modifications. We use the characterization of roots in C,. Recall that
the positive roots of C, consists of the vectors ej t ei j > i > 0 with simple roots
Ei+1
-
ei, i > 0, and 2eo. The coefficient sequences for paths may now include +2.
We describe the points in the type D algorithm that are modified.
The first modification, as in Type B, deals with pairs (P,Q) such that min(Q) =
min(Psucc) = 7. In this case, -y is of the form ej - Ei, j > Z.
1c. Given a branching pair (P, Q), if min(Q) = min(PSUCC) = 7, then construct the
89
pair (P',Q') such that
Pprec
= {Pprec,
}
Q'I= QVY
PS'ucc = PSUCC\-Y
with -y having coefficient 2 in P'. With this pair, go to Step 1.
2b. Given a branching pair (P,Q), if max(Pprec has coefficient 2 in P, then construct
the pair (P',Q') such that
Pprec = Pprec\7
Q'={py,Q}
PSUCe = {Y, Psucc}
with y having coefficient 1 in
Q
and P.
The second modification deals with pairs (P,Q) such that min(min(Q), min(Pucc))
Ej-
and EQ + EPsucc = ej + Ej. In this case, both conditions of lai and laii (or
1bi and 1bii) may hold; therefore go to the following laii (or lbiii).
laiii. Given a branching pair (P, Qi), if EQ1-min(Qi)+EPisuc E R and min(Qi)+
EPi ,u,
E R, then construct the following pairs (P 2 , Q2) and (P 3 , Q3). Here,
90
=
P2 prec
=
Q2
-
P1 prec
Q
min(Q1)
P 2 succ =
{min(Q), Pi succ}
P3 prec
{min(Q), P
and
=
prec}
3 =Qi\ min(Q1)
P3 succ
P1 succ-
With (P 2 , Q2) go to Step 2, and with (P 3 , Q3) go to Step 1.
1biii. Given a branching pair (P 2 , Q 2 ),
if EQ 2
+ EP 2 SUCC
-
min(P 2
succ)
E R and
min(P2 succ)) + EQ 2 E R, then construct the following pairs (P, Qi) and
(P 3 , Q 3). Here,
PI prec
=
P2 prec
Q1 = {min(P 2 succ), Q 2}
P succ
=
P 2 succ\ min(P2 succ)
91
and
P3 prec
=
{P2 prec, min(P2succ)}
Q3=Q2
P3succ = P2succ\
min(P2succ).
With (P1 , Qi), go to Step 2. With (P 3 , Q3) go to Step 1.
2c. Given a branching pair (P 3 , Q3), if max(P
3 prec)
+ EQ 3 E R and max(P 3 prec) +
EP3 suc E R, then construct two pairs (P, Qi) and (P 2 , Q2) such that
P 1 prec
Q1
P1 succ
-
P3 prec\ max(P 3 prec)
= {max(P3 prec),
Q3}
P3 succ
and
P2 prec -
P3 prec\ max(P3 prec)
Q2 = Q3
P2succ = {max(P3 prec), P3succ}.
With (P 1 , Q1 ) and (P 2 , Q2 ), go to Step 2.
The third modification deals with terms that correspond to a single path instead
of a pair. Let S be a path such that ESo,n = Ej - ej and EO,n+1 = 2e.
92
Construct
the pair (P,Q) such that
Pprec =
{---Sn+3, Sn+2}
Q
=
{Sn+1 ,Sn}
Psucc
=
{Sn, Sn- 1 , ... }
Apply Step 2 of the algorithm to (P,Q), The output then corresponds to the term
in the sum that cancels the term corresponding to S.
93
94
Chapter 8
The Matrix Coefficient S(p(g) - q1)
We present some results about the matrix coefficient S(p(g) - #gO). This coefficient
on the (U,
4)
side is analogous to W(p(g) - OK) on the K-fixed side in that it shares
a similar evaluation at torus elements. A full treatment should appear in a separate
document.
Recall that W(p(a) - #,5 .i) is a monomial in C(Xv); this was the main result of
chapter 7. We next show that S(p(a) - U;1) and S(p(a) -OK;1) are also a monomials.
We compare the calculation of S(p(a) - #K;1) to its analogue W(p(a) - oUp;1 '
Theorem 8.0.5 Let a = aoqrX with ao E T(o) and A
J
E X' dominant. Then
2
uz;1 (ka) dk = (1 - q-1)d%1/ X(a) 11
T E R+
K
-q1i-7r.
V
and
K
OK;1(ka) dk = (1
-
where d is the dimension of X'.
95
2
q1)djB 1/
-(a)q-|R+I
Proof The supports of #g .1 and
qK;1
are BIht and BI, respectively; in addition,
BIht C BI. We thus determine when kwrA E BI.
We have, from the Iwahori
decomposition of K,
K
=
B(o)n,(w- 1 (U-) n U- n I)(w- 1 (U-) n U n I).
The dominance of A implies
7r-A (w-1(U-) n U n I)r-' E I.
Therefore, after conjugating by
ArA,
we
are left to determine when
(nw(w-1 (U~) n U-)) n BI
is non-empty. Lemma 8.0.6 says that this set is non-empty only if w = 1.
integral for S(p(a) - #u, .1) now follows from the result for W(p(a) - #
The
1) of chapter
7. The result for S(p(a) - OK;I), however, now follows simply because the measure
of BI is (1
-
q-1)dq-lR+I.
In this case of OK,1, there is only one non-zero value of x
as k ranges over K; contrast to the case of
#
,0;1, where x takes on infinitely many
values as k ranges over K, just as when u ranges over U- in chapter 7.
Lemma 8.0.6 Suppose that the set
(ns(w-(U-) n U-)) n Bn I
is non-empty for some V, w E W. Then f(v) ;> f(w).
96
Proof First let
Sn...S2S1 =W
1 WE
be a reduced word for w-we, where si are (possibly non-distinct) simple reflections
with respect to R+. Then set
Yi = Si...s2s1.
Apply the Iwahori decomposition for G to an element in the set
nw(w-1 (U-) n U-).
This Iwahori decomposition may be achieved using the steps in the explicit Bruhat
decomposition of Section 7.2. It follows that if such an element is in BnI,
then v is
of the form
V = w H(yiyi~_i)ei
for some choice of Ec E {0, 1}. This implies that f(v) ;> f(w) + k, where k is the
number of ei that equal 1.
97
98
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