An exotic Springer correspondence for symplectic groups

0.25 MB·English

by  Syu Kato

An exotic Springer correspondence for symplectic
groups∗
8 Syu Kato †‡
0
0 February 2, 2008
2

n
a
J
Abstract

7
2 Let G be a complex symplectic group. In [K1], we singled out the
nilpotent cone N of some reducible G-module, which we call the (1-)

] exotic nilpotent cone. In this paper, we study the set of G-orbits of
T
the variety N. It turns out that the variety N gives a variant of the
R Springer correspondence for Weyl groups of type C, but shares a similar
. flavor with that of type A case. (I.e. there appears no non-trivial local
h
t systemandthecorrespondenceisbijective.) Asanapplication,wepresent
a
onesufficientconditionforthebijectivityofourexoticDeligne-Langlands
m
correspondence [K1].
[

2 1 Main results
v
8
7 Let G = Sp(2n,C) be a complex symplectic group. Let B and T be its Borel
4 subgroup and a maximal torus of B, respectively. We denote by X∗(T) the
7 character group of T. Let R be the root system of (G,T) and let R+ be its
0 positive part defined by B. Let W := N (T)/T be the Weyl group of (G,T).
G
6 For a group or an algebra H, we put IrrH the set of isomorphism classes of
0
simple H-modules. We embed R and R+ into a n-dimensional Euclid space
/
h E=⊕ Cǫ as:
i i
t
a
m R+ ={ǫ ±ǫ } ∪{2ǫ }⊂{±ǫ ±ǫ }∪{±2ǫ }=R⊂E.
i j i<j i i j i
:
v WedefineV :=C2nandV :=(∧2V )/C. TheserepresentationshaveB-highest
1 2 1
Xi weights ǫ1 and ǫ1+ǫ2, respectively. We put V :=V1⊕V2. For a G-module V,
we define its weight λ-part (with respect to T) as V[λ]. The positive part V+
r
a of V is defined as
V+ := V[λ].
λ∈Q≥M0R+−{0}
∗Thisisarevisedversionofapreprintentitled“Onthegeometryofexoticnilpotentcones”,
available through math.RT/0607478v1. We changed the title in order to stress our central
acheivement.
†Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Meguro Komaba
153-8914, Japan.
‡The author was supported by the JSPS Research Fellowship for young scientists during
thisresearch.
1
We define
F :=G×BV+ ⊂G×BV∼=G/B×V.
Composing with the second projection, we have a map
µ:F −→V.
We denote the image of µ by N. This is the G-variety which we refer as the
exotic nilpotent cone1. By abuse of notation, we may denote the map F → N
also by µ. Basic properties of N are:
Theorem A (Geometric properties of N, cf. [K1] 1.2). We have:
1. The G×(C×)2-action on V descends to N;
2. The defining ideal of N is (C[V]G)C[V]=(C[V ]G)C[V];
+ 2 +
3. The variety N is normal;
4. The map µ is a birational projective morphism;
5. The variety Z :=F × F is a union of equi-dimensional irreducible vari-
N
eties.
This result itself follows rather easily from the results and technique de-
veloped by Kempf, Schwarz, Hesselink, and Dadok-Kac. Here we repeat the
statement for the convenience of readers.
Let P (n) denote the set ofpairs (λ,µ) of partitions suchthat |λ|+|µ|=n.
2
It is well-known that the set P (n) parametrizes IrrW (cf. Macdonald [Mc95] I
2
Appendix B).
Theorem B (= Corollary 6.6). The set of G-orbits of N is in one-to-one
correspondence with P (n).
2
The proof is divided into three steps: First, we introduce a set MP(n) (cf.
2.2) and construct a bijection with P (n). This is done via an intermediate set
2
SP(n) (cf. 3.3). Then, we construct a map MP(n) 7→ G\N and show it is
surjective. This surjectivity statement is an enhancement of a correspondence
given by Sekiguchi-Ohta [Se84, Oh86]. Finally, we deduce the injectivity as a
byproduct of Theorem C explained in the below.
Let O be a G-orbit in N. Let X ∈ O. By means of the Ginzburg theory
[CG97, Gi97] and [K1] 2.13, we equip the vector space
M :=H (µ−1(X),C)
O codimO
withanactionofW. (ThesymbolH alwaysrepresentstheBorel-Moorehomol-
•
ogy group instead of the usual homology group.) Here the RHS is independent
of the choice of X ∈O (as W-modules).
Theorem C (= Theorem6.5). The assignment O7→M establishes a one-to-
O
one correspondence between the set of G-orbits of N and IrrW.
1Fromtheview-pointofinvarianttheory,ourvarietyNisnothingspecial. (Seeeg. Schwarz
[Sc78]andDadok-Kac[DK85])Theauthorconsidersitasan“exoticnilpotentconeoftheLie
algebrag”sinceitsharesmanyrepresentation-theoreticfeatureswiththenilpotentconeofg
inan“exotic” fashion. TheoremCofthispaperrepresentsoneofsuchfeatures.
2
This is an enhancement of Grinberg’s generalized Springer correspondence
[Gr98]. Apart from the Ginzburg theory and a weak form of Theorem B, the
proof depends on two facts: One is the numerical identity |MP(n)| = |IrrW|,
andtheotheristheconnectednessofthestabilizerofG-orbitsofN. Theformer
is a numerical version of a combinatorial bijection P (n) ↔MP(n) defined as
2
above. The latter follows by a result of Igusa (cf. Theorem 6.1).
Foreachλ∈X∗(T)\{0},wefixabasiselementv[λ]∈V[λ]. Then,themost
simple non-trivial example of Theorem C is:
Example D (n = 2). Let G = Sp(4,C) and let N be the nilpotent cone of
sp(4,C). Let x[α] ∈ sp(4,C) be a non-zero T-root vector corresponding to α.
We put α :=ǫ −ǫ . We have:
1 1 2
IrrW G\N P (2) dim G\N
2
sign {0} ( ,∅) 1 {0}
Ssign v[ǫ ] (∅, ) 1 x[2ǫ ]
1 1
Lsign v[α ] ( ,∅) 1 x[α ]
1 1
regular v[α ]+v[ǫ ] ( , ) 2 x[α ]
1 1 1
triv v[α ]+v[ǫ ] (∅, ) 1 x[α ]+x[2ǫ ]
1 2 1 2
Here the sets G\N and G\N specify the corresponding orbits in our result
and the Springer correspondence, respectively. If we denote the usual Springer
resolution by µ , the above table implies:
S
H (µ−1(x[α ]))∼=Lsign⊕regular
2 S 1
and
H (µ−1(v[α ]))∼=Lsign,H (µ−1(v[α ]+v[ǫ ]))∼=regular.
4 1 2 1 1
We set Z :=F × F. For a=(s,q ,q )∈G×(C×)2, we define Na and Za
N 1 2
to be the subvarieties of N and Z consisting of a-fixed points, respectively. We
set G(s):=Z (s). With an aid of Theorem C, we prove:
G
Theorem E (=Theorem7.9). Let a=(s,q ,q )∈G×(C×)2 be a semi-simple
1 2
element such that q is not a root of unity. Then, there exists a one-to-one
2
correspondence
G(s)\Na ↔IrrH (Za),
•
where H (Za) acquires an associative algebra structureby means of convolution
•
operations (cf. §4 or [CG97] §2).
In [K1], we study the representation theory of the affine Hecke algebra H
of type C . It is an associative algebra with three independent parameters
n
(q ,q ,q ). Its quotient by the two-sidedideal generatedby (q +q ) is isomor-
0 1 2 0 1
phic to the extended affine Hecke algebras of type B . In the language of [K1],
n
Theorem E implies the following:
Theorem F (= Corollary 7.10). For the extended affine Hecke algebra of type
B with two-parameters (−q ,q ,q ), the regularity condition of parameters
n 1 1 2
holds automatically unless −q2 6=q±m for 0≤m<n or ql 6=1 for 1≤l <2n.
1 2 2
A large part of the proofs of the above two theorems are borrowed from
Lusztig [Lu95] with severalminor modifications. Most notably, we do not need
3
the odd-term vanishing condition of the homology of fibers by using Borho-
MacPherson’s argument and localization technique. We do not know whether
the odd-term vanishing result holds in this setting. (But it is expected to have
a close connection to the project of Achar-Henderson [AH07].)
Inordertodeepenthesubjectsof[K1]andthispaper,weneedtodetermine
thecorrespondenceofTheoremCexplicitly. Thisisdoneinasubsequentpaper
[K2], where we employ a different method as well as the results of this paper.
2 Preparatory materials
In this section, we collect some extra notation which we use throughout this
paper. For each X ∈V, we write
X :=v[0]+ X(λ)v[λ],
λ∈XX∗(T)
where v[0]∈V[0]. We define the support of X as
kXk:={i∈[1,n]:X(±ǫ )6=0 or X(±ǫ ±ǫ )6=0 for some sign and j ∈[1,n]}.
i i j
The following definition of normalform is a slight enhancement of the good
basis of Ohta [Oh86]. (See also Igusa [Ig73].)
Definition 2.1 (Normalforms). A blockJ(λ) oflengthλ andpositioni is one
of the following vectors in V+:
λ−1
v(j)(λ) :=(1−δ )v[ǫ ]+ v[α ],
i j,0 i+j i+k
k=1
X
where α := ǫ −ǫ , 0≤ j ≤ λ is an integer, and δ is Kronecker’s delta. It
i i i+1 j,0
is clear that kv(j)(λ) k=[i+1,λ+i] or ∅. A normal form of V is a sum
i
n−1
v = v(ji)(λ ) ∈V+
i i
i=0
X
such that kv(ji)(λi)ik∩kv(ji′)(λi′)i′k=∅ if i6=i′.
Definition 2.2 (Marked partitions). A marked partition ~λ = (λ,a) is a par-
tition λ = (λ ≥ λ ≥ ...) of n, together with a sequence a = (a ,a ,...) of
1 2 1 2
integers such that:
1. 0≤a ≤λ for each k;
k k
2. a =0 if λ =λ ;
k k+1 k
3. λ −λ >a −a >0 if p<q and a 6=06=a .
p q p q p q
We denote the set of marked partitions by MP(n). For (λ,a) ∈ MP(n), we
put
J(λ,a):= v(ap)(λp)λ<p,
p≥1
X
where λ< = λ . It is clearly a normal form.
p q<p q
P
4
For a G-variety X, we denote the set of G-orbits of X by O .
X
Remark 2.3 (Ohta [Oh86] §2). Let 0 denote the sequence (0,0,...). Then, the
assignment
P(n)∋λ7→G.J(λ,0)∈O
N∩V2
gives a one-to-one correspondence.
3 Combinatorial correspondence
We retain the setting of the previous section. In this section, we present com-
binatorics which is needed in the sequel.
Definition 3.1 (Bi-partition). Let P (n) denote the set of pairs (λ,µ) of par-
2
titions such that |λ|+|µ|=n.
It is known that IrrW is parametrized by P (n) (cf. [Mc95] I Appendix B).
2
Theorem 3.2. The set P (n) is in one-to-one correspondence with MP(n)
2
Corollary 3.3. The set MP(n) is in one-to-one correspondence with IrrW. 2
The restofthis sectionis devotedtothe constructionofabijectionbetween
MP(n) and P (n). For two partitions λ,µ, we define their sum λ⊙µ as the
2
partition {λ +µ } .
p p p≥1
Definition 3.4. Let I ⊂ Z. A sub-segment [r,s] ⊂ I is called a component
if r −1 6∈ I 6∋ s+1. A pair (λ,I) is called a segmented partition of n if λ
(=(λ ≥λ ...))isapartitionofnandI ⊂[1,λ ](possiblyempty)decomposes
1 2 1
into components
I =[i ,λ ]∪[i ,λ ]∪···
1 j1 2 j2
such that i < λ +1 < i holds for every possible p. We denote the set of
p jp p+1
segmented partitions by SP(n).
Example 3.5 (n=3). We have the following correspondences:
λ (3) (3) (3) (3) (1, 1, 1) (1, 1, 1)
MP(3) 0 (1,0) (2,0) (3,0) 0 (0,0,1)
SP(3) ∅ [1,3] [2,3] [3,3] ∅ [1,1]
P (3) {(3),∅} {∅,(3)} {(1),(2)} {(2),(1)} {(1,1,1),∅} {∅,(1,1,1)}
2
λ (2, 1) (2, 1) (2, 1) (2, 1)
MP(3) 0 (1,0) (2,0) (0,1)
SP(3) ∅ [1,2] [2,2] [1,1]
P (3) {(2,1),∅} {∅,(2,1)} {(1,1),(1)} {(1),(1,1)}
2
Lemma3.6. Thereexistsaone-to-onecorrespondencebetweenthesetsMP(n)
and SP(n).
Proof. Let λ be a partition of n. Let x = (x ,x ,...) be a strictly decreasing
1 2
finite positive integer sequence. We define
MP(λ,x):={(λ,a)∈MP(n):a 6=0⇔p∈x}.
p
5
NoticethatMP(λ,x)=∅ifxcontainspsuchthatλ =λ . LetSP(λ,x)be
p p+1
thesetofsegmentedpartitionssuchthat{λp}p∈x isthesetof(right)boundaries
of components. In other word, we define
SP(λ,x):={(λ,I)∈SP(n):x={p∈Z:λ ∈I 6∋λ +1,λ 6=λ }}.
p p p+1 p
We have
MP(n)= MP(λ,x) and SP(n)= SP(λ,x).
λ∈PG(n),x λ∈PG(n),x
Claim 1. We have #MP(λ,x)=λ #x(λ −λ −1).
x1 i=2 xi xi+1
Proof. We count the number of possibQle markings a = (a ,a ,...). Since we
1 2
haveap =0if p6∈x, werestrictour attensionto the set{ap}p∈x. We countthe
possible choice of a with the knowledge of a ,a ,...,a . If i = 1, then
xi x1 x2 xi−1
the possible choice is 1 ≤ a ≤ λ since a 6= 0. If i > 1, then the possible
x1 x1 x1
choice is
max{a :j <i}<a <min{λ −λ +a :j <i}.
xj xi xi xj xj
By definition, both the maximal and the minimal are attained at j = i−1.
Therefore,the number of possible choice of a is (λ −λ −1) (independent
i xi−1 xi
of {a } ). Therefore, multiplying these yields the result.
j j>i
Claim 2. We have #SP(λ,x)=λ #x(λ −λ −1).
x1 i=2 xi xi+1
Proof. By definition, we have a nonQ-empty segment for each {λq}q∈x. By
definition, the number of choice of segments ended at λ is λ (i = 1) or
xi xi
(λ −λ −1) (i≥2). Multiplying these yields the result.
xi xi+1
By the comparison of two claims, we deduce
#MP(λ,x)=#SP(λ,x).
Hence, it suffices to construct an injective assignment SP(λ,x) → MP(λ,x).
Fix (λ,I) ∈ SP(λ,x). We construct an integer sequence a(λ,I) = (a ,a ,...)
1 2
inductively as follows:
1. If p>n, then we put a :=0;
p
2. Assume that {a } is already defined. Then, we put
q q>p
min{j,j−λ +a −1:q >p,a 6=0} (if [j,λ ]⊂I is a component)
q q q p
a :=
p
(0 (otherwise)
We have
a ≥(λ +2)−λ +a −1>a if p<q and a 6=06=a .
p q q q q p q
Moreover,we deduce
a ≤j−λ +a −1<λ −λ +a
p q q p q q
6
if [j,λ ]⊂I is a component. In particular,
p
λ −a >λ −a if p<q and a 6=06=a .
p p q q p q
Hence, we deduce (λ,a(λ,I)) ∈ MP(λ,x). Thus, it suffices to show a(λ,I) =
a(λ,I′)only ifI =I′. Assume thatI 6=I′. Then, there existsmaximalnumber
q such that 1) a = a′ for every p > q and 2) [j,λ ] is a component of I and
p p q
not a component of I′. Then, we have necessarily a 6= a′. Therefore, the
q q
assignment
SP(n)∋(λ,x)7→(λ,a(λ,I))∈MP(λ,x)
determines an injective map as desired.
Lemma 3.7. There exists a one-to-one correspondence between the sets P (n)
2
and SP(n).
Proof. First, we construct a map P (n) → SP(n). Fix (λ,µ) ∈ P (n). We
2 2
define j+ :=λ +µ and j :=λ +µ +1. Then, we define
p p p p p p+1
I(λ,µ):= [j ,j+].
p p
p≥1
[
Hereweunderstand[j ,j+]=∅whenj+ <j . Weregard[a,b]∪[b+1,c]=[a,c]
p p p p
as segments. We have j+ ∈λ⊙µ. Thus, we conclude (λ⊙µ,I(λ,µ))∈SP(n).
p
Second, we construct a map SP(n)→P (n). Fix (λ,I)∈SP(n). Put
2
µ (λ,I):=#(I ∩[1,λ ]),γ(λ,I):=λ −µ (λ,I).
p p p p
It is clearthat (µ(λ,I),γ(λ,I))∈P (n). These two maps are mutually inverse,
2
which implies the result.
Theorem 3.2 immediately follows from the combination of Lemma 3.7 and
Lemma 3.6.
4 Realization of Weyl groups
We retain the setting of §2. Let R(H) be the complexified representation ring
of an algebraic group H. We have natural identification
Z ∼={(g B,g B,X)∈(G/B)2×V:X ∈g V+∩g V+}.
1 2 1 2
We have an inclusion Z ⊂F ×F. Let p :F3 →F2 be the (i,j)-th projection
ij
(1 ≤ i < j ≤ 3). Since p is proper when restricted to p−1(Z)∩p−1(Z), we
ij 12 23
have a well-defined convolution map
⋆:K(Z)⊗K(Z)∋(F,G)7→ (−1)i[Ri(p ) (p∗ F ⊗Lp∗ G)]∈K(Z).
13 ∗ 12 23
i≥0
X
Theorem 4.1 (Ginzburg). The group K(Z) becomes an algebra via the convo-
lution action. 2
7
Let t:=Lie T. We define
C[t] :=C[t]/ C[t]W ,
0 +
where C[t]W :=C[t]W ∩t∗S(t∗). By the n(cid:10)atural (cid:11)W-action on t, the space C[t]
+ 0
admits a W-action. Hence, we have their amalgamated product
W :=C[W]⊗C[t] ,
0
whose multiplication is given by (w ,f)(w ,g):=(w w ,fw (g)).
1 2 1 2 1
For Z-module A, we set A :=C⊗ A.
C Z
Theorem 4.2. We have an isomorphism K(Z)C ∼=W as algebras.
Proof. Choose an element (1,1,−1,1) ∈ G×(C×)3. It acts on F := G×B
2
(V⊕2 ⊕V )+ with its fixed part isomorphic to F ∼= G×B (V+ ⊕0⊕V+) ⊂
1 2 1 2
F . By [K1] Corollary 2.13 and Remark 2.2 3), we conclude that K(Z) is
2 C
isomorphic to the specialization of the three-parameter Hecke algebra of type
C(1) at q = −q = q = 1 and s = 1 ∈ G. In particular, K(Z) is isomorphic
n 0 1 2 C
to the quotient of the group ring of the affine Weyl group W ⋉X∗(T) by the
ideal generated by the maximal ideal m ⊂ R(G) corresponding to 1 ∈ G. (We
regard X∗(T)∼=Q∨, where Q∨ is the coroot lattice of R.) Here we have
R(T)/mR(T)=R(T)/h[V]−(dimV)[C]:V ∈RepGi∼=C[t]W
0
as W-modules. Since K(Z) =C[W]⊗R(T)/mR(T), the result follows.
C
Corollary 4.3. We have a surjective map W −→→ C[W]. In particular, we
have an inclusion
IrrW ⊂{simple W-modules},
where the RHS is the set of isomorphism classes.
Proof. We retain the setting of the proof of Theorem 4.2. The maximal ideal
m ⊂C[T]correspondingto1∈T isclearlyW-invariant. WehaveC[T]m⊂m .
0 0
It follows that W/Wm ∼=C[W] as desired.
0
Remark 4.4. Asisshownlater(ProofofTheorem6.5),theinclusionofCorollary
4.3 is in fact an equality. We can also deduce it directly from the structure of
W.
5 Rough classification of orbits
We assume the setting of the previous sections. The goal of this section is to
prove:
Theorem 5.1. The map
J:MP(n)∋~λ7→J(~λ)∈G\N
is surjective.
Proposition 5.2 ([K1] 1.9). Every element of N is G-conjugate to a normal
form. 2
8
Remark 5.3. Theorem 5.1 does not follow from Proposition 5.2 immediately
since ImJ does not exhaust the set of normal forms modulo the N (T)-action.
G
LetG ⊂G be the subgroupsuch that1) T ⊂G and 2)the rootsystemof
0 0
(G ,T)is{±(ǫ −ǫ )} ⊂R. We haveG ∼=GL(n,C). LetX ∈LieG ⊂
0 i j i<j 0 ǫi−ǫj 0
gbeanon-zerorootvectorofT-weightǫ −ǫ withstandardnormalization. We
i j
put v :=exp(X )∈G .
i,j ǫi−ǫj 0
Proof of Theorem 5.1. Let λ = (λ,a), where λ is a partition of n and a =
(a ,a ,...) is an integer sequence such that 0 ≤ a ≤ λ for each p. (We put
1 2 p p
~λ=λ if λ is a marked partition.) By Proposition 5.2, we assume
J =J(λ):= J(ai)(λi)λ< =J1+J(λ0)∈V+
i
i
X
withJ ∈V andλ =(λ,0). Leti,j betwodistinctintegerssuchthatλ ≥λ .
1 1 0 i j
Claim 3. There exists elements g+,g− ∈Stab (J(λ )) which induces unipo-
i,j i,j G0 0
tent transforms on V so that
1
v[ǫ ]+v[ǫ ] (0<k ≤λ )
g+v[ǫ ]= λ<i +k λ<j +k j (5.1)
i,j λ<j +k (v[ǫλ<+k] (otherwise)
j
and
v[ǫ ]+v[ǫ ] (0≤k <λ )
gi−,jv[ǫλ<i+1−k]=(v[ǫλλ<i<+1−−kk] λ<j+1−k (otherwisej) . (5.2)
i+1
Proof. It is straight-forwardto see that g± satisfies (5.1) and (5.2). Define
ij
g+ := v ,g− := v ∈G
ij λ<+k,λ<+k ij λ< −k,λ< −k 0
i j j+1 i+1
1≤Yk≤λj 0≤Yk<λj
fixes J(λ ). By a weight comparison, these elements do not depend on the
0
order of the product. The presentation of the transformations on V follow
1
immediately from this expression. We have
g+J(λ )=J(λ )+ X v[ǫ −ǫ ]
i,j 0 0 ǫλ<i +k−ǫλ<j+k λ<j +k λ<j+k+1
1≤Xk<λj
+ X v[ǫ −ǫ ]=J(λ ).
ǫλ<i +k+1−ǫλ<j+k+1 λ<i +k λ<i +k+1 0
1≤Xk<λj
(Remember that the G -action on LieG is the adjoint action.) This proves
0 0
g+ ∈Stab (J(λ )). The case g− follows from a similar equality.
i,j G0 0 i,j
Assume that J =J◦+v[ǫ ]+v[ǫ ] with
1 1 λ<i +ai λ<j +aj
suppJ◦∩([λ<+1,λ< ]∪[λ<+1,λ< ])=∅.
1 i i+1 j j+1
We have
(g+)−1J =J −v[ǫ ].
i,j λ<i +aj
9
If a < a , then there exists an unipotent stabilizer u ∈ GL(λ ) ⊂ G of J(λ )
i j i 0 0
such that
u(g+)−1J =J(~λ )+J◦−v[ǫ ]−v[ǫ ].
i,j 0 1 λ<i +aj λ<j +aj
If a =a , then we have
i j
(g+)−1J =J(~λ )−v[ǫ ].
i,j 0 λ<j +aj
In other words, we have g+u(g+)−1J = J − v[ǫ ] or (g+)−1J = J −
i,j i,j λ<i +ai i,j
v[ǫ ] when a ≤ a . Thus, if a ≤ a , then we have GJ(λ) = GJ(λ′) for
λ<i +ai i j i j
λ′ =(λ,a′), where a′ =a (p6=i) or 0 (p=i).
p p
By using g− instead of g+, we deduce: If λ −a ≤ λ −a , then we have
i,j i,j i i j j
GJ(λ)=GJ(λ′) for λ′ =(λ,a′), where a′ =a (p6=j) or 0 (p=j).
p p
We replace (λ,a) by (λ,a′) when one of the above two inequalities (a ≤ a
i j
or λ −a ≤ λ −a ) occur. Repeating these procedures for all possible pairs
i i j j
(i,j) such that λ ≥ λ , we obtain a marked partition ~µ = (λ,a′′) such that
i j
GJ(~µ)=GJ(µ)=GJ(λ) as desired.
6 An exotic version of Springer correspondence
Weretainthesettingoftheprevioussection. Thefollowingresultisnotexactly
the same as the original, but we can easily deduce it from the proof:
Theorem 6.1 (Igusa [Ig73] Lemma 8). Let ~λ = (λ,0) ∈ P(n) ⊂ MP(n).
Then, the reductive part of the stabilizer of J(~λ) is
L :=Sp(2n ,C)×Sp(2n ,C)×··· ,
λ 1 2
where the sequence (n ,n ,...) are the number of λ ’s which share the same
1 2 i
value. Moreover, we have
ResG V = V(i)⊕λi,
Lλ 1
i≥1
M
where V(i) is the external tensor product of a vector representation of Sp(2n )
i
and trivial representations of Sp(2n ) (j 6=i). 2
j
Corollary 6.2. For each X ∈N, the G-stabilizer of X is connected.
Proof. WeputX =X ⊕X ∈V ⊕V . LetG andG denotetheG-stabilizers
1 2 1 2 1 2
of X and X . We show that G ∩G is connected. Let G = L U be the
1 2 1 2 2 2 2
LevidecompositionofG . ThecomponentgroupofG ∩G isthe sameasthat
2 1 2
of G ∩L . By repeating the argument of [K1] Lemma 7.8, we conclude that
1 2
G ∩L must be connected.
1 2
Theorem 6.3 (cf. [CG97] 8.9.3 and 8.4.8). There exists an algebraic stratifi-
cation Oµ of N such that
µ C [dimF]∼= L ⊠IC(O),
∗ F O
O∈Oµ
M
where L is a vector space and IC(O) is the minimal extension of C [dimO].
O O
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