Nilpotent Cones
Vinoth Nandakumar
An essay submitted in partial fulfillment of
the requirements for the degree of
B.Sc. (Honours)
Pure Mathematics
University of Sydney
August 2010
C ONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Chapter 1.
1.1.
1.2.
1.3.
Classifying nilpotent orbits and centralizers of nilpotents in
classical Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Classification of the nilpotent orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Describing centralizers of nilpotent elements in type A . . . . . . . . . . . . . . 12
Chapter 2.
A resolution of singularities and the closure ordering for
nilpotent orbits Oλ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resolution of singularities in type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Closure ordering in Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resolution of singularities in type C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
25
27
Chapter 3. The enhanced nilpotent cone and some variations . . . . . . . . . . .
3.1. The enhanced nilpotent cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. The exotic nilpotent cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. The 2-enhanced nilpotent cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
35
40
44
2.1.
2.2.
2.3.
Chapter 4. Springer fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1. Irreducible components of Springer fibres . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
iii
Introduction
This essay deals with nilpotent cones, in the framework on complex Lie groups.
There have been many established connections between the structure theory and
representation theory of Lie groups and their nilpotent cones. For instance, the
Springer correspondence relates nilpotent orbits to representations of the Weyl group.
More generally, for any representation of a Lie group it is possible to define its nullcone; the nilpotent cone is the nullcone of the adjoint representation.
Chapter 1 begins with a description of the classical Lie algebras in types A, B, C
and D. Then a definition of what the nilpotent cone is, and what nilpotent orbits are,
is given. This is followed by a description of the nilpotent orbits in type A, which
is a straightforward consequence of the Jordan form theorem. A description of the
nilpotent orbits in type C is then given, which requires more machinery. A description of the nilpotent orbits in types B and D are given without proof, since the proof
is similar in nature to that of type C. Following this, an explicit description of the
centralizers of nilpotent elements in type A is given.
The second chapter of the essay deals with resolution of singularities of nilpotent
orbit closures. Given a nilpotent orbit, its closure will have the structure of a singular algebraic variety. In types A and C, a resolution of singularities is given for
the nilpotent orbit closures; a different technique is required to construct the resolution in each of these cases. As an application of the resolution of singularities
in type A, a description of the closure ordering for nilpotent orbits in type A is given.
In the third chapter, we move on to study some variations of nilpotent cones. In
type A, we study the enhanced nilpotent cone, which is the product of a vector
space with the nilpotent cone of type A. Using a slightly different method to that
originally used, we classify the orbits for the action of the general linear group on
the enhanced nilpotent cone. We next examine Kato’s exotic nilpotent cone in type
C, which is the product of a vector space with a variant of the nilpotent cone in type
C. We describe the orbits of the action of the symplectic group on the exotic nilpotent cone, and describe part of the proof. We then study the 2-enhanced nilpotent
cone in type A, which is the product of two vector spaces with the nilpotent cone of
type A. This is a problem which has not previously been studied, and we describe
some partial results about the orbits in this cases.
In the final chapter, we study Springer fibres in type A. Springer fibres are fibres
of the resolution of singularities for the full nilpotent cone, which is a special case
of the resolution of singularities of nilpotent orbit closures. Following a paper of
iv
I NTRODUCTION
v
Spaltenstein, we describe the irreducible components of Springer fibres.
I expect that this essay should be accessible to a fourth year student with a basic knowledge in Lie theory and algebraic geometry. Complete proofs have been
given, except for the Jacobson-Morozov theorem, and the classification of orbits in
the exotic nilpotent cone.
Acknowledgements
First and foremost, I would like to thank my supervisor, Dr Anthony Henderson.
His guidance and help in all facets of my thesis has been priceless. I am sincerely
grateful for the countless hours he has spent, and the patience he has shown, in explaining mathematics to me and checking my work.
My family deserves a profound thank you for all the emotional support that they
have provided me with throughout this difficult year.
I would also like to thank Rowland Jiang and Clinton Boys for their generous help
with Latex.
vi
C HAPTER 1
Classifying nilpotent orbits and centralizers of nilpotents in
classical Lie algebras
1.1. Definitions
In the following, we work over the field C for convenience; however, almost all of
the results obtained will be valid for any algebraically closed field F in characteristic 0.
Let g denote a reductive Lie algebra, and let G be a connected Lie group having Lie
algebra g. Recall the following definition of the classical Lie groups, and their Lie
algebras. In type A, we have the general linear group; in types B and D we have
the special orthogonal group, and in type C we have the symplectic group.
Type A: Here G = GLn (C), g = gln (C). For convenience, we will work with
GLn (C) as opposed to SLn (C).
Type B: Here G = SO2n+1 (C) = {A ∈ SL2n+1 (C) : At A = I2n+1 }, and
g = so2n+1 (C) = {A ∈ gl2n+1 (C) : A + At = 0}.
Type C: Fix an invertible 2n × 2n matrix J such that J = −J t , so that J specifies a non-degenerate skew-symmetric form h·, ·i on C2n . Here G = Sp2n (C) =
{A ∈ GL2n (C) : At JA = J}, and g = sp2n (C) = {A ∈ gl2n (C) : JA+At J = 0}.
Type D: Here G = SO2n (C) = {A ∈ SL2n (C) : At A = I2n }, and g = so2n (C) =
{A ∈ gl2n (C) : A + At = 0}.
Proposition 1.1. In each of the above cases, we can define a map, known as the
adjoint representation, G → Aut(g), in which g · x = gxg −1 for g ∈ G, x ∈ g.
Proof. We need to check that this action is well-defined, i.e. given g ∈ G and
x ∈ g, gxg −1 ∈ g.
In Type A, the result is clear.
In types B and D, suppose g ∈ SOk (C), so g t = g −1 , and x ∈ sok (C).
1
2
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
gxg −1 + (gxg −1 )t = gxg −1 + (g −1 )t xt g t = gxg −1 + (g t )−1 xt g −1
= gxg −1 + gxt g −1 = g(x + xt )g −1 = 0
Hence gxg −1 ∈ sok (C), as required.
In Type C, suppose g ∈ Sp2n (C), so g t Jg = g, and x ∈ sp2n (C), so Jx + xt J = 0.
Note g t Jg = J implies g t J = Jg −1 , and Jg = (g t )−1 J.
J(gxg −1 ) + (gxg −1 )t J = Jgxg −1 + (g t )−1 xt g t J
= (g t )−1 Jxg −1 + (g t )−1 xt Jg −1
= (g t )−1 (Jx + xt J)g −1 = 0
Hence gxg −1 ∈ sp2n (C), as required.
In the definition above of sp2n (C) as {A ∈ gln (C) : JA + At J = 0}, an equivalent
formulation is {A ∈ gl2n (C) : hAv, wi = − hv, Awi ∀v, w ∈ C2n }.
In a Lie algebra g, recall that an element X ∈ g is “nilpotent" if adX is a nilpotent
endomorphism of the vector space g. In the case of g being a classical Lie algebra,
this is equivalent to X being nilpotent in the sense of matrices.
Definition 1.2. The nilpotent cone of the Lie algebra g, denoted as N , consists of
all nilpotent elements in g.
The group G has a well-defined action on N , since in each of the above cases, G
acts on g by conjugation, and the conjugate of a nilpotent matrix will remain nilpotent.
Definition 1.3. The orbits of the group action of G on N are nilpotent orbits. [3]
Thus the nilpotent cone N is a union of a nilpotent orbits, and two nilpotent elements are in the same nilpotent orbit iff they are conjugate by an element of G.
Alternatively, nilpotent orbits are often referred to as “conjugacy classes", or “adjoint orbits" in N .
Next, we give an explicit description of nilpotent orbits for the classical Lie algebras. Type A (gln ) serves as a straightforward example to illustrate the concepts
above, while type C (sp2n ) takes more work. Types B (so2n+1 ) and type D (so2n )
involve similar ideas to type C, and here we state the results without proof.
1.2. Classification of the nilpotent orbits
1.2.1. Type A. In type A, we have that g = gln (C), and the connected Lie
group with Lie algebra g is G = GLn (C). The nilpotent cone N of gln (C) is then
1.2. C LASSIFICATION OF THE NILPOTENT ORBITS
3
simply the set of all nilpotent matrices in gln (C).
The action of the group G on N is by conjugation, so two nilpotent matrices in
N lie in the same nilpotent orbit iff they are conjugate under GLn (C). Now the
classification of orbits follows from the Jordan form theorem. Keeping in mind that
nilpotent matrices in GLn (C) have all eigenvalues equal to 0, the Jordan form theorem says that any matrix x ∈ N is conjugate to precisely one matrix nλ of the
following form, where λ1 ≥ λ2 ≥ ... ≥ λk are positive integers whose sum is n:


nλ = 


Nλ1
Nλ2



...
Nλk
Here the matrix Ni denotes the following matrix (where there are i rows and columns):

0 1

0 1


..


0 1
0






Now let us make the following definitions:
Definition 1.4. Let P(n) denote the set of all k-tuples (λ1 , ..., λk ), with λ1 ≥ λ2 ≥
P
... ≥ λk satisfying ki=1 λi = n.
Definition 1.5. For each partition λ ∈ P(n), let Oλ denote the nilpotent orbit
containing the matrix nλ .
Using this, it follows that every nilpotent orbit in gln (C) corresponds to a unique
element of P(n). Thus we have proven the following classification of nilpotent orbits in type A:
Proposition 1.6. There is a bijection between the set of nilpotent orbits for the Lie
algebra gln , and the set P(n), given by the Jordan canonical form. Specifically, the
orbit Oλ corresponding to the partition λ consists of all matrices in gln conjugate
to nλ .
If x ∈ Oλ , we refer to λ as the Jordan type of the matrix x. A Jordan basis for
x, is a basis {ei,j } of V , with 1 ≤ i ≤ k, 1 ≤ j ≤ λi , such that xei,j = ei,j−1 if
j > 1, and 0 otherwise. The matrix of x with respect to such a basis, ordered so
that ei,j precedes ei0 ,j 0 if i < i0 or i = i0 and j < j 0 , will then be nλ . It will often
be convenient to represent λ as a diagram of boxes, with λ1 boxes in the first row,
λ2 boxes in the second row, and so forth. Then the boxes in this diagram can be
4
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
thought of as representing the elements of this Jordan basis (with ei,j being the j-th
box in the i-th row), with x moving each box to the box on the left.
1.2.2. Type C. Following Collingwood and McGovern, [3], pg 70-73, we describe the nilpotent orbits in the symplectic Lie algebra sp2n (C). While in the case
of the general linear Lie algebra gln (C), the nilpotent orbits were in 1 − 1 correspondence with the set of all partitions of n, here we will find that the nilpotent
orbits of the symplectic Lie algebra sp2n (C) are in 1 − 1 correspondence with the
set of all partitions of 2n in which every odd part occurs with even multiplicity.
In effect, the problem of classifying of nilpotent orbits in sp2n (C) and proving the
above statement is equivalent to proving the following three propositions:
Proposition 1.7. Given any nilpotent element n ∈ sp2n (C), the Jordan type of n is
a partition λ ∈ P(2n), satisfying the condition that every odd part in λ occurs with
even multiplicity.
Proposition 1.8. Given any partition λ ∈ P(2n) where every odd part in λ occurs
with even multiplicity, there exists a nilpotent element nλ ∈ sp2n (C) such that the
Jordan type of nλ is λ.
Proposition 1.9. Given any two nilpotent elements nλ , n0λ ∈ sp2n (C) which both
have Jordan type λ, then we can find g ∈ Sp2n (C) such that g −1 nλ g = n0λ .
In the proofs of the above propositions, we will use the Jacobson-Morozov Theorem, [3], pg 37:
Jacobson-Morozov Theorem: Given a semisimple Lie algebra g, and a nilpotent
element X in g, then there exists a sl2 triple (H, X, Y ) containing X. (Equivalently,
we can find H, Y ∈ g such that [X, Y ] = H, [H, X] = 2X, [H, Y ] = −2Y ).
To illustrate the Jacobson-Morozov Theorem, we will prove it for g = gln (C).
(Note that here g is not semisimple; nonetheless, the Jacobson-Morozov theorem is
still true in this case). It suffices to find an sl2 -triple containing the nilpotent element
nλ . Let µri = i(r + 1 − i), and define the following three matrices:

Hr+1


=



r
r−2
···
−r + 2
−r

Yr+1
0
 µr 0
 1
···
=



0
µrr 0







,



Xr+1
0 1

0 1


···
=






0 1 
0
1.2. C LASSIFICATION OF THE NILPOTENT ORBITS

Hλ2

H=



Y =


Hλ1

,

···

5

X λ1
X λ2

X=

···
Hλk

Y λ1
Y λ2



X λk



···
Yλk
It is clear by calculation that {Hr+1 , Xr+1 , Yr+1 } is an sl2 -triple, and hence that
{H, X, Y } is an sl2 -triple, with X = nλ . This proves the Jacobson-Morozov Theorem for gln (C). For future use, we prove the following lemma about sl2 -triples in
symplectic groups.
Lemma 1.10. Let V be a vector space with even dimension with a chosen symplecL
tic form h·, ·i, and suppose {H, X, Y } is an sl2 -triple in sp(V ). Let V = sr=1 Vr
be the decomposition of V into irreducible sl2 -modules. If dimVp 6= dimVq , then
Vp ⊥Vq with respect to the symplectic inner product h·, ·i.
Proof. Let dimVp = m, dimVq = n, and without loss of generality assume that
m < n. Let {v1 , v2 , · · · , vn } denote a string basis of Vq , so that the actions of X
and Y on Vq are given by the following (see Section 7.2 of [5] for an exposition of
sl2 -theory):
Xvi = (n + 1 − i)vi−1 and Xvi = 0; vi =
Y vi = ivi+1 and Y vn = 0; vi =
X n−i
vn
(n − i)!
Y i−1
v1
(i − 1)!
Similarly, let {w1 , w2 , · · · , wm } denote a string basis of Vp , so that the actions of X
and Y on Vp are given by the following:
Xwi = (m + 1 − i)wi−1 and Xw1 = 0; wi =
Y wi = iwi+1 and Xwm = 0; wi =
Y i−1
w1
(i − 1)!
Note the following two relations:
Xm
wm = 0
(m − i)!
1
Y m−i+1 wi =
Y m w1 = 0
(i − 1)!
X i wi =
X m−i
wm
(m − i)!
6
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
To prove that Vp ⊥Vq , it suffices to prove hvi , wj i = 0 for 1 ≤ i ≤ n, 1 ≤ j ≤ m. If
n − i ≤ j − 1 and i − 1 ≤ m − j, then n ≤ m, which is a contradiction. So either
n − i > j − 1 or i − 1 > m − j.
If n − i > j − 1:
X n−i
hvi , wj i =
vn , wj
(n − i)!
(−1)n−i vn , X n−i wj
=
(n − i)!
= 0 since n − i ≥ j, X j wj = 0
If i − 1 > m − j:
Y i−1
v1 , wj
hvi , wj i =
(i − 1)!
(−1)i−1 v1 , Y i−1 wj
=
(i − 1)!
= 0 since i − 1 ≥ m − j + 1, Y m−j+1 wj = 0
Thus, in either case, hvi , wj i = 0, as required.
Proof. (of Proposition 1.7)
Given a nilpotent X of type λ, by the Jacobson-Morozov theorem, we can find an
sl2 -triple {H, X, Y } containing X. Decompose C2n as the sum of irreducible sl2 modules V1 , V2 , · · · Vj , with dim Vi = µi , and without loss of generality, we can
assume that µ1 ≥ µ2 ≥ · · · ≥ µj . By sl2 -theory, if vi ∈ Vi is a lowest weight
vector in Vi , Vi is spanned by {vi , Xvi , · · · , X µi −1 vi }, and X µi vi = 0. Thus the set
{X j vi |0 ≤ j ≤ µi − 1} is a Jordan basis for the nilpotent X, and so X has type µ,
and so λ = µ. We have just proven that the dimensions of the irreducible summands
of C2n , considered as an sl2 -module, are λ1 , · · · , λk (so j = k and dim Vi = λi ).
Suppose r is odd. Make the following definition:
M
Vr =
Vi
λi =r
Let Hr−1 be the highest weight space in V r , i.e. the (r − 1)-weight space. It is
clear that dim Hr−1 is the multiplicity of r in the partition λ. Define a form (·, ·) on
Hr−1 by (v, w) = hv, Y r−1 wi. Then the following calculation shows that (·, ·) is a
skew-symmetric form.
(w, v) = w, Y r−1 w = (−1)r−1 Y r−1 w, v
= (−1)r v, Y r−1 w = −(v, w)
1.2. C LASSIFICATION OF THE NILPOTENT ORBITS
7
Next we will prove that the form (·, ·) is non-degenerate. Suppose v ∈ Hr−1 and
(v, w) = 0 ∀w ∈ Hr−1 . As w ranges over Hr−1 , Y r−1 w will range over the lowest
weight space H1−r in V r , by sl2 -theory. Thus v is perpendicular to H1−r , with respect to h·, ·i.
If s 6= 1 − r, then v will be perpendicular to the s-weight space Hs in V r by the
following calculation. Let w ∈ Hs .
hHv, wi = − hv, Hwi
h(r − 1)v, wi = − hv, swi
(r − 1 + s) hv, wi = 0
hv, wi = 0
Since v is perpendicular to Hs for s 6= 1 − r, and to H1−r , v is perpendicular to all
weight spaces in V r , and so v is perpendicular to V r , which is a direct sum of its
weight spaces. Make the following definition:
M
V 0r =
Vi
λi 6=r
By Lemma 1.10, v is perpendicular to V 0r , and hence v is perpendicular to V r ⊕
V 0r = V . Thus v lies in the kernel of the non-degenerate bilinear form h·, ·i, so
v = 0.
Hence (·, ·) is a non-degenerate symplectic form on Hr−1 , so dim Hr−1 is even.
Since dim Hr−1 is equal to multiplicty of r in the partition λ, this proves that every
odd part occurs with even multiplicity in λ.
Proof. (of Proposition 1.8)
First we prove the result for rectangle partitions, λ = (di ). Here if d is odd, i must
be even.
Let V be a vector space of dimension i, with basis {v (1) , v (2) , · · · , v (i) }. If d is even,
define a symmetric form on V via (v (r) , v (s) ) = δrs . If d is odd and i is even, define
a symplectic form on V via (v (r) , v (s) ) = 0 if |r − s| 6= 1; (v (r) , v (r+1) ) = 1 if r is
odd and 0 otherwise; (v (r+1) , v (r) ) = −1 if r is odd and 0 otherwise.
We now construct a 2n-dimensional vector space W , an sl2 -triple {X, Y, H} for
which W is a direct summand of i d-dimensional irreducible sl2 -submodules, and
a symplectic form on W which is invariant under the sl2 action. Let the highest
(1)
(i)
weight space in W have basis {wd−1 , · · · , wd−1 }. For 1 ≤ m ≤ d − 1, 1 ≤ j ≤ i,
m
(j)
(j)
(j)
(j)
define wd−1−2m = Ym! wd−1 , so that {wd−1 , · · · , w1−d } is a basis for an irreducible
sl2 submodule. The superscript denotes which of the i submodules it lies in, and the
8
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
(i)
(i)
subscript denotes the weight. We adopt the convention that wd+1 = w−d−1 = 0.
By sl2 -theory, the actions of X and Y are given by the following:
(j)
(j)
(j)
(j)
Xwd−1−2m = (d − m)wd+1−2m
Y wd−1−2m = (m + 1)wd−3−2m
Define the following form on W :
D
E
(i)
(i0 )
wd−2m−1 , wd−2m0 −1 = 0 unless m + m0 = d − 1
D
E
0
(i)
(i0 )
m d−1
wd−2m−1 , wd−2m0 −1 = (−1)
(v (i) , v (i ) ) if m + m0 = d − 1
m
It is straightforward
to check
·i is a skew-symmetric form. If m+m0 6= d−1,
D
E that h·,
0
(i)
(i )
i0
i
= 0. If m + m0 = d − 1,
then wd−2m−1 , wd−2m0 −1 = − wd−2m
0 −1 , wd−2m−1
0
0
then by construction of the form (·, ·), (v (i) , v (i ) ) = (−1)d (v (i ) , v (i) ). Then we
compute:
D
E
0
(i)
(i0 )
m d−1
wd−2m−1 , wd−2m0 −1 = (−1)
(v (i) , v (i ) )
m
0
m d−1
= (−1)
(−1)d (v (i ) , v (i) )
0
m
0
d−1−m d − 1
= −(−1)
(v (i ) , v (i) )
0
m
0
m0 d − 1
= −(−1)
(v (i ) , v (i) )
0
m
D 0
E
(i )
(i)
= − wd−2m0 −1 , wd−2m−1
Next we check that h·, ·i is invariant under H.
(i0 )
Since Hwd−2m−1
D
(i)
=
(i0 )
(i0 )
(d−2m−1)wd−2m−1 ,
E
wd−2m−1 , Hwd−2m0 −1 = 0 + 0 = 0.
If m + m0 = d − 1:
0
if m+m 6= d−1,
D
(i)
(i0 )
Hwd−2m−1 , wd−2m0 −1
E
+
1.2. C LASSIFICATION OF THE NILPOTENT ORBITS
9
D
E D
E
(i)
(i0 )
(i)
(i0 )
Hwd−2m−1 , wd−2m0 −1 + wd−2m−1 , Hwd−2m0 −1
D
E
(i)
(i0 )
= (d − 2m − 1 + d − 2m0 − 1) wd−2m−1 , wd−2m0 −1
D
E
(i)
(i0 )
= (2(d − 1) − 2(m + m0 )) wd−2m−1 , wd−2m0 −1 = 0
Next we check that h·, ·i is invariant under X.
D
E D
E
(i)
(i0 )
(i)
(i0 )
Xwd−2m−3 , wd−2m0 −1 + wd−2m−3 , Xwd−2m0 −1
D
E D
E
(i)
(i0 )
(i)
(i0 )
= (d − m − 1)wd−2m−1 , wd−2m0 −1 + wd−1−2(m+1) , (d − m0 )wd−1−2(m0 −1)
If m + m0 6= d − 1, then both terms in the above sum are clearly 0. Otherwise, if
m + m0 = d − 1:
D
(i)
(i0 )
Xwd−2m−3 , wd−2m0 −1
E
+
D
(i)
(i0 )
wd−2m−3 , Xwd−2m0 −1
E
E
E
D
(i0 )
(i)
(i)
(i0 )
= (d − m − 1) wd−2m−1 , wd−2m0 −1 + (d − m0 ) wd−1−2(m+1) , wd−1−2(m0 −1)
E
D
E
D
(i0 )
(i)
(i0 )
(i)
= (d − m − 1) wd−2m−1 , wd−2m0 −1 + (m + 1) wd−1−2(m+1) , wd−1−2(m0 −1)
d−1
d−1
0
m (i) (i0 )
= (d − m − 1)
(−1) (v , v ) + (m + 1)
(−1)m+1 (v (i) , v (i ) )
m
m+1
d−1
d−1
= 0, since (d − m − 1)
= (m + 1)
m
m+1
D
Checking that h·, ·i is invariant under Y is a similar calculation that we omit.
It is easy to see that the form h·, ·i is non-degenerate by examining the matrix of
(j)
the
D form with respectEto the basis {wd−1−2m } of W . For a fixed m, j the expression
(j)
(j 0 )
wd−1−2m , wd−1−2m0 will be non-zero for a unique choice of j 0 , m0 by construction of h·, ·i and (·, ·). The matrix of the form h·, ·i will have exactly one non-zero
value in each row and column, and hence be invertible.
Hence X ∈ sp(W ), and X is a nilpotent of type (di ) (this follows using the argument at the start of the proof of Proposition 1.7); this proves Proposition 1.9 in the
case of rectangle partitions.
Now suppose λ = (di11 , · · · , dirr ). For 1 ≤ j ≤ r, let Wj be a vector space of
i
dimension dj ij , and let Xj be a nilpotent of type djj in End(Wj ) which is invariant
for a non-degenerate symplectic form h·, ·ij on Wj . Let W be the direct sum of
10
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
the spaces Wj for 1 ≤ j ≤ r, and let h·, ·i be the non-degenerate symplectic form
obtained as the orthogonal sum of the h·, ·ij . Let X the sum of the matrices {Xj }.
Then X ∈ sp(W ), and X will be a nilpotent of type λ, as required.
Proof. (of Proposition 1.9)
Again we first prove the result for the case of rectangle partitions λ = (di ). Let X
be a nilpotent element in sp(W ), for some vector space W of dimension di and a
symplectic form h·, ·i. By the Jacobson-Morozov Theorem, we can find an sl2 -triple
{X, Y, H} in sp(W ). By the argument at the start of the proof of Proposition 1.7,
W is the sum of i irreducible sl2 -modules of dimension d.
Let Hd−1 denote the highest weight space in W , i.e. the (d − 1)-weight space.
Similarly
toEthe proof of Proposition 1.7, define a form (·, ·) on Hd−1 via (v, w) =
D
Y d−1
v, (d−1)! w .
d−1
Y d−1
Y
d−1
(w, v) = w,
v = (−1)
w, v
(d − 1)!
(d − 1)!
Y d−1
d
= (−1) v,
w = (−1)d (v, w)
(d − 1)!
Thus (·, ·) is symmetric if d is even, and symplectic if d is odd. As proved in
the proof of Proposition 1.7, (·, ·) is non-degenerate. In the case of d being even,
(1)
(i)
pick an orthonormal basis of Hd−1 for the form (·, ·), {wd−1 , · · · , wd−1 }, so that
(i)
(j)
(1)
(i)
(wd−1 , wd−1 ) = δij . In the case of d being odd, pick a basis of Hd−1 , {wd−1 , · · · , wd−1 },
(r)
(s)
(r)
(r+1)
so that (wd−1 , wd−1 ) = 0 if |r − s| =
6 1; (wd−1 , wd−1 ) = 1 if r is odd and 0 oth(r+1)
(r)
erwise; (wd−1 , wd−1 ) = −1 if r is odd and 0 otherwise.
m
(j)
(j)
(j)
(j)
Define wd−2m−1 = Ym! wd−1 for 1 ≤ m ≤ d−1, so that {wd−1 , · · · , w1−d } is a string
basis for an irreducible sl2 -module. Since the r-weight space is orthogonal under
h·, ·i to the s-weight space unless r + s = 0 (proven in the proof of Proposition 1.7):
D
E
(j)
(j)
wd−2m−1 , wd−2m0 −1 = 0 unless m + m0 = d − 1
If m + m0 = d − 1:
1.2. C LASSIFICATION OF THE NILPOTENT ORBITS
D
(j)
(j 0 )
wd−2m−1 , wd−2m0 −1
E
11
Y m (j)
(j 0 )
w ,w
=
0
m! d−1 d−2m −1
E
(−1)m D (j)
(j 0 )
=
wd−1 , Y m w1−d+2m
m!
E
(−1)m (d − m)(d − m + 1) · · · (d − 1) D (j)
(j 0 )
=
wd−1 , w1−d
m!
d−1
Y d−1 (j 0 )
(j)
(j 0 )
(j 0 )
(wd−1 , wd−1 ) (since w1−d =
w )
= (−1)m
(d − 1)! d−1
m
(j 0 )
(j 0 )
(j 0 )
In the above, Y m w1−d+2m = (d−m)(d−m+1) · · · (d−1)w1−d since Y w1−d+2m =
(j 0 )
(j 0 )
(j 0 )
(d − m)w1−d+2m−2 , Y w1−d+2m−2 = (d − m + 1)w1−d+2m−4 , and so on until
(j 0 )
(j 0 )
Y w1−d+2 = (d − 1)w1−d .
Hence we have proven above, that given two nilpotents X and X 0 of type (di ) in
(j)
0(j)
sp(W ), we can construct bases of W , {wd−2m−1 }, {wd−2m−1 }, with 0 ≤ m ≤
(j)
d − 1, 1 ≤ j ≤ i, such that the matrix of X with respect to the basis {wd−2m−1 } is
0(j)
the same as the matrix of X 0 with respect to the basis {wd−2m−1 }. Further, we have
that:
D
(j)
(j 0 )
wd−2m−1 , wd−2m0 −1
(j)
E
=
D
0(j)
0(j 0 )
wd−2m−1 , wd−2m−1
E
0(j)
Let g be the matrix such that gwd−2m−1 = wd−2m−1 . By the above condition,
g ∈ Sp(W ); and the matrices X and X 0 are conjugate by g. This proves that any
two nilpotents X and X 0 of type (d(i) ) are conjugate by g, proving the claim in the
case of rectangle partitions.
In the case of an arbitrary partition λ = (di11 , · · · , dirr ), given a nilpotents of type
λ, X ∈ sp(W ), by the Jacobson-Morozov Theorem we can find an sl2 -triple
{X, Y, H} in sp(W ). Let Wj be the sum of the ij dj -dimensional sl2 -submodules.
By the Lemma 1.10, Wj is orthogonal under h·, ·i to Wk if j 6= k. Now we can construct bases of the spaces Wj , and put them together to construct a basis of the space
W , such that the matrix of X with respect to the basis, as well as the values of the
form h·, ·i applied to pairs of basis elements, depend solely on the partition λ (and
not on X). Given another nilpotent X 0 of type λ, we can do the same thing. Then
X 0 will be conjugate to X via the matrix g taking the first basis to the second, since
the matrix of X with respect to the first basis is the same as the matrix of X 0 with
respect to the second, and g ∈ Sp(W ) since the inner products of the corresponding
pairs of basis elements is the same.
Definition 1.11. If λ ∈ P(2n) is a partition in which every odd part occurs with
even multiplicity, then define Oλ to be the set of all nilpotents in sp2n (C) of type λ.
12
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
1.2.3. Types B and D. Here we state the results without proofs. The proofs are
similar in nature to that of type C.
Proposition 1.12. Given any nilpotent X ∈ so2n+1 (C), the Jordan type λ of X
satisfies the condition that every even part in λ occurs with even multiplicity. Conversely, for any such partition λ of 2n + 1, there exists a nilpotent X ∈ so2n+1 (C)
with Jordan type λ. Any two such nilpotents X, X 0 ∈ so2n+1 (C) are conjugate by
an element of the special orthogonal group SO2n+1 (C). Thus the nilpotent orbits
in type B are in bijection with partitions of 2n + 1 in which every even part occurs
with even multiplicity.
Proposition 1.13. Given any nilpotent X ∈ so2n (C), the Jordan type λ of X satisfies the condition that every even part in λ occurs with even multiplicity. Conversely,
for any such partition λ of 2n, there exists a nilpotent X ∈ so2n (C) with Jordan
type λ. For such a partition λ of 2n, the set of nilpotents of type λ are a single orbit
under the action of SO2n (C); except when λ is a partition containing only even
parts, in which case the set of nilpotents of type λ splits up into two orbits under the
action of SO2n (C).
1.3. Describing centralizers of nilpotent elements in type A
Let λ = (λ1 , λ2 , ..., λk ). Recall that the representative nλ for the nilpotent
Pn−1 orbit Oλ
takes the following form, where Ni ∈ Matn (C) is defined by Ni = j=1 ej,j+1 (as
in Section 1.2.1).


Nλ1

nλ = 

Nλ2



...
Nλk
First we compute the centralizer CMn (C) (nλ ) of nλ in the matrix algebra Mn (C),
then the centralizer CGLn (C) (nλ ) in GLn (C).
Proposition 1.14. Given x ∈ Mn (C), let x consist of k 2 blocks xi,j , where xi,j has
size λi × λj , as follows.

x11
 x21
x=

xk1

x1k
x2k 


...
xkk
Then x ∈ CMn (C) (nλ ) precisely when the blocks xi,j take the following form. If
λi ≥ λj , the entries must satisfy the following (here 1 ≤ k ≤ λi , 1 ≤ l ≤ λj ):
1.3. D ESCRIBING CENTRALIZERS OF NILPOTENT ELEMENTS IN TYPE A
xij,kl = 0 if k > l
xij,kl = xij,1,l−k+1 if k ≤ l
If λi < λj , the entries must satisfy the following (here 1 ≤ k ≤ λi , 1 ≤ l ≤ λj ):
xij,kl = 0 if l − k < λj − λi
xij,kl = xij,1,l−k+1 otherwise
Proof. Since x ∈ CMn (C) (nλ ), we have xnλ = nλ x:

x11
 x21


...
xk1

Nλ1

Nλ2



x1k
Nλ1


x2k  
Nλ2

...
xkk
Nλk

x11
x1k
  x21
x
2k


...
...
xk1
xkk
Nλk


=





Simplifying this expression,

Nλ1 x11
 Nλ2 x21


Nλk xk1
···


Nλ1 x1k
x11 Nλ1


Nλ2 x2k 
 x21 Nλ1
 = 
Nλk xkk
xk1 Nλ1
···

x1k Nλk
x2k Nλk 


xkk Nλk
Thus we require that Nλi xij = xij Nλj , for all i, j ∈ {1, · · · , k}.

0 1

0 1


...


0 1
0

xij,11



...


xij,λi 1

xij,11





xij,1λj


=


...
xij,λi 1

xij,1λj
xij,λi λj
xij,λi λj
0 1

0 1


...


0 1
0
Expanding the above expression gives the following:







13
14
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS

(1.15)
xij,21




 xij,λi 1
0
xij,2λj
...
xij,λi λj
0


0 xij,11
 
 
=
 
 
xij,1,λj −1





...
0 xij,λi 1

xij,λi ,λj −1
Case 1: If λi ≥ λj , equating the entries in (1.15) now gives the following; here
1 ≤ k ≤ λi , 1 ≤ l ≤ λj :
xij,kl = 0 if k > l
xij,kl = xij,1,l−k+1 if k ≤ l
To see this, we can illustrate with an example where λi = 4, λj = 3:

 
xij,21 xij,22 xij,23
0 xij,11
 xij,31 xij,32 xij,33   0 xij,21

 
 xij,41 xij,42 xij,43  =  0 xij,31
0
0
0
0 xij,41

xij,12
xij,22 

xij,32 
xij,42
xij,11 = xij,22 = xij,33 ,xij,12 = xij,23
xij,21 = xij,32 = xij,43 = 0, xij,31 = xij,42 = 0, xij,41 = 0


xij,11 xij,12 xij,13
 0
xij,11 xij,12 

xij = 
 0
0
xij,11 
0
0
0
Case 2: If λi < λj , equating the entries in (1.15) gives the following (here 1 ≤ k ≤
λi , 1 ≤ l ≤ λj ):
xij,kl = 0 if l − k < λj − λi
xij,kl = xij,1,l−k+1 otherwise
To see this, consider the below example where λi = 3, λj = 4:
1.3. D ESCRIBING CENTRALIZERS OF NILPOTENT ELEMENTS IN TYPE A
15

 

xij,21 xij,22 xij,23 xij,24
0 xij,11 xij,12 xij,13
 xij,31 xij,32 xij,33 xij,34  =  0 xij,21 xij,22 xij,23 
0
0
0
0
0 xij,31 xij,32 xij,33
xij,11 = xij,22 = xij,33 = 0
xij,21 = xij,32 = 0, xij,31 = 0
xij,12 = xij,23 =xij,34 , xij,13 = xij,24


0 xij,12 xij,13 xij,14
0
xij,12 xij,13 
xij =  0
0
0
0
xij,12
Now that we know the structure of the matrices xij which form the matrix x, effectively we have computed the centralizer CMn (C) (nλ ) of nλ inside the matrix algebra
Mn (C). As an example of what this centralizer looks like in practice, we examine
the example λ = (3, 22 , 1). A typical element x ∈ CMn (C) (nλ ) has the following
form:
x11,11 x11,12 x11,13 x12,11

x11,11 x11,12

x11,11


x21,12 x21,13 x22,11


x21,12


x31,12 x31,13 x32,11


x31,12
x41,13


x12,12 x13,11 x13,12 x14,11

x12,11
x13,11



x22,12 x23,11 x23,12 x24,11 

x22,11
x23,11

x32,12 x33,11 x33,12 x34,11 


x32,11
x33,11
x42,12
x43,12 x44,11
Now we will compute the centralizer CGLn (C) (nλ ) by describing precisely when a
matrix x ∈ CMn (C) (nλ ) is invertible.
Proposition 1.16. Let x ∈ CMn (C) (nλ ), and let λ = (ab11 , ab22 , · · · , abt t ). Define the
following square matrices, where cs = b1 + · · · + bs , and 0 ≤ s ≤ t − 1. Then
x ∈ CGLn (C) (nλ ) precisely when the matrices xs are invertible.
xcs +1,cs +1,11 · · ·
..
=
.
xcs +bs+1 ,cs +1,11 · · ·

xs+1

xcs +1,cs +bs+1 ,11
..

.
xcs +bs+1 ,cs +bs+1 ,11
16
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
Proof. We start with the above example λ = (3, 22 , 1). We may consider the rows
and columns as being labelled by the set {e1,1 , e1,2 , e1,3 , e2,1 , e2,2 , e3,1 , e3,2 , e4,1 }.
Consider re-labelling the rows and columns instead with the set {e1,1 , e2,1 , e3,1 , e4,1 ,
e1,2 , e2,2 , e3,2 , e1,3 }. The matrix now takes the following form:
x11,11 x12,11 x13,11

x22,11 x23,11

x32,11 x33,11









x14,11 x11,12
x24,11 x21,12
x34,11 x31,12
x44,11
x11,11
x12,12
x22,12
x32,12
x42,12
x12,11
x22,11
x32,11
x13,12
x23,12
x33,12
x43,12
x13,11
x23,11
x33,11
x11,13
x21,13
x31,13
x41,13
x11,12
x21,12
x31,12
x11,11











The matrix is now block upper-triangular and will be invertible if and only if x11,11 6=
0, x44,11 6= 0, and the following matrix is invertible:
x22,11 x23,11
x32,11 x33,11
The above example makes it clear that in the general case λ = (ab11 , ab22 , · · · , abt t ),
after re-labeling the rows and columns of the matrix x as indicated above (so that
ei,k precedes ej,l either if k < l or if k = l and i < j) to obtain a matrix x0 , the
following square matrices (as defined above) will occur on the diagonal of x0 . Here
cs = b1 + · · · + bs , where 0 ≤ s ≤ t − 1.
xcs +1,cs +1,11
..
=
.

xs+1
···
xcs +bs+1 ,cs +1,11 · · ·

xcs +1,cs +bs+1 ,11
..

.
xcs +bs+1 ,cs +bs+1 ,11
The matrix xs will occur as times. The entry in the row corresponding to ei,k and
column corresponding to ej,l will occur in one of the copies of the matrix xs precisely when k = l and λi = λj . It remains to prove that all the other entries of
the matrix x0 will lie in the upper half of the matrix, so that x0 will be block uppertriangular with blocks xs ; it will then follow that x is invertible precisely when all
the matrices xs are invertible.
First consider an entry xij,kl of the matrix x0 when i ≤ j (so λi ≥ λj ). The entry
xij,kl lies in the row corresponding to ei,k , and the column corresponding to ej,l . As
shown above, the entry xij,kl can only be non-zero if k ≤ l. If k < l, then ei,k will
precede ej,l in the labelling of the rows and columns in x0 , and hence xij,kl will lie in
the upper half of x0 . If k = l and i < j, again ei,k will precede ej,l in the labelling of
the rows and columns, and xij,kl will lie in the upper half of x0 . If k = l and i = j,
then ei,k = ej,l , and xij,kl will lie on the diagonal of the matrix x0 , and will lie in one
1.3. D ESCRIBING CENTRALIZERS OF NILPOTENT ELEMENTS IN TYPE A
17
of the matrices xs .
Next consider an entry xij,kl of the matrix x0 , when i > j (so λi ≤ λj ). The entry
xij,kl will lie in the row corresponding to ei,k , and the column corresponding to ej,l .
As proven above xij,kl can only be nonzero if l − k ≥ λj − λi . If k < l, then ei,k
will precede ej,l in the labelling of the rows and columns in x0 , and hence xij,kl will
lie in the upper half of x0 . If k = l, then 0 ≥ λj − λi ≥ 0, so λi = λj . In this case
the entry xij,kl will occur in one of the matrices xs .
This shows that every non-zero entry in x0 occurs in the upper half, or in of the
diagonal blocks xs , proving that x0 is block upper triangular with blocks xs . It
follows that x is invertible precisely when all the matrices xs are invertible.
We now compute the dimension of the orbit Oλ . We first compute the dimension
of the centralizer CGLn (C) (nλ ) in the following two Lemmas. Given a partition λ,
define its transpose partition µ by letting µi be #{j|λj ≥ i}. Diagramatically, the
transpose partition µ is obtained from λ by reflecting λ along the diagonal containing the first box in the first row, the second box in the second row, and so on.
Lemma 1.17. Given a partition λ = (λ1 , λ2 , · · · , λk ) and its transpose partition
µ = (µ1 , µ2 , · · · , µl ), the following identity holds:
k
X
j=1
(2j − 1)λj =
l
X
µ2j
j=1
Proof. Proof by induction on the number of parts of µ.
Suppose λ1 = λ2 = · · · = λi > λi+1 , so µl = i. Consider the partition λ0 =
(λ1 − 1, · · · , λi − 1, λi+1 , · · · , λk ). By induction the result is true for λ0 :
(λ1 − 1) + · · · + (2i − 1)(λi − 1) + (2i + 1)λi+1 + · · · + (2k − 1)λk =
l−1
X
µ2j
j=1
Adding µ2l = i2 = 1 + 3 + · · · + (2i − 1) to each side now gives the required identity
for λ0 :
k
X
(2j − 1)λj = λ1 + · · · + (2i − 1)λi + · · · + (2k − 1)λk
j=1
= µ21 + µ22 + · · · + µ2l
Lemma 1.18. The dimension of the centralizer CGLn (C) (nλ ) is
Pl
j=1
µ2j .
18
1. N ILPOTENT ORBITS IN THE CLASSICAL L IE ALGEBRAS
Proof. The dimension of the centralizer is clearly equal to the number of free variables xij,kl in the description of the centralizer. We use the example λ = (3, 22 , 1) to
illustrate the ideas. In this case, a typical element x of the centralizer CGLn (C) (nλ )
takes the following form:
x11,11 x11,12 x11,13 x12,11

x11,11 x11,12

x11,11


x21,12 x21,13 x22,11


x21,12


x31,12 x31,13 x32,11


x31,12
x41,13


x12,12 x13,11 x13,12 x14,11

x12,11
x13,11



x22,12 x23,11 x23,12 x24,11 

x22,11
x23,11

x32,12 x33,11 x33,12 x34,11 


x32,11
x33,11
x42,12
x43,12 x44,11
The matrix x consists of blocks xij for 1 ≤ i, j ≤ k. The number of free variables
in the block xi,j is min(λi , λj ). Hence the dimension of the centralizer is equal to
the following sum:
X
min(λi , λj )
1≤i,j≤k
In the above sum, the term λj occurs with multiplicity 2j−1, since min(λi , λj ) = λj
when i ≤ j, so λj = min{λp , λq } when (p, q) ∈ {(1, j), (2, j), · · · , (j, j), (j, j −
P
1), · · · , (j, 1)}. Hence the dimension of the centralizer is equal to kj=1 (2j − 1)λj ;
the conclusion now follows from Lemma 2.8.
Given the standard nilpotent nλ ∈ Oλ , clearly Oλ = G.nλ . Hence dim(G.nλ ) =
dim(G) − dim(Stab(nλ )). Since the stabilizer of nλ in G is simply the centralizer
CGLn (C) (nλ ), using Lemma 1.18 we now compute:
dim(Oλ ) = dim(G) − dim(CGLn (C) (nλ ))
2
=n −
l
X
i=1
µ2i
C HAPTER 2
A resolution of singularities and the closure ordering for
nilpotent orbits Oλ
In this section, given a nilpotent orbit Oλ we describe how to construct a resolution of singularities for the algebraic variety Oλ (here Oλ denotes the closure of the
quasi-affine algebraic variety Oλ ). As a consequence of this resolution of singularities, we derive the closure ordering on the nilpotent orbits Oλ . The resolution
of singularities for the general linear group (type A) and for the symplectic group
(type C) are somewhat different in nature.
2.1. Resolution of singularities in type A
Let x ∈ gln (C) be nilpotent, corresponding to a partition λ ∈ P(n). In this section,
we will construct a resolution of singularities for the algebraic variety Oλ = Gx,
where G = GLn (C).
Consider the following flag of vector spaces inside V ∼
= Cn (V being the vector
space of dimension n on which gln (C) acts):
0 ⊂ ker(x) ⊂ ker(x2 ) ⊂ · · · ⊂ ker(xm ) = V
Define P to be the parabolic subgroup of the Lie group GLn (C) stabilizing the
above flag:
P := {u ∈ GLn (C) : u(kerxi ) ⊂ kerxi }
Define p to be the Lie algebra of P , i.e. the parabolic Lie algebra stabilizing the
above flag:
p := {u ∈ gln (C) : u(kerxi ) ⊂ kerxi }
Define n to be the Lie algebra moving each vector space in the above flag into the
one below:
n := {u ∈ gln (C) : u(kerxi ) ⊂ kerxi−1 }
One can show that n is the nil-radical of the parabolic Lie algebra P . The below
example shows the structure of n and p for some specific nilpotent elements:
19
2. R ESOLUTION OF SINGULARITIES FOR Oλ
20
Example 2.1. Consider the partition λ = (3, 2, 1). Pick x to be the following
matrix, of type (3, 2, 1). In this example will compute p and n.

0
1
0







1


0


0
1 

0
0
Then by calculation, we see the following:









ker(x) = 








∗
∗
∗
0
0
0








 ,















ker(x2 ) = 








∗
∗
∗
∗
∗
0















We claim that p and n are given by the following:
∗ ∗ ∗ ∗
 ∗ ∗ ∗ ∗

 ∗ ∗ ∗ ∗
p=
∗


∗


0
∗
∗ 
 0


∗ 
 0
, n = 
∗ 


∗ 
∗
In the above, the corresponding parabolic subgroup P
section of p with GLn (C), as sets.

∗
∗
∗
∗
∗
0 0 ∗ ∗
0 0 ∗ ∗
0 0 ∗ ∗
0 0
0 0

∗
∗ 

∗ 

∗ 
∗ 
0
of GLn (C) will be the inter-
To see that p and n have this form, it is clear by inspection that for all u ∈ p,
u(ker(x)) ⊂ ker(x), u(ker(x2 )) ⊂ ker(x2 ), and that for all v ∈ n, v(ker(x)) =
0, v(ker(x2 )) ⊂ ker(x). It remains to see that there are no other matrices in gln (C)
satisfying this property. The first column of a matrix u in p is equal to ue1 , and
hence must lie in ker(x). By a similar argument, the second and third columns of
u must lie in ker(x), the fourth and fifth columns must lie in ker(x2 ), and the last
column of u must lie in ker(x3 ). The first column of a matrix v in n is equal to
ve1 and hence must be 0. By a similar argument, the second and third columns of
v must be 0, the fourth and fifth columns of v must lie in ker(x), and the last column of v must lie in ker(x2 ). This shows that p and n have the form specified above.
In general, given a nilpotent element x corresponding to a partition λ, the following
result gives the dimensions of the spaces ker(xi ).
Lemma 2.2. If λ = (λ1 , · · · , λk ), suppose µ = λt is the transpose partition with
parts µ = (µ1 , · · · , µl ). Then for 0 ≤ s ≤ l, dim(ker(xs )) = µ1 + · · · + µs , and for
s > l, dim(ker(xs )) = n.
2.1. R ESOLUTION OF SINGULARITIES IN TYPE A
21
Proof. Since x ∈ Oλ , there exists a basis ei,j of V , with 1 ≤ i ≤ k, 1 ≤ j ≤ λi ,
such that xei,j = ei,j−1 if j ≥ 1, and xei,1 = 0. Then it follows by a quick induction
that xs ei,j = 0 if j ≤ s, and xs ei,j = ei,j−s if j > s. We claim that ker(xs ) is
spanned by {ei,j |j ≤ s}. To prove this, it is clear that the vectors {ei,j |j ≤ s}
are linearlyP
independent and lie inside ker(xs ). Conversely, suppose a ∈ ker(xs ),
where a = i,j ai,j ei,j . Then:
xs a =
X
=
X
ai,j xs ei,j
i,j
ai,j ei,j−s = 0
i;j>s
P
Hence if j > s, ai,j = 0, so a =
i;j≤s ai,j ei,j , and a lies in the span of the
vectors {ei,j |j ≤ s}. This proves that ker(xs ) is spanned by {ei,j |j ≤ s}. It follows that the basis vectors spanning ker(xs ) consists of the basis vectors spanning
ker(xs−1 ), together with {ei,j |j = s}. The cardinality of the set {ei,j |j = s} is
equal to the maximum value of i for which λi ≥ s; this is in turn equal to µs .
Thus dim(ker(xs )) − dim(ker(xs−1 )) = µs . It then follows by a quick induction
that dim(ker(xs )) = µ1 + · · · + µs . If s > l, then µl+1 = · · · = µs = 0, so
dim(ker(xs )) = µ1 + · · · + µl = n.
Now we will compute dim p and dim n. From Example 2.1 and Lemma 2.2, it is
clear that for a suitable choice of x ∈ Oλ (so that ker(xi ) will consist of the vectors
in V with the first µ1 + · · · + µi co-ordinates non-zero), p will consists of matrices
that are block upper-triangular, with blocks of sizes µ1 , · · · , µl . The subalgebra n
of p will consist of those matrices in p with all diagonal blocks equal zero.
dim(p) =
X
i≥j
l
l
X
1X 2 1 X 2
µi µj =
µi + (
µi + 2
µi µj )
2 i=1
2 i=1
i>j
l
l
l
1X 2 1 X 2 1X 2 1 2
=
µ + (
µi ) =
µ + n
2 i=1 i 2 i=1
2 i=1 i 2
dim(n) = dim(p) −
l
X
i=1
l
µ2i
1
1X 2
= n2 −
µ
2
2 i=1 i
Definition 2.3. Let G ×P n to be the quotient of the Cartesian product space G × n
by the equivalence relation (g, y) ∼ (gp−1 , p.y) = (gp−1 , pyp−1 ), where p ∈ P .
Recall that the quotient G/P is the partial flag variety, under the identification
gP → {0 ⊂ g ker(x) ⊂ g ker(x2 ) ⊂ · · · ⊂ g ker(xl )}:
22
2. R ESOLUTION OF SINGULARITIES FOR Oλ
G/P = {0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vl = V |dim(Vi ) = dim(ker(xi ))}
The partial flag variety is a projective variety, since there is a natural inclusion of
G/P into the product of the Grassmanians Gra1 (V ) × Gra2 (V ) × · · · × Gral−1 (V )
(where ai = dim(ker(xi ))), and each Grassmanian is a projective variety.
Proposition 2.4. Define the map φ : G ×P n → G/P × gln (C) by φ(g, y) =
(gP, gyg −1 ). Then the map φ is well-defined and injective, hence giving G ×P n the
structure of an algebraic variety (as a closed subvariety of G/P ×gln (C)). Further,
im(φ) = {((Vi ), y)|yVi ⊂ Vi−1 }.
Proof. To check that φ is well-defined, it suffices to check that φ(g, y) = φ(gp−1 , pyp−1 ).
This is a straightforward calculation:
φ(gp−1 , pyp−1 ) = (gp−1 P, (gp−1 )(pyp−1 )(gp−1 )−1 )
= (gP, (gp−1 )(pyp−1 )(pg −1 ))
= (gP, gyg −1 ) = φ(g, y)
To check that φ is injective, suppose φ(g1 , y1 ) = φ(g2 , y2 ), so that (g1 P, g1 y1 g1−1 ) =
(g2 P, g2 y2 g2−1 ). Since g1 P = g2 P , p = g2−1 g1 ∈ P . Since g1 y1 g1−1 = g2 y2 g2−1 ,
g2−1 g1 y1 g1−1 g2 = y2 so py1 p−1 = y2 . Hence (g2 , y2 ) = (g1 p−1 , py1 p−1 ) ∼ (g1 , y1 )
by definition of the equivalence relation in G ×P n.
It is clear from the definition of φ that im(φ) = {(gP, y) : g −1 yg ∈ n}. Since this is
a closed condition, this shows that im(φ) is a closed subvariety of G/P × gln (C).
The element gP ∈ G/P corresponds to the partial flag {0 ⊂ g ker(x) ⊂ · · · ⊂
g ker(xl )}. Since g −1 yg ∈ n, g −1 yg ker(xi+1 ) ⊂ ker(xi ), so then yg ker(xi+1 ) ⊂
g ker(xi ). Hence any (gP, y) ∈ im(φ) corresponds to an element ((Vi ), y) with
yVi ⊂ Vi−1 . Conversely, given some ((Vi ), y) such that yVi ⊂ Vi−1 , suppose
for each i, Vi = g ker(xi ) for some g ∈ G. Then yg ker(xi ) ⊂ gker(xi−1 ), so
g −1 yg ker(xi ) ⊂ ker(xi−1 ) for each i, and g −1 yg ∈ n. Hence ((Vi ), y) corresponds
to (gP, y) where g −1 yg ∈ n, and hence ((Vi ), y) lies in im(φ) if yVi ⊂ Vi−1 . This
proves that im(φ) = {((Vi ), y)|yVi ⊂ Vi−1 }. Since φ is injective, G ×P n can be
identified with this variety.
Corollary 2.5. G ×P n is a vector bundle over G/P , with fibres isomorphic to n.
Proof. By Proposition 2.4, identify G×P n with the variety {((Vi ), y)| yVi ⊂ Vi−1 }.
Let p1 denote the projection onto the first factor; p1 is clearly a surjective map
onto G/P . Given a fixed element gP = (Vi ) ∈ G/P , the fibre p−1
1 (Vi ) is {y ∈
gln (C)|yVi ⊂ Vi−1 }. This fibre is simply a conjugate of the Lie algebra n, and
hence is isomorphic to n as vector spaces.
2.1. R ESOLUTION OF SINGULARITIES IN TYPE A
23
Define the map π : G×P n → gln (C), by π(g, y) = gyg −1 . As proved in Proposition
2.4, π is well-defined. If we instead consider G ×P n as the set {((Vi ), y)|yVi ⊂
Vi−1 }, the map π is projection onto the second factor.
Corollary 2.6. The map π is a projective morphism of varieties.
Proof. By definition, a map π : X → Y is a projective morphism if we can find a
projective variety Z, and a closed embedding φ : X → Z × Y , so that π = p2 ◦ φ,
where p2 : Z × Y → Y is the projection onto the second factor. In this case, let
Z = G/P be the partial flag variety, and let φ : G ×P n → G/P × gln (C) be
the map from Proposition 2.4. It is clear that Z is a projective variety. Since φ is
an injective map (proven in Proposition 2.4), whose image is a closed-subvariety of
G/P × gln (C) (namely the set {(gP, y)|g −1 yg ∈ n}), φ is a closed embedding. It is
clear that p2 ◦ φ = π, since p2 (φ(g, y)) = p2 (gP, gyg −1 ) = gyg −1 = π(g, y). This
proves that π is a projective morphism.
Now we can state the first main result of this section: the map π is a resolution
of singularities for the orbit closure Oλ . At this point, it is hardly clear the above
map even surjects onto Oλ ; this, among a couple other things are what needs to be
verified in order to check that this map is a resolution of singularities.
Theorem 2.7. The map π : G ×P n → gln (C) is a resolution of singularities for
the orbit closure Oλ . Equivalently, the following four statements are true:
(i) The image of the map π in gln (C) is precisely the orbit closure Oλ .
(ii) The map π is injective when restricted to π −1 (Oλ ). The inverse map, when restricted to Oλ , π −1 : Oλ → G ×P n is a morphism of algebraic varieties.
(iii) As an algebraic variety, G ×P n is smooth and irreducible.
(iv) The map π is a proper morphism of algebraic varieties.
Proof. (of Theorem 2.7 (iii), (iv))
The variety G ×P n is irreducible, since it is a vector bundle over the partial flag variety G/P , and the partial flag variety is an irreducible variety. The variety G ×P n
is smooth, since it is a vector bundle over the G/P , and G/P is smooth as it is a
homogenous space.
The fact that π is a proper morphism of varieties follows from the fact that it is a
projective morphism of varieties (Corollary 2.6).
Proof. (of Theorem 2.7 part (i))
The image of the map π contains Oλ , since given n ∈ Oλ , n = gxg −1 for some
g ∈ GLn (C), so if we know that x ∈ n, then π(g, x) = n. To verify that x ∈ n,
using the definition of n, it suffices to verify that x(ker(xi )) ⊂ (ker(xi−1 )). This is
24
2. R ESOLUTION OF SINGULARITIES FOR Oλ
clear, since if u ∈ (ker(xi )), xi u = xi−1 (xu) = 0; i.e. xu ∈ (ker(xi−1 )).
Since the map π is a projective morphism, the image of π is a closed subvariety of
gln (C). Since the image of π contains Oλ , it follows that the image of π contains
Oλ . As proven above, the variety G ×P n is an irreducible variety, so its image
under the morphism φ will be an irreducible variety.
Next, it is clear that the image of the map π is stable under conjugation by elements
of G. If v ∈ im(π), i.e. v = gyg −1 for g ∈ G, y ∈ n, and if w = hvh−1 (i.e. w
lies in the same G-orbit as v), it follows that w = h(gyg −1 )h−1 = (hg)y(hg)−1 , so
w ∈ im(π). It is also clear that every element in the image of the map π is nilpotent,
since if v = gyg −1 lies in the image of π, y is nilpotent since every element in n is
nilpotent, and hence v is nilpotent. This means that the image of the map im(π) is
a union of nilpotent orbits of G. However, we already know that im(π) is a closed
algebraic variety, which means that im(π) is a union of some family of nilpotent
orbit closures. Since im(π) is an irreducible algebraic variety, it follows that im(π)
is a single orbit closure.
Since im(π) contains Oλ , and since the dimension of the image of the morphism
π is at most the dimension of the domain G ×P n, we have the following chain of
inequalities:
dim(Oλ ) ≤ dim(im(π)) ≤ dim(G ×P n)
Using the fact that the variety G ×P n is a vector bundle over the partial flag variety
G/P with fibres isomorphic to n, we can compute its dimension. The dimension
of G/P is equal to dim(G) − dim(P ), where G and P are quasi-affine algebraic
varieties (following the notation of Hartshorne). The dimension of the Lie groups
G and P will be equal to the dimension of the corresponding Lie algebras gln (C)
and p. Since we have computed dim(p) and dim(n) previously, we can now compute
dim(G ×P n):
dim(G ×P n) = dim(G) − dim(P ) + dim(n)
= dim(gln (C)) − dim(p) + dim(n)
l
l
1
1X 2
1
1X 2
= n2 − ( n2 +
µi ) + ( n2 −
µ)
2
2 i=1
2
2 i=1 i
2
=n −
l
X
µ2i
i=1
On the other hand, dim(Oλ ) = dim(Oλ ) (since the dimension of a closure of an
algebraic variety is equal to the dimension of the algebraic variety). As proven in
2.2. C LOSURE ORDERING IN T YPE A
Section 1.3, dim(Oλ ) = n2 −
Pl
i=1
µ2i . Hence dim(Oλ ) = n2 −
25
Pl
i=1
µ2i .
Hence dim(G ×P n) = dim(Oλ ), which means we must have equality in the chain
of inequalities dim(Oλ ) ≤ dim(im(π)) ≤ dim(G ×P n). Hence im(π) is a single
nilpotent orbit closure containing Oλ , and has the same dimension, so im(π) is
equal to Oλ .
Proof. (of Theorem 2.7 part (ii))
First we need to check that the map π is injective when restriced to π −1 (Oλ ).
Consider G ×P n as being the variety {((Vi ), y)|yVi ⊂ Vi−1 } and the map π as
being projection onto the second factor. We are required to prove that given a
y ∈ Oλ , there exists a unique flag (Vi ) with dim(Vi ) = dim(ker(xi )) such that
yVi ⊂ Vi−1 . Suppose that (Vi ) is such a flag. Since yV1 = 0, V1 ⊂ ker(y); since
y 2 V2 ⊂ yV1 = 0, V2 ⊂ ker(y 2 ); by a quick induction it follows that Vi ⊂ ker(y i ).
But dim(Vi ) = dim(ker(xi )) = dim(ker(y i )), since x, y ∈ Oλ . Hence Vi = ker(y i ).
Conversely, it is clear that (ker(y i )) is a flag with the desired properties. This shows
that π is injective when restricted to π −1 (Oλ ).
Next we need to show that the map π −1 : Oλ → G ×P n is a morphism of varieties. From the above paragraph, when restricted to Oλ , π −1 (y) = {((ker(y i )), y)}.
Hence we must prove that the map α : Oλ → G/P , defined by α(y) = (ker(y i )) is
a morphism of varieties.
Let Gx = CGLn (C) (x). We claim that Gx ⊂ P . If v ∈ Gx then vx = xv, and a
quick induction shows that vxi = xi v for all i. Then v(ker(xi )) ⊂ ker(xi ), since
if w ∈ ker(xi ) for some w ∈ Cn , then xi .w = 0 so xi .(vw) = v(xi .w) = 0, i.e.
v.w ∈ ker(xi ). Since v(ker(xi )) ⊂ ker(xi ) for all i, v ∈ P ; thus Gx ⊂ P . This
means that now we can define a morphism of varieties θ : G/Gx → G/P by declaring that θ(uGx ) = uP .
Since Oλ = Gx, the orbit-stabilizer theorem gives a bijection between Oλ and
G/Gx , via uxu−1 → uGx . This bijection is an isomorphism since we are working over C. We claim now that the maps α and θ can be identified under this
bijection. To prove this, under the map α, the element uxu−1 is sent to the flag
(ker(uxu−1 )i ) = (ker(uxi u−1 )). Under the map θ, the corresponding element uGx
is sent to uP , which corresponds to the flag (uker(xi )). Thus we need to prove that
ker(uxi u−1 ) = uker(xi ). This is a straightforward check: v ∈ ker(uxi u−1 ) ⇐⇒
uxi u−1 v = 0 ⇐⇒ xi u−1 v = 0 ⇐⇒ u−1 v ∈ ker(xi ) ⇐⇒ v ∈ uker(xi ). Hence
α and θ can be identified, proving that α is a morphism of varieties.
2.2. Closure ordering in Type A
In this section, given two nilpotent orbits Oλ and Oλ0 , we give a criterion for Oλ0 ⊂
Oλ . Recall the definition of the dominance partial ordering on the set of partitions
P(n) of n:
26
2. R ESOLUTION OF SINGULARITIES FOR Oλ
Definition 2.8. λ ≥ λ0 if for each i, λ1 + · · · + λi ≥ λ01 + · · · + λ0i .
Theorem 2.9. Oλ0 ⊂ Oλ precisely when λ0 ≤ λ.
Proof. By Theorem 2.7, Oλ is the image of the π, so y ∈ Oλ if and only if there
exists a flag (Vi ) with dimVi = dim(ker(xi )) = µ1 + · · · + µi such that yVi ⊂ Vi−1 .
Suppose Oλ0 ⊂ Oλ . Pick y ∈ Oλ0 ; then there exists a flag (Vi ) with dimVi =
µ1 + · · · + µi such that yVi ⊂ Vi−1 . It is clear that y i Vi = 0, so Vi ⊂ ker(y i ),
so dimVi ≤ dim(ker(y i )). If µ0 is the transpose partition to λ0 , dim(ker(y i )) =
µ01 + · · · + µ0i , as seen in Proposition 2.2. Hence µ1 + · · · + µi ≤ µ01 + · · · + µ0i
for each i, so µ ≤ µ0 . Since the operation of transposition is an order-reversing
involution on P(n) (see pg 6-7 of [7]), it follows that λ ≥ λ0 .
Conversely, suppose y ∈ Oλ0 , for some λ0 ≤ λ; we will prove that y ∈ Oλ , by
constructing a flag of subspaces (Vi ) with dimVi = µ1 + · · · + µi , such that yVi ⊂
Vi−1 . Let µ0 denote the transpose partition for λ0 . To construct this flag of subspaces,
pick a Jordan basis for y, and represent it by a diagram for the partition λ0 , so that
y moves each box to the box immediately to the left of it. As an example, let
λ0 = (42 , 22 ), λ = (5, 3, 2), µ0 = (4, 23 ), µ = (32 , 2, 12 ). In the below diagram, we
have filled up the partition diagram for λ0 with µ1 1-s, µ2 2-s, µ3 3-s, µ4 4-s and µ5
5-s, so that the numbers strictly increase across the rows. Now let Vi be spanned by
the boxes filled with numbers smaller than or equal to i. Then the spaces Vi will
have the correct dimensions; and will have the property that yVi ⊂ Vi−1 , due to
strict increase across rows.
The above example shows that it is sufficient to prove that there exists a way of
filling up the diagram for the partition λ0 with µ1 1-s, µ2 2-s, and so on, such that
the numbers strictly increase across the rows. By transposing the diagram, this
is equivalent to proving that there exists a way of filling up the diagram for the
partition µ0 with µ1 1-s, µ2 2-s, and so on, such that the numbers strictly increase
down the columns. The slightly stronger statement, that there exists a semistandard
tableaux of shape µ0 and content µ if µ ≤ µ0 is a standard fact; see pg 26 of Fulton
[4]. Here a semistandard tableaux of shape µ0 and content µ is a diagram of the
partition µ0 filled with µ1 1-s, µ2 2-s, and so on, such that the numbers weakly
increase across rows and strictly increase down columns.
2.3. R ESOLUTION OF SINGULARITIES IN TYPE C
27
2.3. Resolution of singularities in type C
Here we use the theory of the Jacobson-Morozov resolution to construct a resolution of singularities for the nilpotent orbit closure Oλ in the symplectic Lie algebra
sp2n (C). Denote G = Sp2n (C), g = sp2n (C). Let X be a nilpotent of type λ in
Oλ . By the Jacobson-Morozov Theorem, we can find an sl2 -triple {H, X, Y 0 } in
g. Then we can consider g as sl2 -module, via the adjoint action. By sl2 -theory, we
have the following:
g=
M
ga , where ga = {Y ∈ g|[H, Y ] = aY }
a∈Z
The following calculation shows that [ga , gb ] ⊆ ga+b . Let u ∈ ga , v ∈ gb :
[H, [u, v]] = [[H, u], v] + [[v, H], u]
= [au, v] + [−bv, u]
= (a + b)[u, v]
[u, v] ∈ ga+b
L
Let g≥i = a≥i ga , and let p = g≥0 . Since [ga , gb ] ⊆ ga+b , p is closed under the Lie
bracket, and hence is a Lie sub-algebra of g. Let P be the corresponding connected
Lie subgroup of G.
Since there is a natural action of G on a vector space V of dimension 2n, we may
also consider V as an sl2 -module. By the argument at the start of Proposition 1.7,
the dimensions of the irreducible sl2 -submodules of V are λ1 , λ2 , · · · , λk . The
weights of V as an sl2 -submodule will thus be λ1 − 1, λ1 − 3, · · · , 3 − λ1 , 1 −
λ1 , λ2 − 1, λ2 − 3, · · · , 3 − λ2 , 1 − λ2 , · · · . By sl2 -theory, we have the following:
V =
M
Va , where Va = {v ∈ V |H · v = av}
a∈Z
The following calculation shows that ga · Vb ⊆ Va+b . Let u ∈ ga , v ∈ Vb , so that
[H, u] = au, H · v = bv.
a(u · v) = [H, u] · v = H(u · v) − u(H · v)
= H(u · v) − u(bv)
= H(u · v) − b(u · v)
H(u · v) = (a + b)(u · v), so u · v ∈ Va+b
28
2. R ESOLUTION OF SINGULARITIES FOR Oλ
We now construct a partial flag in V as follows:
M
V≥b =
Va
a≥b
0 ⊆ · · · ⊆ V≥2 ⊆ V≥1 ⊆ V≥0 ⊆ V≥−1 ⊆ V≥−2 ⊆ · · · ⊆ V
X λi − b = db (λ)
dim V≥b =
2
1≤i≤k,λ >b
i
⊥
= V≥1−b . To prove
We now claim that this partial flag is isotropic, i.e. that V≥b
this, first recall from the proof of Proposition 1.7 that the r-weight space Vr will
be orthogonal to the s-weight space Vs unless r + s = 0. This shows that Vr will
be perpendicular to Vs if r ≥ b, s ≥ 1 − b since then r + s ≥ 1 > 0, and hence
V≥b is perpendicular to V≥1−b . Further, dim V≥b is equal to the number of weights
in V which are at least b, counted with multiplicity, and dim V≥1−b is equal to the
number of weights in V which are at least 1 − b, counted with multiplicity. Since
the weights in V are symmetric about 0, the number of weights in V which are at
least 1 − b is equal to the number of weights in V which are at most b − 1. Since
every weight is either at most b − 1 or at least b, and since there are 2n weights in
⊥
total, this shows that dimV≥b + dimV≥1−b = 2n. It now follows that V≥b
= V≥1−b .
Proposition 2.10. P is the stabilizer of this partial isotropic flag in Sp2n (C).
Proof. It suffices to prove that the stabilizer of this flag in sp2n (C) is p. To see why
this is true, if A stabilizes this flag, then for each t, exp(tA) will stabilize this flag.
Since P is generated is by the one-parameter subgroups exp(tA) as A ranges over
p, it follows that P will stabilize this flag.
Conversely we must show that the stabilizer of this flag in Sp2n (C) is not larger
than P ; so suppose that it is P 0 , and has Lie algebra p0 . Since the stabilizer of an
isotropic flag is connected, and the Lie algebra determines the Lie group if the Lie
group is connected, it suffices to prove that p0 ⊆ p, which will imply P 0 ⊆ P and
hence P 0 = P . Given any A ∈ p0 , for all t, exp(tA) ∈ P 0 , so exp(tA) will stabilize
the flag, and so will exp(tA)−1. Defining the norm ||X|| of a matrix to be the maximum of the absolute values of its entries, recall that the power series for ln(1 + X)
will converge if ||X|| < 1. Choosing t arbitrarily small so that || exp(tA) − 1|| < 1,
then tA = ln(1 + exp(tA) − 1) will stabilize the flag; hence A will stabilize the flag,
so A ∈ p. This shows that p0 ⊆ p; hence it is sufficient to show that the stabilizer of
the isotropic flag in sp2n (C) is p.
Let q be the stabilizer of the flag in sp2n (C).
q = {Y ∈ g|Y V≥b ⊆ V≥b ∀b}
= {Y ∈ g|Y Vb ⊆ V≥b ∀b}
2.3. R ESOLUTION OF SINGULARITIES IN TYPE C
29
To see the equality of the above two lines, clearly Y V≥b ⊆ V≥b implies Y Vb ⊆ V≥b .
Conversely, if Y Vb ⊆ V≥b for all b, then if a ≥ b, Y Va ⊆ V≥a ⊂ V≥b , so
Y V≥b ⊆ V≥b .
Since gi Vb ⊆ Vb+i ⊂ V≥b if i ≥ 0, it follows that gi ⊂P
q for each i ≥ 0, and thus
p ⊆ q. Conversely, suppose x ∈ q, x ∈
/ p. Let x = i xi , with xi ∈ gi . Since
x∈
/ p, for some j < 0, xP
j 6= 0. Then xj vl 6= 0 for some vl ∈ Vl ; we know that
xj vl ∈ Vj+l . Then xvl = i xi vl , and xi vl ∈ Vi+l ; since xj vl 6= 0, and j + l < l, it
follows xvl ∈
/ V≥l . This contradicts the fact that xVl ⊆ Vl . Hence q ⊆ p, implying
that p = q, as required.
Since P is the stabilizer of an isotropic flag, it follows that P is a parabolic subgroup of Sp2n (C). Since [ga , gb ] ⊂ ga+b , [p, g≥2 ] ⊆ g≥2 . It follows that P acts by
conjugation on g≥2 . To see this, given X ∈ p, g≥2 is stable under ad(X), and hence
under exp(ad(X)). But we have that exp(ad(X)) = Ad(exp(X)), so g≥2 is stable
under conjugation by elements of the form exp(X) for X ∈ p. Since P is generated
by elements of the form exp(X) for X ∈ p, it follows that P acts by conjugation
on g≥2 .
Definition 2.11. Let G ×P g≥2 be the quotient of the Cartesian product G × g≥2 by
the equivalence relation (g, y) ∼ (gp−1 , p · y) = (gp−1 , pyp−1 ) where p ∈ P .
P is the stabilizer of the isotropic flag {· · · ⊆ V≥1 ⊆ V≥0 ⊆ V≥−1 ⊆ · · · }, and
G acts transitively on the set {(Wb )| dim Wb = db (λ), Wb⊥ = W1−b } (this follows
from the stronger fact that Sp2n (C) acts transitively on the set of flags {0 = V0 ⊂
V1 ⊂ · · · ⊂ V2n−1 ⊂ V2n = V |Vi⊥ = V2n−i }). Thus under the identification
gP → {· · · ⊆ gV≥1 ⊆ gV≥0 ⊆ gV≥−1 ⊆ · · · }, G/P is the partial flag variety:
G/P = {(Wb )| dim Wb = db (λ), Wb⊥ = W1−b }
The partial flag variety G/P is a projective variety,
since there is a natural inclusion
Q
of G/P into the product of the Grassmanians b Grdb (λ) (V ), and each Grassmanian
is a projective variety.
Proposition 2.12. Define the map φ : G ×P g≥2 → G/P × g by φ(g, Y ) =
(gP, gY g −1 ). Then φ is well-defined and injective. Then we have that im(φ) =
{((Wb ), U )|U Wb ⊆ Wb+2 }. Thus im φ is a closed subvariety of G/P × g, giving
G ×P g≥2 the structure of an algebraic variety.
Proof. The proof that φ is a well-defined and injective map is the same as in the
proof of the Proposition 2.4, and is omitted.
It is clear by definition of φ that im(φ) = {(gP, U )|U ∈ gg≥2 g −1 }. The condition
that U ∈ gg≥2 g −1 is a closed condition, and thus im(φ) is a closed subvariety of
G/P × g. Suppose (gP, U ) ∈ im(φ), so that U = gY g −1 for some Y ∈ g≥2 . The
element gP ∈ G/P corresponds to the flag (gV≥b ), so gY g −1 (gV≥b ) = gY V≥b ⊆
gV≥b+2 (here Y V≥b ⊆ V≥b+2 since Y ∈ g≥2 ). Hence the (gP, U ) corresponds to an
30
2. R ESOLUTION OF SINGULARITIES FOR Oλ
element ((Wb ), U ) such that U Wb ⊆ Wb+2 .
Conversely, suppose we have an element (gP, U ) = (gP, gY g −1 ) which corresponds to an element ((Wb ), U ) such that U Wb ⊂ Wb+2 ; we will prove that Y ∈
g≥2 . Since Wb = gV≥b , we have that gY g −1 (gV≥b ) ⊂ gV≥b+2 , i.e. gY V≥b ⊂
gV≥b+2 , or Y V≥b ⊂ V≥b+2 . In the proof of Proposition 2.10, we proved that
Y V≥b ⊆ V≥b for each b implies that Y ∈ g≥0 . A similar argument will show
that if Y V≥b ⊂ V≥b+2 for each b, then Y ∈ g≥2 .
Corollary 2.13. G ×P g≥2 is a vector bundle over G/P , with fibres isomorphic to
g2 .
Proof. By Proposition 2.12, identify G ×P g≥2 with the variety {(gP, U )|U ∈
gg≥2 g −1 }. Let p1 denote the projection onto the first factor; p1 is clearly a surjective map onto G/P . Given a fixed element gP ∈ G/P , the fibre p−1
1 (gP ) is
{U ∈ gg≥2 g −1 }. This fibre is simply a conjugate of the vector space g≥2 , and hence
is isomorphic to g≥2 as vector spaces.
Define the map π : G ×P g≥2 → g by π(g, y) = gyg −1 . As proved in Proposition
2.12, π is a well-defined map. If we instead consider G×P g≥2 as {((Wb ), U )|U Wb ⊆
Wb+2 }, π is projection onto the second factor. By the exact same argument used in
Corollary 2.6, we may prove that π is a projective morphism of varieties. Now we
state the main result of this section: the map π is a resolution of singularities for the
orbit closure Oλ .
Theorem 2.14. The map π : G ×P g≥2 → g is a resolution of singularities for the
orbit closure Oλ . Equivalently, the following four statements are true:
(i) The image of the map π in g is precisely the orbit closure Oλ .
(ii) The map π is injective when restricted to π −1 (Oλ ). The inverse map, when restricted to Oλ , π −1 : Oλ → G ×P g≥2 is a morphism of algebraic varieties.
(iii) As an algebraic variety, G ×P g≥2 is smooth and irreducible.
(iv) The map π is a proper morphism of algebraic varieties.
Proof. (of Theorem 2.14 (iii), (iv)) The variety G ×P g≥2 is smooth and irreducible,
since it is a vector bundle over the partial flag variety G/P , which is smooth and
irreducible since it is a homogeneous space.
The fact that π is a proper morphism of varieties follows from the fact that it is a
projective morphism of varieties.
Proof. (of Theorem 2.14 (i))
We first prove that im(π) contains Oλ . Since π is a projective morphism, its image
is a closed subvariety of g, so it suffices to prove that im(π) contains Oλ . Since
2.3. R ESOLUTION OF SINGULARITIES IN TYPE C
31
X ∈ g2 ⊂ g≥2 , X = π(1, X), so X is contained in im(π). If Y ∈ Oλ , then
Y = gXg −1 for some g ∈ G, so Y = π(g, X) and so Y is contained in im(π). This
proves that im(π) contains Oλ .
Next, it is clear that im(π) is stable under conjugation by elements of G. If v ∈
im(π), i.e. v = gyg −1 for g ∈ G, y ∈ g≥2 , and if w = hvh−1 (i.e. w lies in the same
G-orbit as v), it follows that w = h(gyg −1 )h−1 = (hg)y(hg)−1 , so w ∈ im(π). It
is also clear that every element of g≥2 is nilpotent. To see this, g = g≥i for some i
sufficiently small; [g≥2 , g≥i ] ⊂ [g≥i+2 ], so if x ∈ g≥2 , ad(x)j g≥i ∈ g≥i+2j = 0 if j
is picked sufficiently large; this proves that ad(x) is a nilpotent endomorphism of g,
and hence x is nilpotent. Since every element in im(π) is conjugate to an element in
g≥2 , every element in im(π) is nilpotent. Since im(π) is stable under conjugation, it
follows that im(π) is a union of nilpotent orbits. Since im(π) is closed, it is a union
of nilpotent orbit closures. As im(π) is an irreducible variety, it is a single nilpotent
orbit closure.
As im(π) contains Oλ , and since the dimension of the image of the morphism π
is at most the dimension of its domain G ×P g≥2 , we have the following chain of
inequalities:
dim(Oλ ) ≤ dim(im(π)) ≤ dim(G ×P g≥2 )
Since G ×P g≥2 is a vector bundle over G/P with fibres isomorphic to g≥2 , we may
compute its dimension as follows:
dim(G ×P g≥2 ) = dim(G) − dim(P ) + dim(g≥2 )
= dim(g) − dim(p) + dim(g≥2 )
= dim(g) − dim(g≥0 ) + dim(g≥2 )
= dim(g) − dim(g0 ) − dim(g1 )
To compute dim(Oλ ), first note that dim(Oλ ) = dim(Oλ ) = dim(G.X) = dim(G)−
dim(GX ). Here GX denotes the centralizer of X in G, which has the same dimension as its Lie algebra, the centralizer of X in g. To compute this dimension, consider g as an sl2 -module, under the adjoint action. Then g has a basis of weight
vectors for H, with the weights consisting of the strings {µ1 − 1, µ1 − 3, · · · , 3 −
µ1 , 1 − µ1 }, {µ2 − 1, · · · , 1 − µ2 }, and so-on (where µi are the dimensions of the
irreducible summands of g as an sl2 -module). By sl2 -theory, X will annihilate the
highest weight vector in each string, and act non-trivially on all other vectors in each
string. Thus the centralizer of X in g will be spanned by the highest weight vectors
in each string, and its dimension will be equal to the number of weight strings. Each
weight string will either pass through 0 (if the highest weight in the string is even)
32
2. R ESOLUTION OF SINGULARITIES FOR Oλ
or pass through 1 (if the highest weight in the string is odd). Thus the dimension of
the centralizer of X in g will be equal to dim(g0 ) + dim(g1 ). Thus we have:
dim(Oλ ) = dim(Oλ )
= dim(G) − dim(GX )
= dim(G) − dim(gX )
= dim(G) − dim(g0 ) − dim(g1 )
Hence dim(Oλ ) = dim(G ×P g≥2 ), and we must have equality in the chain of
inequalities dim(Oλ ) ≤ dim(im(π)) ≤ dim(G ×P g≥2 ). Since im(π) is a single
nilpotent orbit closure containing Oλ , and with the same dimension, im(π) = Oλ .
Proof. (of Theorem 2.14 (ii))
We first need to check that the map π is injective when restricted to π −1 (Oλ ). Considering G×P g≥2 as the set {((Wb ), U )|U Wb ⊆ Wb+2 } and π as the projection onto
the second factor, we must prove that given U ∈ Oλ , there exists a unique flag (Wb )
such that U Wb ⊆ Wb+2 . It suffices to prove this for U = X; indeed suppose we
know the statement for U = X and suppose for some U = gXg −1 with g ∈ G, we
have two flags (Wb ) and (Wb0 ) with the required property. Then the flags (g −1 Wb )
and (g −1 Wb0 ) both have the required property with respect to X, and must be the
same (since there is only one flag which has the property for X). This then forces
(Wb ) and (Wb0 ) to be the same, proving the statement for U .
So we need to show that if (Wb ) is an isotropic flag with XWb ⊆ Wb+2 , then
Wb = V≥b . We first illustrate with the example λ = (32 , 2). The below partition
diagram shows a basis of V , where X takes each box to the box on its left, and the
numbers represent the weights of the basis vectors. Reading off the diagram, we
have that d3 (λ) = 0, d2 (λ) = 2, d1 (λ) = 3, d0 (λ) = 5, d−1 (λ) = 6, d−2 (λ) = 8.
Since XV = XW−2 ⊂ W0 , and dim XV = 5, dim W0 = d0 (λ) = 5, it follows that
XV = W0 ; by inspection we also have that XV = V≥0 , so W0 = V≥0 . Similarly,
since X 2 V = X 2 W−2 ⊂ W2 , but dim X 2 V = 2 and dim W2 = d2 (λ) = 2;
by inspection we also have that X 2 V = V≥2 , so W2 = V≥2 . Now that we have
W2 = V≥2 and W0 = V≥0 , since both flags are isotropic we have W−1 = V≥−1 and
W1 = V≥1 . This now proves that the two flags are the same, as required.
2.3. R ESOLUTION OF SINGULARITIES IN TYPE C
33
To show the statement in general, we proceed by induction on the size of the largest
part λ1 . The base case is when λ = (1l ), for some l even. Here, d0 (λ) = l and
d1 (λ) = 0, so we are dealing with a flag W1 ⊂ W0 with dim W1 = 0, dim W0 = l.
This forces W1 = 0 = V≥1 , W0 = V = V≥0 , proving the statement in this case.
We will first prove that Wλ1 −1 = V≥λ1 −1 = im(X λ1 −1 ) (note Wλ1 −1 is the smallest non-zero subspace in the flag). Since V = W1−λ1 , applying the fact that
XWb ⊆ Wb+2 λ1 − 1 times, we get that X λ1 −1 V ⊆ Wλ1 −1 . But dim X λ1 −1 is
equal to the multiplicity of λ1 in λ (this can be seen by looking at how X λ1 −1
acts on a Jordan basis for X); and dim Wλ1 −1 is equal to the number of weights
which are greater than or equal to λ1 − 1, which is also equal to the multiplicity
of λ1 in λ. This shows that X λ1 −1 V = Wλ1 −1 . The same argument also proves
that X λ1 −1 V = V≥λ1 −1 , and hence Wλ1 −1 = V≥λ1 −1 . Since the flag is isotropic,
⊥
Wλ⊥1 −1 = W2−λ1 and V≥λ
= V≥2−λ1 , and so W2−λ1 = V≥2−λ1 .
1 −1
Consider V 0 = V≥2−λ1 /V≥λ1 −1 (as proved in the last paragraph, this is equal to
W2−λ1 /Wλ1 −1 ). Define a symplectic form on this space as follows:
hv + V≥λ1 −1 , v 0 + V≥λ1 −1 i = hv, v 0 i
This form is well-defined since V≥2−λ1 is perpendicular to V≥λ1 −1 , so v and v 0 will
be orthogonal to any vector in V≥λ1 −1 . This form is clearly skew-symmetric. It is
non-degenerate, since if v + V≥λ1 −1 lies in the kernel of the form, then v will be or⊥
thogonal to everything in V≥2−λ1 , so v ∈ V≥2−λ
= V≥λ1 −1 , so then v + V≥λ1 −1 = 0.
1
The endomorphism X will induce a nilpotent endomorphism X 0 in V 0 of type λ0 ,
where λ0 is the partition with parts λ1 − 2, · · · , λr − 2, λr+1 , · · · , λk , arranged in
decreasing order (where r is the multiplicity of λ1 in λ). To see this, suppose V has
basis ei,j , where 1 ≤ i ≤ k, 1 ≤ j ≤ λi , such that Xei,j = ei,j−1 if j > 1 and 0
otherwise. Then V≥λ1 −1 = im(X λ1 −1 ) will be spanned by {e1,1 , · · · , er,1 }, while
⊥
V≥2−λ1 = im(X λ1 −1 ) will be spanned by all of the ei,j -s excluding {e1,λ1 , · · · , er,λ1 }.
Thus V 0 is spanned by the images of all ei,j -s satisfying 2 ≤ j ≤ λ1 −1 if 1 ≤ i ≤ r,
and X acts as before; this shows that X 0 has type λ0 . Consider the two flags induced
by V≥b and Wb in V 0 :
0 ⊂ Wλ1 −2 /V≥λ1 −1 ⊂ · · · ⊂ W3−λ1 /V≥λ1 −1 ⊂ V 0
0 ⊂ V≥λ1 −2 /V≥λ1 −1 ⊂ · · · ⊂ V≥3−λ1 /V≥λ1 −1 ⊂ V 0
It is easy to check that the above flags are isotropic; we will check that the first
flag is isotropic (the same method will work for the second flag). Given 3 − λ1 ≤
b ≤ λ1 − 2, we need to check that (Wb /V≥λ1 −1 )⊥ = W1−b /V≥λ1 −1 . Since Wb⊥ =
W1−b , the two spaces are orthogonal. It suffices to check that dim(Wb /V≥λ1 −1 ) +
dim(W1−b /V≥λ1 −1 ) = dim V 0 . This is true because dim(V≥λ1 −1 ) = r, so LHS =
34
2. R ESOLUTION OF SINGULARITIES FOR Oλ
dim(W≥b ) + dim(W≥1−b ) − 2r = dim V − 2r = dim V 0 = RHS.
Since X(Wb ) ⊆ Wb+2 , we have X 0 (Wb /V≥λ1 −1 ) ⊆ Wb+2 /V≥λ1 −1 , and similarly
X 0 (V≥b /V≥λ1 −1 ) ⊆ V≥b+2 /V≥λ1 −1 . By the induction hypothesis applied to X 0 , it
now follows that Wb /V≥λ1 −1 = V≥b /V≥λ1 −1 , and hence Wb = V≥b . This gives the
required result, proving that the map π is injective when restricted to π −1 (Oλ ).
Next we need to prove that the map π −1 : Oλ → G ×P g≥2 is a morphism of
varieties. Given U = gXg −1 ∈ Oλ , from the above paragraphs, there exists a
unique flag (Wb ) with U Wb ⊆ Wb+2 , and it is clear that (gV≥b ) is such a flag. Thus
π −1 (gXg −1 ) = ((gV≥b ), gXg −1 ). To prove that π −1 is a morphism, it suffices to
prove that the mapα, defined by α(gXg −1 ) = (gV≥b ) is a morphism of varieties.
Let GX = CSp2n (C) (X). Since Oλ = Gx, the orbit-stabilizer theorem gives a bijection between Oλ and G/GX via uXu−1 → uGX . This bijection is an isomorphism
of varieties since we are working over C. We also have that GX ⊂ P ; to see this,
given g ∈ GX , then g.(V≥b ) will be an isotropic flag with X(gV≥b ) ⊂ gV≥b+2 . Since
there is a unique flag with this property, this forces g ∈ P , and hence GX ⊂ P . This
means we have a morphism θ : G/GX → G/P , defined via θ(uGX ) = uP . We
claim that θ and α can be identified under the bijection between Oλ and G/GX . To
prove this, under the map α, the element gXg −1 is sent to (gV≥b ). Under the map θ,
the corresponding element gGX is sent to gP , which corresponds to the flag (gV≥b ),
as required. This proves that α is a morphism of varieties.
C HAPTER 3
The enhanced nilpotent cone and some variations
3.1. The enhanced nilpotent cone
The enhanced nilpotent cone for GLn (C) (see Achar-Henderson, [1]), V × N , is
defined to be the product of the vector space V = Cn with the nilpotent cone N ;
there is a natural action of GLn (C) on V , which gives a natural action of GLn (C)
on V × N . In this section, we classify the orbits of GLn (C) on V × N , and prove
that the orbits are in 1 − 1 correspondence with bi-partitions of n, defined below.
Definition 3.1. An ordered pair of partitions (µ, ν) with |µ| + |ν| = n is called a
bi-partition of n. The set of all bi-partitions of n will be denoted Q(n). Call (µ, ν)
a bi-partition of λ, if µ + ν = λ.
Definition 3.2. Given a vector v ∈ V , for 1 ≤ j ≤ k, define the j-th “portion"
of v to be the vector obtained by considering the λ1 + · · · + λj−1 + 1-th, · · · , λ1 +
· · · λj−1 + λj -th coordinates of V .
The below theorem, which re-phrases Proposition 2.3 from [1], is the main Theorem
in this section. The proof given here is a variation of the proof given in [1].
Theorem 3.3. The orbits of GLn (C) on V × N are in one-to-one correspondence
with the set Q(n). The partition µ + ν = λ of n specifies the Jordan type of the
nilpotent element n ∈ N . Suppose λ = (λ1 , · · · , λk ), µ = (µ1 , · · · , µk ), ν =
(ν1 , · · · , νk ). Then (vµ , nλ ) is an orbit representative for the orbit corresponding to
(µ, ν), where nλ is as described in section 2.2.1, and vµ is constructed as follows:
for each 1 ≤ j ≤ k, the j-th portion has a 1 in the µj -th coordinate, and 0-s
elsewhere.
The problem of computing the orbits of GLn (C) on V × N is equivalent to the
problem of computing the orbits of CGLn (C) (nλ ) on V . To see this, if two elements
(v1 , n1 ), (v2 , n2 ) ∈ V × N are in the same GLn (C)-orbit, then n1 and n2 are nilpotent elements with the same Jordan type, say corresponding to the partition λ. Then
it suffices to check when (v1 , nλ ), (v2 , nλ ) are in the same GLn (C)-orbit. This happens when there exists a g with gv1 = v2 with gnλ g −1 = nλ , i.e. if v1 , v2 lie in the
same CGLn (C) (nλ )-orbit.
Example 3.4. Let λ = (3, 2, 2). A typical element g ∈ CGL7 (C) (nλ ) takes the
following form, using the structure of the centralizer CGLn (C) (nλ ) as described in
Section 2.3.1. In the case of the bi-partition (µ, ν) = ((2, 1, 1), (1, 1, 1)), vµ will be
as follows.
35
36
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS





g=



λ1
λ2
λ1
λ8
λ10
λ3 λ4
λ2
λ1
λ9 a11
λ8
λ11 a21
λ10





gvµ = 



λ5
λ4
λ6
λ12 a12
a11
λ14 a22
a21

 
λ7
0
λ6 
 1 

 

 0 

 
λ13  , vµ =  1 

 
a12 
 0 
 1 
λ15 
a22
0
λ2 + λ4 + λ6
λ1
0
λ8 + a11 + a12
0
λ10 + a21 + a22
0









Definition 3.5. Given a vector v, say that it is of type (α, β), where α = (α1 , · · · , αk ),
β = (β1 , · · · , βk ) and αi + βi = λi for all 1 ≤ i ≤ k, if the maximal t such that
the t-th component in the i-th portion is non-zero, is αi for each 1 ≤ i ≤ k. Let
Wα be the set of vectors of type (α, β). Let vα be constructed as follows: for each
1 ≤ j ≤ k, the j-th portion has a 1 in the αj -th coordinate, and 0-s elsewhere.
˙ on pairs of compositions (α, β) such that
Definition 3.6. Define the ordering >
0
˙
α + β = λ, by declaring (α, β)>(α , β 0 ) iff αj ≥ αj0 for all 1 ≤ j ≤ k. In particular
this ordering restricts to bi-partitions λ. Define a map φ from pairs of compositions
adding up to λ, to bi-partitions of λ as follows: let φ(α, β) be the unique minimal
bi-partition lying above the pair of compositions (α, β).
In the above definition, it is not immediately clear why there must exist a unique
minimal bi-partition above the pair of compositions (α, β). We justify this in the
following proposition, by explicitly constructing φ(α, β) in terms of (α, β).
Proposition 3.7. The unique minimal bi-partitition (γ, δ) = φ(α, β) lying above
the pair (α, β) is given by the following:
γi = max({αj | j ≥ i}∪{λi −βj | j < i}), δi = min({λi −αj | j ≥ i}∪{βj | j < i})
Proof. In order to prove the proposition, we must check that:
a) (γ, δ) ∈ Q(n) and γ + δ = λ,
˙
b) (α, β)<(γ,
δ)
˙ 0 , δ 0 ) for some (γ 0 , δ 0 ) ∈ Q(n) with γ 0 +δ 0 = λ, then (γ, δ)<(γ
˙ 0 , δ 0 ).
c) If (α, β)<(γ
To check (a), it is clear from the definitions that γ + δ = λ, so it suffices to check
that γi ≥ γi+1 and δi ≥ δi+1 for all 1 ≤ i ≤ k − 1. To show that γi ≥ γi+1 , we must
show that
max{αi , · · · , αk , λi −β1 , · · · , λi −βi−1 } ≥ max{αi+1 , · · · , αk , λi+1 −β1 , · · · , λi+1 −βi }
3.1. T HE ENHANCED NILPOTENT CONE
37
To do this, for every element u in the set {αi+1 , · · · , αk , λi+1 − β1 , · · · , λi+1 − βi },
we will show that there exists some element u0 in the set {αi , αi+1 , · · · , αk , λi −
β1 , · · · , λi − βi−1 } with u0 ≥ u. If u = αj for some i + 1 ≤ j ≤ k, we can
simply pick u0 = u. If u = λi+1 − βj for some 1 ≤ j ≤ i − 1, then we can pick
u0 = λi −βj . If u = λi+1 −βi , then we can pick u0 = αi (since λi+1 ≤ λi = βi +αi ).
This concludes the proof that γi ≥ γi+1 . To show that δi ≥ δi+1 , we must show that
min{λi −αi , · · · , λi −αk , β1 , · · · , βi−1 } ≥ min{λi+1 −αi+1 , · · · , λi+1 −αk , β1 , · · · , βi }
To do this we will prove that for any v in the set {λi −αi , · · · , λi −αk , β1 , · · · , βi−1 },
we can find a v 0 in the set {λi+1 − αi+1 , · · · , λi+1 − αk , β1 , · · · , βi } with v ≥ v 0 .
If v = λi − αj with i + 1 ≤ j ≤ k, then we can simply pick v 0 = λi+1 − αj . If
v = λi − αi = βi , we can pick v 0 = βi . If v = βj for 1 ≤ j ≤ i − 1, we can pick
v 0 = βj . This concludes the proof that δi ≥ δi+1 , and hence that of (a).
To check (b), it suffices to show that αi ≤ γi for all 1 ≤ i ≤ k. This follows from
the definition γi = max{αi , αi+1 , · · · , αk , λi − β1 , · · · , λi − βi−1 }.
˙ 0 , δ 0 ), i.e. that
To check (c), we must show that (γ, δ)<(γ
γi0 ≥ γi = max{αi , αi+1 , · · · , αk , λi − β1 , · · · , λi − βi−1 }
Thus we must first show that γi0 ≥ αj for i ≤ j ≤ k. This is true because γi0 ≥ γj0 ≥
αj . We must also show that γi0 ≥ λi − βj for 1 ≤ j ≤ i − 1. This is equivalent to
showing that βj ≥ λi − γi0 = δi0 . The reason that this is true is because βj ≥ δj0 ≥ δi0 ,
(where βj ≥ δj0 is true since γj0 ≥ αj and γj0 + δj0 = αj + βj = λj ).
Definition 3.8. Let Vµ be the set of vectors v ∈ V such that the type (α, β) of v
satisfies φ(α, β) = (µ, ν).
It is clear that vµ ∈ Wµ ⊂ Vµ . We claim that Vµ = CGLn (C) (nλ ).vµ . This is sufficient to prove the Theorem 3.3, since V is the disjoint union of all Vµ , as (µ, ν)
ranges over Q(n). The fact Vµ is a single CGLn (C) (nλ )-orbit is a consequence of the
following three Lemmas. The reason for this is that Lemma 3.9 implies that Vµ is a
union of CGLn (C) (nλ )-orbits on V , while Lemma 3.10 applied to α = µ, along with
Lemma 3.11, imply that CGLn (C) (nλ ) acts transitively on Vµ .
Lemma 3.9. The set Vµ is stable under the action of CGLn (C) (nλ ).
Lemma 3.10. Any vector in Wα can be mapped to any other vector in Wα by some
element of CGLn (C) (nλ ).
Lemma 3.11. Given a vector v of type (α, β), there exists another v 0 of type φ(α, β)
in the same CGLn (C) (nλ )-orbit as v.
Proof. (of Lemma 3.10)
It suffices to show that vα can be mapped to any other vector in Wα (since it is clear
that vα ∈ Wα ). To illustrate the idea behind this proof, we revisit Example 3.4 with
λ = (3, 2, 2), α = (2, 1, 1). Consider the following choice of g ∈ CGLn (C) (nλ ):
38
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
λ1 λ2 λ3
λ1 λ2


λ1


a11 λ12
g=

a11


a22 λ15
a22











,
v
=
 α 






0
1
0
1
0
1
0










,
gv
=


α






λ2
λ1
0
a11
0
a22
0









The only constraints on the matrix g ∈ CGLn (C) (nλ ) above is that λ1 , a11 , a12 6=
0, so the image gvα ranges across all vectors of the form above subject to this
constraint. But this is precisely how Wα is defined, concluding the proof in the case
of this example. It is clear how to generalize this proof for λ = (3, 2, 2) to arbitrary
λ = (λ1 , · · · , λk ).
Proof. (of Lemma 3.11)
If φ(α, β) = (µ, ν), we shall prove that vα is in the same orbit as some vector
v 0 ∈ Vµ ; this suffices since by Lemma 3.10 any vector of type (α, β) is in the same
CGLn (C) (nλ )-orbit as vα . We will first prove that there exists g ∈ CMn (C) (nλ ) such
that gvα ∈ Wµ .
By Proposition 3.7, µi = max({αj |j ≥ i} ∪ {λi − βj |j < i}). We will show that,
for any s ∈ {αj |j ≥ i} ∪ {λi − βj |j ≤ i}, we can choose g, so that gvα has a
non-zero value in the s-th co-ordinate in the i-th portion, and all higher entries in
the i-th portion are 0.
If s = αj , for some j ≥ i, pick any g with a non-zero entry in the αj -th row and αj th column of the (i, j)-th block of g. Since j ≥ i, λi ≥ λj , so this will be possible,
since it permissible to have a non-zero entry in the r-th row and s-th column of the
(i, j)-th block of g when r ≤ s in this case, by Proposition 1.14. We stipulate also
that for all j 0 , all other entries in the (i, j 0 )-th blocks of g, apart from those forced
to be equal to this entry, be 0. It follows then that gvα will have a non-zero entry in
the αj -th coordinate of the i-th portion, as required. The fact that all higher entries
in the i-th portion are 0 follows from the fact that all entries higher than αj in the
j-th portion of vα are 0.
Otherwise if s = λi − βj for some j ≤ i, pick a g with a non-zero entry in the
(λi −βj )-th row and αj -th column of the (i, j)-th block of g. Since j ≤ i, λi ≤ λj , it
is permissible to have a non-zero entry in the r-th row and s-th column of the (i, j)th block only if r −s ≤ λi −λj ; thus this is permissible since λi −βj −αj = λi −λj .
We stipulate also that all other entries in the (i, j 0 )-th blocks of g, apart from those
already forced to be equal to this entry, be 0. It follows then that gvα will have a
non-zero entry in the (λi − βj )-th coordinate of the i-th portion, as required. The
fact that all higher entries in the i-th portion are 0 follows from the fact that all entries higher than αj in the j-th portion of vα are 0.
3.1. T HE ENHANCED NILPOTENT CONE
39
In the above paragraphs, we have shown by imposing certain restrictions on the
coordinates in the (i, j)-th blocks of g, for fixed i, we can ensure that gvα has a
non-zero value in the µi -th co-ordinate in the i-th portion, and all higher entries in
the i-th portion are 0. Since there are no dependencies between the entries in the
(i, j)-th block and (i0 , j 0 )-th block of g unless i = i0 , j = j 0 , we can impose all of
these restrictions simultaneously, and find a g so that gvα will have a non-zero value
in the µi -th coordinate of the i-th portion, and zero values in all higher co-ordinates
for each i. Equivalently, we have found a g so that gvα has type (µ, ν).
Now note that g + aI ∈ CMn (C) (nλ ), for all a ∈ C, and (g + aI)(vα ) = gvα + avα .
It is clear that for all a with a finite number of exceptions, gvα + avα will have type
(µ, ν), and that for all a with a finite number of exceptions, g + aI will be invertible.
Taking a value of a outside the union of these two finite sets of exceptions, g + aI ∈
CGLn (C) (nλ ) will have the required property.
Proof. (of Lemma 3.9)
The proof of Lemma 3.9 uses Lemmas 3.10 and Lemma 3.11. Suppose that Vµ is not
stable under the action of CGLn (C) (nλ ); i.e. suppose that for given some v1 ∈ Vµ1 for
some bi-partition (µ1 , ν1 ), we can find some v2 ∈ Vµ2 for some bi-partition (µ2 , ν2 )
with v1 , v2 being in the same CGLn (C) (nλ ) orbit. Then by Lemmas 3.10 and 3.11,
v1 is in the same orbit as vµ1 , and v2 is in the same orbit as vµ2 . Thus vµ1 and vµ2
are in the same orbit; we will now show that this is not possible.
We claim that for g ∈ CGLn (C) (nλ ), the lth co-ordinate of the ith portion of g.vµ is
0 for l > µi . (The computation in Example 3.4 of gvµ illustrates this claim.) This
will then imply that vµ1 and vµ2 cannot be in the same CGLn (C) (nλ )-orbit, since for
some i, since µ1 6= µ2 , WLOG we must have (µ1 )i > (µ2 )i . If vµ1 = g.vµ2 , then the
vector vµ1 will have a non-zero entry, 1, in the (µ1 )i -th position, which contradicts
the above claim since (µ1 )i > (µ2 )i .
To show that the lth co-ordinate of the ith portion of g.vµ is 0 for l > µi , we must
examine the structure of the centralizer CGLn (C) (nλ ). We are required to prove that
the (λ1 + · · · + λi−1 + l)-th coordinate of g.vµ is 0; accordingly, examine the entries
in the (λ1 + · · · + λi−1 + l)-th row of the matrix g ∈ CGLn (C) (nλ ). The non-zero
entries in the column vector vµ occur in the co-ordinates (λ1 + · · · + λj−1 + µj )-th
positions for j = 1, · · · , k. Thus to prove that the (λ1 + · · · + λi−1 + l)-th coordinate
of g.vµ is 0, it suffices to prove that the (λ1 + · · · + λj−1 + µj )-th coordinate in the
(λ1 + · · · + λi−1 + l)-th row of g is 0 for all l > µi and all j. To do this, we must
separately examine the cases i ≤ j and i > j.
If i ≤ j, note that in the (i, j)-th block of the matrix g, which is of size λi × λj , the
(u, v)-th coordinate is 0 if u > v, by Proposition 1.14. Here we are interested in the
co-ordinate in the (λ1 + · · · + λi−1 + l) row and the (λ1 + · · · + λj−1 + µj ) column
of g. This lies in the l-th row and µj -th column of the (i, j)-th block of the matrix
40
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
g. But l > µi ≥ µj since i < j, which concludes the proof of the claim in this case.
If i > j, then note that in the (i, j)-th block of the matrix g, which is of size
λi × λj , the (u, v)-th coordinate is 0 if v − u < λj − λi , by Proposition 1.14. Again,
here we are interested in the co-ordinate in the (λ1 + · · · + λi−1 + l)-th row and the
(λ1 +· · ·+λj−1 +µj )-th column of g, which lies in the l-th row and the µj -th column
of the (i, j)-th block of g. But µj − l < µj − µi ≤ λj − λi ; where µj − µi ≤ λj − λi
since λi − µi ≤ λj − µj (i.e. νi ≤ νj , which is true since i > j). This concludes the
proof of the claim in the second case, and hence the proof of Lemma 3.9.
3.2. The exotic nilpotent cone
In this section, we study Kato’s exotic nilpotent cone, N. We state the classification
of the orbits in the exotic nilpotent cone, and describe part of the proof of this classification. Let V ∼
= C2n be a vector space of dimension 2n, with a symplectic form
t
h·, ·i, hv, wi = v Jw corresponding to the following 2n × 2n matrix J:
1

···






1
−1
···







−1
Following [2], define S, N as follows (here N denotes N (gl(V )), the set of nilpotent matrices in End(V )). N is the exotic nilpotent cone.
S = {x ∈ End(V )| hxv, wi = hv, xwi , ∀v, w ∈ V }
N = V × (S ∩ N )
The following calculation shows that there is a well-defined action of Sp(V ) on S
by conjugation. Let g ∈ Sp(V ), so that hgv, gwi = hv, wi for all v, w, and let
x ∈ S:
gxg −1 v, w = gxg −1 v, gg −1 w = xg −1 v, g −1 w
= g −1 v, xg −1 w = gg −1 v, gxg −1 w
= v, gxg −1 w
Since Sp(V ) clearly acts by conjugation on N , Sp(V ) acts by conjugation on S∩N .
There is also a natural action of Sp(V ) on V , and hence there is an action of Sp(V )
on the exotic nilpotent cone N = V × (S ∩ N ).
3.2. T HE EXOTIC NILPOTENT CONE
41
There is a natural inclusion map φ2 : N → V × N , of the exotic nilpotent cone into
the enhanced nilpotent cone of size 2n. We next construct a map φ1 : V1 ×N1 → N,
where V1 ∼
= Cn is a vector space of dimension n, and N1 = N (gl(V1 )) is the set of
nilpotent matrices in End(V1 ); thus V1 ×N1 is the enhanced nilpotent cone of size n.
Lemma 3.12. The following map φ1 : V1 × N1 → N is well-defined. Here x ∈ N1 ,
and xt denotes its transpose along the skew diagonal, so that xti,j = xn+1−j,n+1−i .


v1
 .. 


 .  v1


v
x


.
n
φ1 ( ..  , x) = (
)
,
xt
 0 
 . 
vn
 .. 
0
Proof. Let X = ( x xt ). It is clear that if x is nilpotent, then xt is nilpotent, and so
X is nilpotent, i.e. X ∈ N . Thus it suffices to prove that X ∈ S, i.e. hXv, wi =
hv, Xwi for all v, w ∈ V . Letting ei be the vector with a 1 in the i-th co-ordinate
and zeroes elsewhere, since {e1 , · · · , e2n } is a basis for V , it sufficient to prove that
hXei , ei0 i = hei , Xei0 i. By construction of h·, ·i, recall that:


if i + i0 6= 2n + 1,
0
hei , ei0 i = 1
if i < i0 , i + i0 = 2n + 1,

−1 if i > i0 , i + i0 = 2n + 1
If 1 ≤ i, i0 ≤ n, then Xei , Xei0 ∈ Span{e1 , · · · , en }, and hXei , ei0 i = hei , Xei0 i =
0. Similarly if n + 1 ≤ i, i0 ≤ 2n, hXei , ei0 i = hei , Xei0 i = 0. If 1 ≤ i ≤ n, n + 1 ≤
i0 ≤ 2n, then we compute hXei , ei0 i , hei , Xei0 i as follows. Let i0 = j + n, with
1 ≤ j ≤ n.
hXei , ej+n i = hx1,i e1 + · · · + xn,i en , ej+n i = xn+1−j,i
hei , Xej+n i = ei , xt1,j en+1 + · · · + xtn,j e2n = xtn+1−i,j
By the definition of xt , xn+1−j,i = xtn+1−i,j , so hXei , ej+n i hei , Xej+n i in this case.
A similar argument works in the case when n + 1 ≤ i ≤ 2n, 1 ≤ i0 ≤ n. Hence
X ∈ S, as required.
Lemma 3.13. Suppose (v, n1 ) and (w, n2 ) are in the same GL(V1 ) orbit of V1 ×N1 .
Then φ(v, n1 ) and φ(w, n2 ) are in the same Sp(V )-orbit of N.
g
Proof. Suppose g(v, n1 ) = (w, n2 ). Define g1 = ( (gt )−1 ). We will first check that
g1 ∈ Sp(V ), and then verify that g1 (φ(v, n1 )) = φ(w, n2 ).
42
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
To check that g1 ∈ Sp(V ), it is necessary to check that g1T Jg1 = J (g1T denotes the
0
ordinary transpose). Here J = ( −I 0 I ); where I 0 denotes the n × n matrix with 1’s
along the skew diagonal. To help with this computation, note first that g T I 0 = I 0 g t ,
since by inspection both matrices will have gn+1−j,i as their (i, j)-th co-ordinate.
Then we obtain that g T I 0 (g t )−1 = I 0 ; and transposing the equation we also have
T
T
I 0 g = g t I 0 , so (g t )−1 I 0 g = I 0 . Now we compute:
T g
I0
g
(g t )−1
−I 0
(g t )−1
T
g
I 0 (g t )−1
=
T
−I 0 g
(g t )−1
g T I 0 (g t )−1
I0
=
=
T
−I 0
−(g t )−1 I 0 g
g
Hence g1 ∈ Sp(V ). Next, it is clear that if gv = w, then ( (gt )−1 )( v0 ) = ( w0 ). Since
g(v, n1 ) = (w, n2 ), we have gn1 g −1 = n2 . Since (AB)t = B t At (one way of seeing
this is that, from above, xT I 0 = I 0 xt so xt = I 0−1 xT I 0 , and this identity holds for the
ordinary transpose), taking the transpose of gn1 g −1 = n2 gives (g t )−1 nt1 g t = nt2 .
Now we compute:
g
gt
=
−1
−1
g
nt1
g
gt
=
−1
n1
gt
n1
−1
g −1
gt
nt1
gn1 g −1
(g t )−1 nt1 g t
=
n2
nt2
This proves that g1 (φ(v, n1 )) = φ(w, n2 ), concluding the proof.
From results in section 3.1, we know that the orbits of GL(V1 ) on V1 × N1 are in
bijection with bi-partitions of n; call Oµ,ν the orbit corresponding to the bi-partition
(µ, ν). By the above Lemma, any two elements in φ(Oµ,ν ) will be in the same
Sp(V ) orbit of N.
Definition 3.14. Define Oµ,ν to be the Sp(V )-orbit in N containing φ1 (Oµ,ν ).
Given a partition λ = (λ1 , · · · , λk ), let λ∪λ denote the partition (λ1 , λ1 , · · · , λk , λk ).
Proposition 3.15. We have that φ2 (φ1 (Oµ,ν )) ⊆ Oµ∪µ,ν∪ν .
Proof. As in the proof of Theorem 3.3, let (vµ , nλ ) be an orbit representative for
Oµ,ν in V1 × N1 . It suffices to prove that φ2 (φ1 (vµ , nλ )) ∈ Oµ∪µ,ν∪ν . Consider the
following example, where λ = (2, 1), µ = (1, 1), ν = (1, 0).
3.2. T HE EXOTIC NILPOTENT CONE




φ2 (φ1 (vµ , nλ )) = (


1
0
1
0
0
0
43
0 1
0
 
 
0
 
,
0
 
 
0 1
0
 




)


This example makes it clear that while nλ consists of the blocks Nλ1 , · · · , Nλk arranged in that order, ntλ will consists of the same blocks arranged in the reverse
order: Nλk , · · · , Nλ1 . Since nλ and ntλ have the same Jordan type, it follows then
that we can find a matrix g, such that gnλ g −1 = ntλ . Let g 0 be a 2n × 2n matrix
consisting of the four n × n quadrants, with the top left quadrant being the identity
matrix, the bottom right quadrant being g, and the other two quadrants empty. Acting on the above pair by the matrix g 0 , g 0 will stabilize the vector and transform the
matrix into a matrix with nλ occurring twice:




(


1
0
1
0
0
0
0 1
0
 
 
0
 
,
0 1
 
 
0

 



)


0
Now act on the above pair by the matrix h which will conjugate the matrix above to
the standard nilpotent nλ∪λ . It is clear what effect this will have on the vector.




(


1
0
0
0
1
0
0 1
0
 
 
0 1
 
,
0
 
 
0
 




)


0
It is clear how this example generalizes: we have shown that φ2 (φ1 (vµ , nλ )) is in the
same orbit as (vµ0 , nλ∪λ ), where vµ0 is a vector with its (2i − 1)-st portion consisting
of a 1 in the µi -th coordinate and 0-s elsewhere, and its 2i-th portion zero for all
1 ≤ i ≤ k. We can now use the results in Section 3.1 to compute the orbit in V × N
that it lies in.
The type of the vector vµ0 is (α, β), where
α = (µ1 , 0, µ2 , 0, · · · , µk , 0), β = (ν1 , λ1 , ν2 , λ2 , · · · , νk , λk ).
44
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
Using Proposition 3.7 and Lemmas 3.8 − 3.10, the orbit which (vµ0 , nλ∪λ ) lies in
will be given by the bi-partition φ(α, β) = (γ, δ). We have that
γ2i = max({αj | j ≥ 2i} ∪ {λi − βj |j < 2i})
Inspecting the values of {αj |j ≥ 2i}, it is clear that max{αj |j ≥ 2i} = µi+1 .
Inspecting the values of {βj |j < 2i}, it is clear that we have min{βj |j < 2i} =
min(ν1 , λ1 , ν2 , λ2 , . . . , νi ) = νi , so max{λi − βj |j < 2i} = λi − νi = µi . Hence
γ2i = max(µi+1 , µi ) = µi . Next, we have
γ2i−1 = max({αj |j ≥ 2i − 1} ∪ {λi − βj |j < 2i − 1})
Inspecting the values of {αj |j ≥ 2i − 1}, it is clear that max{αj |j ≥ 2i − 1} = µi .
Inspecting the values of {βj |j < 2i − 1}, it is clear that min{βj |j < 2i − 1} =
min(ν1 , λ1 , ν2 , λ2 , . . . , νi−1 , λi−1 ) = νi−1 , so max{λi − βj |j ≤ 2i − 1} = λi − νi−1 .
Hence γ2i−1 = max(µi , λi −νi−1 ) = µi , since νi−1 ≥ νi = λi −µi , so µi ≥ λi −νi−1 .
Thus γ2i−1 = γ2i = µi , proving that γ = µ∪µ, and hence δ = ν ∪ν. This concludes
the proof that φ2 (φ1 (vµ , nλ )) ∈ Oµ∪µ,ν∪ν .
Corollary 3.16. The orbits Oµ,ν and Oµ0 ,ν 0 are distinct if (µ, ν) 6= (µ0 , ν 0 ) are
distinct bi-partitions.
Proof. This follows from the above Lemma, since φ2 (Oµ,ν ) and φ2 (Oµ0 ,ν 0 ) belong
to Oµ∪µ,ν∪ν and Oµ0 ∪µ0 ,ν 0 ∪ν 0 , respectively.
Theorem 3.17. The orbits of Sp(V ) on N 0 are in bijection with the bi-partitions of
n, with the bipartition (µ, ν) corresponding to the orbit Oµ,ν .
Proof. All that remains to be proved is that every point in N lies in one of the orbits
Oµ,ν . For this, we refer the reader to Theorem 1.14 in [6].
3.3. The 2-enhanced nilpotent cone
Here we describe some partial results regarding the orbits of G = GLn (C) on the
2-enhanced nilpotent cone, defined below.
Definition 3.18. The 2-enhanced nilpotent cone is defined to be V × V × N .
The action of G on the 2-enhanced nilpotent cone is clear, since G acts naturally
on V , and acts by conjugation on N . This is the first instance so far in which we
will find that there are infinitely many orbits. The problem of computing the orbits
of GLn (C) on V × V × N is equivalent to the problem of computing the orbits
of CGLn (C) (nλ ) on V × V . To see this, if two elements (v1 , w1 , n1 ), (v2 , w2 , n2 ) ∈
V × V × N are in the same GLn (C)-orbit, then n1 and n2 are nilpotent elements
with the same Jordan form, say corresponding to the partition λ. Thus it suffices to
check when two vectors (v10 , w10 , nλ ) and (v20 , w20 , nλ ) are in the same GLn (C)-orbit.
This happens iff there exists g with g −1 nλ g = nλ such that g(v10 , w10 ) = (v20 , w20 );
i.e. iff (v10 , w10 ), (v20 , w20 ) lie in the same CGLn (C) (nλ ) orbit.
3.3. T HE 2- ENHANCED NILPOTENT CONE
45
Example 3.19. Let λ = (4, 2, 1), and let µ = (2, 2, 1), ν = (2, 0, 0), so that (µ, ν)
is a bi-partition of λ. Pick v ∈ V arbitrary, and g ∈ CGLn (C) (nλ ). In the following
calculation, v should be considered as the union of the 1st portion (v1 , v2 , v3 , v4 ),
the 2nd portion (v5 , v6 ), and the 3rd portion (v7 ). It is clear that the 7 coordinate
functions specifying the vector gv can be expressed as bilinear functions in the
variables λi specifying the coordinates of g, and the variables vi describing the
vector v. For 1 ≤ j ≤ 7, define fj to be the coordinate function describing the j-th
coordinate of the vector gv. Using the below example as a starting point, we will
make an observation about which variables precisely the functions fj depend upon.
λ1 λ2 λ3

λ1 λ2

λ1


g=

λ8








gv = 



∗
∗
v3
v4
v5
v6
v7

∗
∗
λ1 v3 + λ2 v4
λ1 v4
λ8 v3 + λ9 v4 + λ5 v5 + λ6 v6 + λ15 v7
λ8 v4 + λ5 v6
λ10 v4 + λ11 v6 + λ7 v7

λ4 λ12 λ13 λ14
λ3
λ12
λ2
λ1
λ9 λ5 λ6 λ15
λ8
λ5
λ10
λ11 λ7










,v = 






















In the above example, we note that the functions f3 , f4 , f5 , f6 , f7 do not depend on
the variables v1 and v2 (the variables v1 and v2 have been left out to emphasize this
fact). Equivalently, the 3rd, 4th, 5th, 6th and 7th coordinates of gv can be expressed
as linear functions of the 3rd, 4th, 5th, 6th and 7th coordinates of v. This leads to
the question: what is so special about those particular co-ordinates which results in
this phenomena? To answer this question, the co-ordinate v3 and v4 are the last 2
coordinates in the 1st portion of v, v5 and v6 are the last 2 co-ordinates in the 2nd
portion of v, and the coordinate v7 is the last 1 coordinate in the 3rd portion of v;
and the triple (2, 2, 1) specifies the bi-partition (µ, ν) of λ. The generalization of
this phenomenon is stated in the below Proposition.
Proposition 3.20. Let the vector v have portions of sizes (λ1 , · · · , λk ), let (µ, ν) be
a bi-partition of λ, and let Sµ,ν = {µ1 +1, · · · , λ1 , λ1 +µ2 +1, · · · , λ1 +λ2 , · · · , λ1 +
· · · + λk−1 + µk + 1, · · · , λ1 + · · · + λk } (here |Sµ,ν | = |ν|). If g ∈ CGLn (C (nλ ),
then for j ∈ Sµ,ν , the j-th coordinate of the vector gv can be expressed as a sum of
products of coefficients of g and co-ordinates vi as i ranges over Sµ .
Proof. Since j ∈ Sµ,ν , let j = λ1 + · · · + λs−1 + p, for some µs + 1 ≤ p ≤ λs
(here 1 ≤ s ≤ k). We will examine the j-th row of the matrix g. It is sufficient to
prove that the entry in the i-th column and j-th row of the matrix g is 0 if i ∈
/ Sµ,ν .
46
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
The required result will then follow since the j-th coordinate of gv, which is the
product of the row vector formed by the j-th row of g with the column vector v,
will only contain terms involving the co-ordinates vi for i ∈ Sµ,ν . Since i ∈
/ Sµ,ν ,
let i = λ1 + · · · + λt−1 + q, for some 1 ≤ q ≤ µt . If we divide up the matrix g
into k 2 blocks (with the (u, v)-th block having dimension λu × λv ), as described
in Section 2.3.1, the entry in the j-th row and i-th column will be the p-th row and
q-th column in the (s, t)-th block. We consider the two cases where s ≤ t and s > t.
If s ≤ t, we have that λs ≥ λt ; and the co-ordinate in the p-th row and the q-th
column will be zero if p > q. We know that p ≥ µs + 1 > µt ≥ q; here p ≥ µs
and q ≤ µt are from the above paragraph, while µs ≥ µt is because s ≤ t. This
completes the proof when s ≤ t.
If s > t, we have that λs ≤ λt ; and the co-ordinate in the p-th row and the q-th
column will be zero if q − p < λt − λs . Since q ≤ µt , µs + 1 ≤ p, q − p < µt − µs ≤
λt − λs . The last inequality is because νs ≤ νt (since s > t), so λs − µs ≤ λt − µt ,
so µt − νs ≤ λt − λs . This completes the proof when s > t.
Definition 3.21. Given a bi-partition (µ, ν) of λ, let Tµ,ν = {v ∈ V |vi = 0 for all i ∈
Sµ,ν }.
Definition 3.22. Let Xµ,a0 ,a1 ,··· ,aν1 −1 = {(v, w) ∈ V × V |(a0 + a1 nλ + · · · +
aν1 −1 nνλ1 −1 )v − w ∈ Tµ,ν }.
Proposition 3.23. The set Xµ,a0 ,a1 ,··· ,aν1 −1 ⊂ V × V is stable under the action of
CGLn (C) (nλ ).
Example 3.24. Let λ = (4, 2, 1). To illustrate the above concepts, the following is a
description of the sets Tµ,ν and Xµ,a0 ,··· ,aν1 −1 , for the bi-partition µ = (2, 0, 0), ν =
(2, 2, 1). In the below examples, all vectors should be consisted as consisting of
the first portion (the first 4 co-ordinates), the second portion (the following 2 coordinates), and the third portion (the last co-ordinate).

Tµ,ν




= {



∗
∗
0
0
0
0
0
a0 a1


a0 a1


a0 a1




a0
},
a
+
a
n
=
 0

1 λ


a0 a1




a0











a0
3.3. T HE 2- ENHANCED NILPOTENT CONE





If v = 



∗
∗
v3
v4
v5
v6
v7










 , (a0 + a1 nλ )v = 







Hence (v, w) ∈ Xµ,a0 ,a1




iff w = 



∗
∗
a0 v3 + a1 v4
a0 v4
a0 v5 + a1 v6
a0 v6
a0 v7
w1
w2
a0 v3 + a1 v4
a0 v4
a0 v5 + a1 v6
a0 v6
a0 v7
47














 for some w1 , w2 .



Now we return to the proof of Proposition 3.23.
Proof. (of Proposition 3.23)
Suppose (v, w) ∈ Xµ,a0 ,a1 ,··· ,aν1 −1 ; we need to prove that (gv, gw) ∈ Xµ,a0 ,a1 ,··· ,aν1 −1
for all g ∈ CGLn (C) (nλ ). This is equivalent to saying that (a0 + a1 nλ + · · · +
aµ1 −1 nνλ1 −1 )gv − gw ∈ Tµ,ν .
Since g commutes with nλ , g commutes with all powers of nλ , so (a0 + a1 nλ +
· · · + aν1 −1 nνλ1 −1 )gv = g(a0 + a1 nλ + · · · + aν1 −1 nνλ1 −1 )v. Thus we must prove that
g[(a0 + a1 nλ + · · · + aν1 −1 nνλ1 −1 )v − w] ∈ Tµ,ν . Since (v, w) ∈ Xµ,a0 ,a1 ,··· ,aν1 −1 ,
(a0 + a1 nλ + · · · + aµ1 −1 nµλ1 −1 )v − w ∈ Tµ,ν . Thus it is sufficient for us to show
that Tµ,ν is stable under the action of CGLn (C) (nλ ).
The fact that Tµ,ν is stable under the action of CGLn (C) (nλ ) is clear using Proposition
3.20 by the following argument. Suppose v ∈ Tµ,ν , i.e. that vi = 0 for all i ∈ Sµ,ν .
Given g ∈ G, the i-th co-ordinate of gv is a linear combination of co-ordinates
from g and co-ordinates vj for j ∈ Sµ,ν , and since vj = 0 for j ∈ Sµ,ν , the i-th
co-ordinate of gv is 0 for i ∈ Sµ,ν . Hence gv ∈ Tµ,ν , completing the proof.
Thus we have found certain sets which are stable under the action of CGLn (C) (nλ ).
Using a variant of this method, we next describe the orbits of CGLn (C) (nλ ) on V ×V
in the case where λ = (pq ) is a rectangle partition. (Here (pq ) = (p, p, · · · , p),
where p occurs q times.)
Using Theorem 3.3, we can assume that the second vector is in standard form, vµ
for some bi-partition (µ, ν) = (rq , (p − r)q ) of λ. Then the problem is equivalent to classifying the orbits of G(vµ ,nλ ) on V , where G(vµ ,nλ ) is the stabilizer in
G = GLn (C) of the pair (vµ , nλ ). For convenience, let Gr = G(nλ ,vµ ) . Below we
48
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
describe the structure of Gr . Recall from Proposition 1.14 that for any g ∈ Gnλ , g
can be divided into q 2 p × p blocks gi,j , with 1 ≤ i, j ≤ q, with gi,j upper triangular
with entries on the diagonals equal. Call the entries in the first row of the block gi,j ,
g1i,j , g2i,j , · · · , gpi,j . The matrix g must satisfy the condition that the matrix (g1i,j ) is
invertible.
Lemma 3.25. A matrix g of the form above, will lie in Gr if and only if it satisfies
the following additional conditions, for 1 ≤ i ≤ q:
q
q
X
X
i,j
gk = 1 if k = 1,
gki,j = 0 if 2 ≤ k ≤ r
j=1
j=1
If r = 0 there are no additional restrictions on Gr .
Proof. First consider the example p = 4, q = 2, r = 3. For convenience, we use
different notation for the entries of g to that above.
a1 b 1 c 1

a1 b 1

a1




 a3 b 3 c 3

a3 b 3


a3

d 1 a2
c1
b1
a1
d 3 a4
c3
b3
a3

0
b2 c2 d2
a2 b 2 c 2   0

a2 b 2   1

a2   0

b4 c4 d4   0

a4 b 4 c 4 
 0

 1
a4 b 4
0
a4


 
 
 
 
 
=
 
 
 
 
c1 + c2
b1 + b 2
a1 + a2
0
c3 + c4
b3 + b4
a3 + a4
0











Equating the coefficients gives a1 + a2 = a3 + a4 = 1, b1 + b2 = b3 + b4 =
0, c1 + c2 = c3 + c4 = 0. This proves it in the case of this example, and it is clear
how this example will generalize.
Definition 3.26. Let v ∈ V , v = (v1,1 , · · · , v1,p , v2,1 , · · · , v2,p , · · · , vq,1 , · · · , vq,p ).
For 1 ≤ i ≤ p, let vi = (v1,i , v2,i , · · · , vq,i ). Define the height h of v to be the
maximal h such that vh is non-zero (so vh+1 = · · · = vp = 0, vh 6= 0).
It is clear the set of all vectors of a fixed height h will be stable under the action of
Gnλ , and hence Gr . To see this, the set of all vectors of a fixed height h is the set
Tµ1 ,ν1 − Tµ2 ,ν2 , where (µ1 , ν1 ) = (hq , (p − h)q ), (µ2 , ν2 ) = ((h − 1)q , (p − h + 1)q ).
The result then follows from the fact that Tµ1 ,ν1 and Tµ2 ,ν2 are stable under the action
of Gnλ , which is shown in the proof of Proposition 3.23.
Definition 3.27. Let the type of v be a sequence (λ1 , · · · , λj ) defined as follows. Let
j = min(j 0 , r), where j 0 ≤ h is maximal such that vh , vh−1 , · · · , vh+1−j 0 ∈ Cw0 ;
here w0 = (1, 1, · · · , 1) (with q ‘1’s). For 1 ≤ i ≤ j, let λi be the scalar such that
vh+1−i = λi w0 .
Note that in the above definition, j = 0 is allowed, in which case the type will be
empty. If j > 0, then λ1 6= 0, since vh = λ1 w0 and vh 6= 0 by definition of h.
Lemma 3.28. The set of all vectors of height h and type (λ1 , · · · , λj ) is stable under
the action of Gr .
3.3. T HE 2- ENHANCED NILPOTENT CONE
49
Proof. It suffices to show that the set of all vectors with height h and satisfying
vh = λ1 w0 , · · · , vh+1−j = λj w0 is stable under the action of Gr . The reason why
this suffices is as follows: if we have a vector v1 of type (λ1 , · · · , λj ) in the same
orbit as a vector v2 , then if v2 does not have type (λ1 , · · · , λj ) then v2 is forced to
have type (λ1 , · · · , λj , λj+1 , · · · ); but then v1 , which is in the same orbit as v2 , will
now be forced to have type (λ1 , · · · , λj , λj+1 , · · · ), which is a contradiction.
Let us consider the example p = 4, q = 2, r = 3, h = 3, j = 2. Here y = λ1 , z =
λ2 .

∗
b2 c2 d2
a2 b 2 c 2   z

a2 b 2   y

a2   0

b4 c4 d4   ∗

a4 b 4 c 4 
 z

 y
a4 b 4
0
a4
  
∗
∗
(a1 + a2 )z + (b1 + b2 )y   z 
  
(a1 + a2 )y
  y 
  
0
  0 
= 
∗
  ∗ 
 
(a3 + a4 )z + (b3 + b4 )y 
  z 
  y 
(a3 + a4 )y
0
0
a1 b 1 c 1

a1 b 1

a1




 a3 b 3 c 3

a3 b 3


a3







=




d 1 a2
c1
b1
a1
d 3 a4
c3
b3
a3











Above we have used the relations in Gr , a1 + a2 = a3 + a4 = 1, b1 + b2 = b3 + b4 =
0, c1 + c2 = c3 + c4 = 0. It is fairly clear how this example will generalize. The
reason it is necessary to define j so that j ≤ r, is that by Lemma 3.25 there are
precisely r equations among its entries which govern its structure, and if j > r then
these equations will run out, and there is no assurance that the conditon vh−r =
λr+1 w will be preserved by the action of Gr .
Lemma 3.29. If j < r, then any two vectors of height h and type (λ1 , · · · , λj ) are
in the same Gr -orbit.
Proof. Let v be the vector with height h such that v1 = v2 = · · · = vh−j−1 = 0,
vh−j = (0, · · · , 0, 1) (here there are q − 1 0-s), vh+1−j = λj w0 , · · · , vh = λ1 w0 .
Then v clearly has height h and type (λ1 , · · · , λj ). We will prove that any other
vector of height h and type (λ1 , · · · , λj ) lies in the same Gp -orbit as v; this will
suffice, since by Lemma 3.25, Gp ⊂ Gr . Let us consider the example with λ =
(42 ), µ = (32 ), j = 2, h = 4.
50
3. T HE ENHANCED NILPOTENT CONE AND SOME VARIATIONS
a1 b 1 c 1

a1 b 1

a1




 a3 b 3 c 3

a3 b 3


a3

d 1 a2
c1
b1
a1
d 3 a4
c3
b3
a3

b2 c2 d2
0
a2 b 2 c 2   0

a2 b2   λ2

a2   λ1

b4 c4 d4   0

a4 b 4 c 4 
 1
a4 b4   λ2
a4
λ1


 
 
 
 
 
=
 
 
 
 
b2
a2
λ2
λ1
b4
a4
λ2
λ1











Here we have used the relations in Gp , a1 + a2 = 1, b1 + b2 = c1 + c2 = d1 + d2 = 0.
In the vector on the right above, the value of the pair (a2 , a4 ) can be an arbitrary
element of C2 not lying in Cw0 . To see this, the invertible 2 × 2 matrix ( aa13 aa24 ) acts
on C2 × C2 , and can take any pair of vectors forming a rank 2 matrix, to any other
pair of vectors forming a rank 2 matrix. In particular, it can take the pair of vectors
(0, 1), (1, 1) to (a2 , a4 ), (1, 1) if a2 6= a4 . The condition that the matrix stabilizes
(1, 1) imposes the conditions that we have on the matrix, a1 + a2 = a3 + a4 = 1,
and the matrix will take (0, 1) to (a2 , a4 ). The variables b2 and b4 are free to vary,
because the equations relate them to b1 and b3 which do not occur. This then means
the set of vectors on the right ranges over all vectors of height 4 and type (x, y),
proving the statement in the case of the example.
It is clear how this example will generalize. Here we require that j < r; if j = r,
in the above example for instance if r = 2 instead, there will be vectors of type
(λ1 , λ2 ) with a2 = a4 , which is not accounted for in the above argument.
Lemma 3.30. If j = r, then any two vectors of height h and type (λ1 , · · · , λr ) are
in the same Gr -orbit.
Proof. Let v be the vector with height h such that v1 = v2 = · · · = vh−r = 0,
vh+1−r = λr w0 , · · · , vh = λ1 w0 . Then v is clearly of type (λ1 , · · · , λr ), so it suffices to prove that any other vector of type (λ1 , · · · , λr ) lies in the same orbit as v.
Let us consider the example with λ = (42 ), µ = (22 ), h = 4. Here x = λ1 , y = λ2 .
a1 b 1 c 1

a1 b 1

a1




 a3 b 3 c 3

a3 b 3


a3

d1 a2
c1
b1
a1
d3 a4
c3
b3
a3

b2 c 2 d 2
0
a2 b 2 c 2   0

a2 b 2   y

a2   x

b 4 c4 d 4   0

a4 b4 c4 
 0
a4 b 4   y
a4
x


 
 
 
 
 
=
 
 
 
 
(c1 + c2 )y + (d1 + d2 )x
(c1 + c2 )x
y
x
(c3 + c4 )y + (d3 + d4 )x
(c3 + c4 )x
y
x











Here we have used the relations in Gr , a1 + a2 = 1, b1 + b2 = 0. In the above, the
vector on the right can be made to be an arbitrary vector of type (x, y). To see this,
since x 6= 0, and there are no restrictions on c1 , c2 , c3 and c4 , the values of (c1 + c2 )x
3.3. T HE 2- ENHANCED NILPOTENT CONE
51
and (c3 + c4 )x can be made arbitrary. Since there are no restrictions on d1 , d2 , d3
and d4 , the values of (c1 + c2 )y + (d1 + d2 )x and (c3 + c4 )y + (d3 + d4 )x can be
made arbitrary. This proves the statement in the case of the example. It is clear that
this example will generalize.
Combining the previous three Lemmas, we can now state the classification of orbits
of Gr on V .
Theorem 3.31. The orbits of Gr on V are given by the height h of the vector v, and
its type (λ1 , · · · , λj ).
As an example to illustrate the theorem, the following are the orbits of Gr on V in
the case where λ = (32 ), µ = (22 ).




































a
0
0
b
0
0



















 |a 6= b ; 













 

a
λ
0
b
λ
0







∗
a
λ
∗
b
λ


















∗
a
0
∗
b
0



















∗
∗
b
∗
∗
a













 |a 6= b ; 













 

0
0
0
0
0
0
















λ 

0 



0 
 , λ 6= 0
λ 


0 


0




λ2 








 λ1 










 0 

6 0
|a
=
6
b
,λ
=
6
0;
 , λ1 =


 λ2 


















 λ1 
0




∗












 λ2 








 λ1 
|a
=
6
b
,λ
=
6
0;
6 0


 , λ1 =



 ∗ 














 λ2 

λ1












 |a 6= b ; 














 








C HAPTER 4
Springer fibres
In this section, we will define Springer fibres, and study some properties of these
Springer fibres. We describe the irreducible components of the Springer fibres associated to a nilpotent of type λ, and prove that they are in bijection with the set of
standard tableaux of type λ. Here we follow the presentation in Spaltenstein, [8].
4.1. Irreducible components of Springer fibres
Recall that in Section 2.1, we constructed a map π : G ×P n → Oλ . Consider
the case when λ = (n). Using results in Section 2.2, Oλ0 ⊂ Oλ when λ0 ≤ λ.
Since λ0 ≤ (n) for any partition λ0 , it follows that O(n) = N in our case. The space
G×P n can be viewed as {(x, (Vi ))|xVi ⊆ Vi−1 } with dim Vi = µ1 +· · ·+µi (where
µ is the transpose partition of (n)); here since µ = (1n ), µ1 + · · · + µi = i, and the
parabolic subgroup P will become a Borel subgroup B. Hence the space G ×B n
can be viewed as {(x, (0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V ))|x(Vi ) ⊆ Vi−1 }, with
dim Vi = i, and the map π, which is known as the Springer resolution, is projection
onto the first factor.
Definition 4.1. Given x ∈ Oλ , the Springer fibre Fx is defined as follows:
Fx = π −1 (x) = {0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V |x(Vi ) ⊂ Vi−1 }
Let Sλ denote the set of standard tableaux associated to the partition λ; here a “standard tableau” is a way of filling up the Young diagram for the partition λ with the
numbers 1, 2, 3, · · · , n in such a way that the numbers are strictly increasing along
rows and down columns. Given a σ ∈ Sλ , let σi denote the number of the column which i lies in. It is fairly clear that σ can be reconstructed from the numbers
σ1 , σ2 , · · · , σn , since proceeding inductively, the entry i must be inserted into the
highest entry in the σi -th column which has not yet been occupied by the entries
1, 2, · · · , i − 1 (to ensure that entries are increasing down columns). Define a total
ordering on Sλ as follows (known as the “reverse lexicographic" ordering): given
σ, σ 0 ∈ Sλ , suppose that σ < σ 0 if for some i, σi < σi0 but σj = σj0 if i < j ≤ n.
Suppose we have a Jordan basis ei,j for x, with 1 ≤ i ≤ µ1 , 1 ≤ j ≤ λi , and
xei,j = ei,j−1 if j ≥ 1, and 0 otherwise. Consider an example with λ = (4, 32 , 2).
The diagram below shows the partition λ, and the basis elements ei,j can be identified with the boxes of λ in an obvious way. Let Wi = span(e1,1 , e2,1 , · · · , ei,1 ).
In the case of our example, the below diagram shows that ker x = W4 , ker x ∩
im x3 = W1 , ker x ∩ im x2 = W3 , ker x ∩ im x = W4 . This shows that in general
ker x ∩ im xi−1 = Wµi (here µi is the length of the i-th column of λ). In particular,
Wµi is independent of the choice of Jordan basis (although Wi for general i is not).
52
4.1. I RREDUCIBLE COMPONENTS OF S PRINGER FIBRES
53
Lemma 4.2. Consider the flag (V1 /V1 , V2 /V1 , · · · , Vn /V1 ). The nilpotent x clearly
induces a nilpotent transformation x0 of Vn /V1 , and moves each element of this flag
into the previous flag. Then the Jordan type of the nilpotent x0 corresponds to the
partition obtained by deleting the corner at the bottom of the j-th column of λ,
where j is such that V1 ∈ Wµj but V1 ∈
/ Wµj+1 .
Proof. We first consider the example above, λ = (4, 32 , 2), with j = 3. Then
V1 ∈ span (e1,1 , e2,1 , e3,1 ) but V1 ∈
/ span (e1,1 ). The first case is when V1 =
span (ae1,1 + e2,1 ). Then the following set will be a second Jordan basis for x:
{e1,1 , e1,2 , e1,3 , e1,4 }, {ae1,1 +e2,1 , ae1,2 +e2,2 , ae1,3 +e2,3 }, {e3,1 , e3,2 , e3,3 }, {e4,1 , e4,2 }.
Letting the image of ei,j in V /V1 be e0i,j , note that in V /V1 , ae01,1 + e02,1 = 0. It follows that a possible Jordan basis for x on V /V1 is given by {e01,1 , e01,2 , e01,3 , e01,4 },
{ae01,2 + e02,2 , ae01,3 + e02,3 }, {e03,1 , e03,2 , e03,3 }, {e04,1 , e04,2 }. Thus the Jordan type of x0
on V /V1 is (4, 3, 22 ), which is obtained from λ by deleting the last box in the third
column, proving the claim in the first case of this example.
The second case is when V1 = span (ae1,1 + be2,1 + e3,1 ). Then the following set will be a second Jordan basis for x: {e1,1 , e1,2 , e1,3 , e1,4 }, {e2,1 , e2,2 , e2,3 },
{ae1,1 + be2,1 + e3,1 , ae1,2 + be2,2 + e3,2 , ae1,3 + be2,3 + e3,3 }, {e4,1 , e4,2 }. Letting
the image of ei,j in V /V1 be e0i,j , note that in V /V1 , ae01,1 + be02,1 + e03,1 = 0. It follows that one possible Jordan basis for x0 on V /V1 is given by {e01,1 , e01,2 , e01,3 , e01,4 },
{e02,1 , e02,2 , e02,3 }, {ae01,2 + be02,2 + e03,2 , ae01,3 + be02,3 + e03,3 }, {e4,1 , e4,2 }. Thus the
Jordan type of x0 on V /V1 is (4, 3, 22 ), which is obtained from λ by deleting the last
box in the third column, proving the claim in the second case of the example, and
finishing the example.
In the general case, it is straightforward to see how to generalize this construction
and construct the necessary Jordan basis for x0 on V /V1 .
By induction on dim V , we will show that there exists a map θ : Fx → Sλ . Since
by induction this is possible for the smaller flag (V1 /V1 ⊂ V2 /V1 ⊂ · · · ⊂ Vn /V1 ),
we have a way of associating a standard tableau σ 1 of size n − 1 to the action of
x0 on this flag, where σ 1 has shape λ0 , which is obtained by deleting the corner at
the bottom of the j-th row of λ (by the above Lemma). Then, to the action of x on
the flag (V0 ⊂ V1 ⊂ · · · ⊂ Vn ), we will associate the standard tableaux σ of shape
λ, by adding to σ 1 the number n in the corner at the bottom of the j-th row (this
is a standard tableaux since this box is a corner). This gives us the required map
θ : Fx → Sλ . By the construction of θ : Fx → Sλ , given F ∈ Fx , the shape of the
54
4. S PRINGER FIBRES
standard tableau containing the entries 1, · · · , n − s and ignoring all higher entries
in θ(F ), corresponds to the Jordan type of x on V /Vs . For σ ∈ Sλ , let Fσ = θ−1 (σ).
It is clear by induction that Fσ is non-empty. The following theorem is the main
result of this section.
Theorem 4.3. The irreducible components of the Springer fibre Fx are the closures
Fσ , and hence are in bijection with the set of standard tableaux Sλ .
Given a vector space W , let P(W ) denote the set of all one-dimensional subspaces
of W ; then P(W ) is a projective algebraic variety. Let Bi = P(Wi ) − P(Wi−1 )
and B1 = P(W1 ), so that P(ker x) is the disjoint union of Bi with 1 ≤ i ≤ µ1 .
Define a map p : Fx → P(ker x), which takes a flag (V0 ⊂ V1 ⊂ · · · ⊂ Vn )
to V1 . Define Xi = p−1 (Bi ). Then Fx is the disjoint union of X1 , X2 , · · · , Xµ1 .
Let Xi0 = p−1 (Cei,1 ). The variety Xi0 can be considered as a “n − 1 dimensional
version” of Fx , since if x0 is the induced nilpotent in V /Cei,1 , Xi0 can be identified
with the set of all complete flags in this quotient space with x0 moving each flag
into the previous flag. Keeping in mind that Xi is the set of all flags with their 1dimensional component in Bi , and Xi0 is the set of all flags with their 1 dimensional
component being a specific element of Bi , the following Proposition is plausible:
Proposition 4.4. We can find an isomorphism of algebraic varieties f : Xi →
Bi × Xi0 , such that if f (F ) = (bi , F 0 ) for some flags F, F 0 and bi ∈ Bi , then
θ(F ) = θ(F 0 ).
Proof. First note that Wi−1 is isomorphic to Bi = P(Wi ) − P(Wi−1 ). To see
this, Wi = Wi−1 ⊕ Cei,1 , so every one-dimensional subspace in Bi can be written as C(ei,1 + wi−1 ) for some wi−1 ∈ Wi−1 , uniquely. It follows that the map
wi−1 → C(ei,1 + wi−1 ) is an isomorphism from Wi−1 to Bi . This means it suffices
to prove the Proposition with Bi replaced by Wi−1 .
P
Suppose wP∈ Wi−1 , so w = 1≤s≤i−1 as es,1 for some constants as . For j ≤ λi−1 ,
let wj =
1≤s≤i−1 as es,j . Note that xwj = wj−1 (here define w0 = 0 for convenience). Define a linear map gw : V → V by letting gw (ei0 ,j ) = ei0 ,j if i0 6= i,
and gw ei,j = ei,j + wj . It is clear that gw is invertible, since the matrix of gw with
respect to the Jordan basis for x is unitriangular. We next check that gw x = xgw .
If i0 6= i, then gw xei0 ,j = gw ei0 ,j−1 = ei0 ,j−1 , while xgw ei0 ,j = xei0 ,j = ei0 ,j−1 , so
gw xei0 ,j = xgw ei0 ,j (for all i0 , let ei0 ,0 = 0 to deal with the boundary case). Next,
gw xei,j = gw ei,j−1 = ei,j−1 + wj−1 , while xgw ei,j = x(ei,j + wj ) = ei,j−1 + wj−1
since xwj = wj−1 , so gw xei,j = xgw ei,j .
Consider the map g : Wi−1 × Xi0 → Xi defined by g(w, F ) = gw (F ). We first need
to show that g is a well-defined map, i.e. that gw (F ) lies in Xi . If F = (V0 ⊂ V1 ⊂
· · · ⊂ Vn ), then gw (F ) = (V0 ⊂ gw (V1 ) ⊂ · · · ⊂ gw (Vn−1 ) ⊂ Vn ), so xgw (Vi ) =
gw x(Vi ) ⊂ gw (Vi−1 ), which means gw (F ) ∈ Fx . If F = (V0 ⊂ V1 ⊂ · · · ⊂ Vn ) lies
in Xi0 , then p(gw (F )) = gw (V1 ) = gw (Cei,1 ) = C(ei,1 + w1 ) = C(ei,1 + w). Since
C(ei,1 + w) ∈ Bi , it follows that gw (F ) ∈ Xi ; and hence g is a well defined map.
To check that g is surjective, suppose F 0 = (V00 ⊂ V10 ⊂ · · · ⊂ Vn0 ) is some flag
in Xi . Suppose V10 = C(ei,1 + w). Since gw (C(ei,1 )) = C(ei,1 + w) we have
4.1. I RREDUCIBLE COMPONENTS OF S PRINGER FIBRES
55
that gw−1 (V10 ) = C(ei,1 ). Since xgw = gw x, xgw−1 = gw−1 x, and so xgw−1 (Vi0 ) =
0
gw−1 x(Vi0 ) = gw−1 Vi−1
, which means gw−1 (F 0 ) ∈ Xi0 . It then follows that F 0 =
g(w, gw−1 (F 0 )), proving that g is surjective. To prove that g is injective, suppose we
have that gw (F ) = F 0 for some w ∈ Wi−1 , F ∈ Xi0 , F 0 ∈ Xi . If F 0 = (V00 ⊂
V10 ⊂ · · · ⊂ Vn0 ), F = (V0 ⊂ V1 ⊂ · · · ⊂ Vn ), then gw (V1 ) = gw (C(ei,1 )) =
C(ei,1 + w1 ) = V10 . The condition C(ei,1 + w1 ) = C(ei,1 + w) = V10 determines
w uniquely from F 0 . Then gw (F ) = F 0 , so F = gw−1 (F 0 ), which determines F
uniquely from F 0 . This shows that g is injective. If we define f to be the inverse of
g, f will an isomorphism between Xi and Wi−1 × Xi0 .
To show the remainder of the proposition, it is enough to show that θ(F ) = θ(gw (F )),
for F = (V0 ⊂ V1 ⊂ · · · ⊂ Vn ) ∈ Xi0 . Recall that the shape of the standard tableau containing the entries 1, · · · , n − s and ignoring all higher entries in
θ(F ), corresponds to the Jordan type of x on V /Vs . It thus suffices to prove that
the Jordan type of x on V /Vs is the same as the Jordan type of x on V /gw (Vs )
(here x really means the induced transformation on the quotient). To see this, suppose am,n + gw (Vs ) is a Jordan basis for x on V /gw (Vs ). Then consider the basis
gw−1 am,n + Vs for V /Vs . Clearly xVs ⊂ Vs , and gw−1 xgw = x, so x(gw−1 am,n + Vs ) =
(gw−1 xgw )(gw−1 am,n ) + Vs = gw−1 xam,n + Vs = gw−1 am,n−1 + Vs ; hence gw−1 am,n + Vs
is a Jordan basis for x on V /Vs . This shows that θ(F ) = θ(gw (F )), as required.
Proposition 4.5.
a) The set Fσ is a locally closed subset of Fx , and thus we
can give Fσ the structure of an algebraic variety.
P
b) The variety Fσ is irreducible, and has dimension i µi (µ2i −1) .
Proof. (a) It suffices to prove that the set ∪σ0 ≥σ Fσ0 is closed for each σ. To see
this, since the ordering > is a total ordering, we can find a standard tableaux
σ + which comes immediately before σ, so that σ 0 > σ means σ 0 ≥ σ + . Then
Fσ = ∪σ0 ≥σ Fσ0 − ∪σ0 ≥σ+ Fσ0 . It would then follow that Fσ is an intersection of an
open set and a closed set, and is hence locally closed.
Let σn = j. Make the following definitions:
[
Fσ0
A=
0 ≥j+1
σn
Z=
[
Fσ 0
σ 0 ≥σ
B=
[
Fσ 0
0 ≥j
σn
Then by the way the ordering on Sλ is defined, we have A ⊂ Z ⊂ B. It is clear
that B = p−1 (P(Wµj )), and hence B is closed in Fx . Similarly A is closed in Fx .
Since B is closed in Fx , to show that Z is closed in Fx it suffices to prove that Z is
closed in B. Equivalently, it is sufficient to prove that B − Z is open in B. Since A
is closed, A is closed in B, so B − A is open in B. Thus to show B − Z is open in
56
4. S PRINGER FIBRES
B, it will suffice to show that B − Z is open in B − A, since B − A is open in B.
The spaces B − Z and B − A are the following:
[
Fσ0
B−Z =
0 =j
σ 0 <σ,σn
B−A=
[
Fσ 0
0 =j
σn
Pick an element F in B − Z; so suppose F ∈ Fτ for some standard tableaux τ with
τ < σ and τn = j. We will construct an open neighbourhood of F in B − A.
Consider the isomorphism Xµj ∼
= Bµj × Xµ0 j defined in Proposition 4.4, and view
Xµ0 j as Fx0 0 , an “n − 1 dimensional version” of Fx , corresponding to the action of
x0 on the flag variety F 0 consisting of quotient spaces (the Jordan type of x0 is the
partition obtained by deleting the last box in the j-th column of λ). If the flag in Xµj
lies in Fσ0 for some σ 0 < σ, σn0 = j, then by the property stated in Proposition 4.4,
the flag in Xµ0 j will also lie in Fσ0 , and hence will lie in Fσ0 01 when Xµ0 j is considered
as Fx0 0 (here σ 01 is obtained by deleting the n at the end of the j-th column of σ 0 ).
We thus have the following:
Xµ j ∩
Under the isomorphism Xµj ∼
= Bµj × Xµ0 j ,
[
[
Fσ0 corresponds to Bµj × (
Fσ0 01 )
0 =j
σ 0 <σ,σn
σ 01 <σ 1
S
By induction, we may assume that σ01 <σ1 Fσ0 01 is open in Xµ0 j = Fx0 0 . This will
S
S
imply that Bµj × σ01 <σ1 Fσ0 01 is open in Bµj × Xµ0 j , so Xµj ∩ σ0 <σ,σn0 =j Fσ0 is
open in Xµj .
S
Note next that Xµj+1 ∪ · · · ∪ Xµj = σn0 =j Fσ0 = B − A by considering the possibilities for the first subspace in the flag. It follows that Xµj+1 ∪ · · · Xµj −1 =
p−1 (P(Wµj −1 )) ∩ (B − A) is closed in B − A, so Xµj is open in B − A.
S
This now means that Xµj ∩ σ0 <σ,σn0 =j Fσ0 is open in Xµj , which is in turn open in
B − A; hence it is open in B − A, and it is also clearly contained in B − Z. We will
be done if we can show that it contains F (since this will then be the required open
neighbourhood of F in B − A). To show that it contains F , we are required to show
that F is contained in Xµj . F is chosen to be in some Fτ for some τ with τn = j; so
the first subspace in the flag lies in Wµj but not in Wµj+1 . We require that the first
subspace in the flag F lies in Wµj but not in Wµj −1 . To do this, we have freedom in
choosing the Jordan basis for x, so we can simply choose the Jordan basis for x so
that the first subspace in the flag lies in Wµj but not in Wµj −1 .
4.1. I RREDUCIBLE COMPONENTS OF S PRINGER FIBRES
57
(b) Continue to denote σn by j. Consider the morphism p : Fσ → P(Wµj ) −
P(Wµj+1 ). We have seen that the fibre of this morphism over any point can be identified with Fσ0 1 . This means that Fσ is a fibre bundle over P(Wµj ) − P(Wµj+1 ) with
fibres isomorphic to Fσ0 1 . We can assume by induction that Fσ0 1 is an irreducible variety. Since P(Wµj ) − P(Wµj+1 ) is an open subset of the irreducible variety P(Wµj ),
it is an irreducible variety. Since a fibre bundle over an irreducible variety with irreducible fibres is irreducible, it follows that Fσ is irreducible.
P
Now assume by induction that dim Fσ0 1 = ( i6=j 21 µi (µi − 1)) + 12 (µj − 1)(µj − 2).
The base case for this induction is when σ 1 is the standard tableaux with one box
(containing 1). In this case, the Springer fibre corresponds to a point (the flag 0 =
V0 ⊂ V1 = V ), which has dimension 0, as required. For the induction step, clearly
dim(P(Wµj ) − P(Wµj+1 )) = µj − 1. Since Fσ is a fibre bundle over P(Wµj ) −
P(Wµj+1 ) with fibres isomorphic to Fσ0 1 we have:
dim Fσ = dim Fσ0 1 + dim(P(Wµj ) − P(Wµj+1 ))
X1
1
=(
µi (µi − 1)) + (µj − 1)(µj − 2) + µj − 1
2
2
i6=j
X1
1
=(
µi (µi − 1)) + (µj − 1)(µj )
2
2
i6=j
X1
=
µi (µi − 1)
2
i
This now proves the required dimension formula for Fσ .
Now we are in a position to prove Theorem 4.3.
Proof. It is clear that Fx = ∪Fσ . Since the closure of an irreducible variety is
irreducible, (b) of Proposition 4.5 gives us that Fσ is irreducible. Since (b) of
Proposition 4.5 tells us that the dimensions of the varieties Fσ are all equal, it is
not possible for any one of these varieties to contain another (since a proper closed
subvariety of an irreducible variety has strictly smaller dimension). Finally, it is not
possible for Fσ = Fσ0 for distinct σ, σ 0 , since then Fσ and Fσ0 would be disjoint
nonempty open subsets (since Fσ is locally closed, hence open in its closure), and
it is not possible for an irreducible variety to have disjoint nonempty open subsets.
This now completes the proof.
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