Double Kostka polynomials and Hall bimodule

0.29 MB·English

by  Liu Shiyuan

Double Kostka polynomials and Hall bimodule
Liu Shiyuan and Toshiaki Shoji
5
1
0
2 Abstract. Double Kostka polynomials K (t) are polynomials in t, indexed
λ,µ

n by double partitions λ,µ. As in the ordinary case, Kλ,µ(t) is defined in terms
a of Schur functions s (x) and Hall-Littlewood functions P (x;t). In this paper,
λ µ
J
we study combinatorial properties of K (t) and P (x;t). In particular, we
λ,µ µ
4
show that the Lascoux-Schu¨tzenbergertype formula holds for K (t) in the case
2 where µ = (−;µ′′). Moreover, we show that the Hall bimodulλe,µM introduced

] by Finkelberg-Ginzburg-Travkinis isomorphic to the ring of symmetric functions
T (with two types of variables) and the natural basis u of M is sent to P (x;t)
λ λ
R (up to scalar) under this isomorphism. This gives an alternate approachfor their
. result.
h
t
a
m
Introduction
[

1
Kostka polynomials K (t), indexed by double partitions λ,µ, were introduced
v λ,µ
6 in [S1, S2] as a generalization of ordinary Kostka polynomials K (t) indexed by
λ,µ
9
partitionsλ,µ. Inthispaper, wecallthemdoubleKostkapolynomials. LetΛ = Λ(y)
9
5 be the ring of symmetric functions with respect to the variables y = (y1,y2,...) over
0 Z. We regard Λ ⊗ Λ as the ring of symmetric functions Λ(x(1),x(2)) with respect
.
1 to two types of variables x = (x(1),x(2)). Schur functions {s (x)} gives a basis of
λ
0
Λ⊗Λ. In [S1, S2], the function P (x;t) indexed by a double partitionµ was defined,
5 µ
1 as a generalization of the ordinary Hall-Littlewood function P (x;t) indexed by a
µ
v: partition µ. {Pµ(x;t)} gives a basis of Z[t] ⊗Z (Λ ⊗ Λ), and as in the ordinary
i case, K (t) is defined as the coefficient of the transition matrix between two basis
X λ,µ
{s (x)} and {P (x;t)}.
r λ µ
a After the combinatorial introduction of K (t) in [S1, S2], Achar-Henderson
λ,µ
[AH] gave a geometric interpretation of double Kostka polynomials in terms of the
intersection cohomology associated to the closure of orbits in the enhanced nilpo-
tent cone, which is a natural generalization of the classical result of Lusztig [L1]
that Kostka polynomials are interpreted by the intersection cohomology associated
to the closure of nilpotent orbits in gl . At the same time, Finkelberg-Ginzburg-
n
Travkin [FGT] studied the convolution algeba associated to theaffine Grassmannian
in connection with double Kostka polynomials and the geometry of the enhanced
nilpotent cone. In particular, they introduced the Hall bimodule M (the mirabolic
Hall bimodule in their terminology) as a generalization of the Hall algebra, and
showed that M is isomorphic to Λ⊗Λ over Z[t,t−1], and P (x;t) is obtained as the
λ
image of the natural basis u of M.
λ
In this paper, we study the combinatorial properties of K (t) and P (x;t). In
λ,µ µ
particular, we show that the Lascoux-Schu¨tzenberger type formula holds for K (t)
λ,µ
1
2 LIU ANDSHOJI
in the case where µ = (−;µ′′) (Theorem 3.12). Moreover, in Theorem 4.7, we give
a more direct proof for the above mentioned result of [FGT] (in the sense that we
don’t appeal to the convolution algerba associated to the affine Grassmannian).
In the appendix, we give tables of double Kostaka polynomials for 2 ≤ n ≤ 5,
where n is the size of double partitions. The authors are grateful to J. Michel for
the computer computation of those polynomials.
1. Double Kostka polynomials
1.1. First we recall basic properties of Hall-Littlewood functions and Kostka
polynomials in the original setting, following [M]. Let Λ = Λ(y) = Λn be the
n≥0
ring of symmetric functions over Z with respect to the variables y = (y ,y ,...),
1 2
L
where Λn denotes the free Z-module of symmetirc functions of degree n. We put
Λ = Q⊗ Λ,Λn = Q⊗ Λn. Forapartitionλ = (λ ,λ ,...,λ ), put|λ| = k λ .
Q Z Q Z 1 2 k i=1 i
Let P be the set of partitions of n, i.e. the set of λ such that |λ| = n. Let s
n λ
be the Schur function associated to λ ∈ P . Then {s | λ ∈ P } gives aPZ-baisis
n λ n
of Λn. Let p ∈ Λn be the power sum symmetric funtion associated to λ. Then
λ
{p | λ ∈ P } gives a Q-basis of Λn. For λ = (1m1,2m2,...) ∈ P , define an
λ n Q n
integer z by
λ
(1.1.1) z = imim !.
λ i
i≥1
Y
Following [M, I], we introduce a scalar product on Λ by hp ,p i = δ z . It is
Q λ µ λµ λ
known that {s } form an orthonormal basis of Λ.
λ
1.2. Let P (y;t) be the Hall-Littlewood function associated to a partition
λ
λ. Then {P | λ ∈ P } gives a Z[t]-basis of Λn[t] = Z[t] ⊗ Λn, where t is an
λ n Z
indeterminate. P enjoys a property that
λ
(1.2.1) P (y;0) = s , P (y;1) = m ,
λ λ λ λ
where m (y) is a monomial symmetric function associated to λ. Kostka polynomials
λ
K (t) ∈ Z[t] (λ,µ ∈ P ) are defined by the formula
λ,µ n
(1.2.2) s (y) = K (t)P (y;t).
λ λ,µ µ
µX∈Pn
Recallthedominanceorderλ ≤ µinP , whichisdefinedbytheconditionλ ≤ µ
n
if and only if i λ ≤ i µ for each i ≥ 1. For each partition λ = (λ ,...,λ ),
j=1 j j=1 j 1 k
we define an integer n(λ) by n(λ) = k (i − 1)λ . It is known that K (t) = 0
P P i=1 i λ,µ
unless λ ≥ µ, and that K (t) is a monic of degree n(µ)−n(λ) if λ ≥ µ ([M, III,
λ,µ
P
(6.5)]).
For λ = (λ ,...,λ ) ∈ P with λ > 0, we define z (t) ∈ Q(t) by
1 k n k λ
(1.2.3) z (t) = z (1−tλi)−1,
λ λ
i≥1
Y
DOUBLEKOSTKAPOLYNOMIALS 3
where z is as in (1.1.1). Following [M, III], we introduce a scalar product on
λ
Λ (t) = Q(t) ⊗ Λ byhp ,p i= z (t)δ . Then P (y;t) form an orthogonal basis
Q Z λ µ λ λ,µ λ
of Λ[t] = Z[t]⊗ Λ. In fact, they are characterized by the following two properties
Z
([M, III, (2.6) and (4.9)]);
(1.2.4) P (y;t) = s (x)+ w (t)s (x)
λ λ λµ µ
µ<λ
X
with w (t) ∈ Z[t] , and
λµ
(1.2.5) hP ,P i= 0 unless λ = µ.
λ µ
1.3. Let Ξ = Ξ(x) = Λ(x(1)) ⊗ Λ(x(2)) be the ring of symmetric functions
over Z with respect to variables x = (x(1),x(2)), where x(1) = (x(1),x(1),...),x(2) =
1 2
(x(2),x(2),...). We denote it as Ξ = Ξn, similarly to the case of Λ. Let
1 2 n≥0
P be the set of double partitions λ = (λ′,λ′′) such that |λ′| + |λ′′| = n. For
n,2
λ = (λ′,λ′′) ∈ P , we define a Schur fuLnction s (x) ∈ Ξn by
n,2 λ
(1.3.1) sλ(x) = sλ′(x(1))sλ′′(x(2)).
Then {s | λ ∈ P } gives a Z-basis of Ξn. For an integer r ≥ 0, put p(1) =
λ n,2 r
p (x(1))+p (x(2)), andp(2) = p (x(1))−p (x(2)), where p isthepower sum symmetric
r r r r r r
function in Λ. . For λ ∈ P , we define p (x) ∈ Ξn by
n,2 λ
(1) (2)
(1.3.2) p = p p ,
λ λ′ λ′′
i j
i j
Y Y
where λ = (λ′,λ′′) such that λ′ = (λ′,λ′,...,λ′ ), λ′′ = (λ′′,λ′′,...,λ′′ ) with
1 2 k′ 1 2 k′′
λ′ ,λ′′ > 0. Then {p | λ ∈ P } gives a Q-basis of Ξn. For λ ∈ P , we define
k′ k′′ λ n,2 Q n,2
functions z(1)(t),z(2)(t) ∈ Q(t) by
λ λ
k′ k′′
(1.3.3) z(1)(t) = (1−tλ′j)−1, z(2)(t) = (1+tλ′j′)−1.
λ λ
j=1 j=1
Y Y
Forλ ∈ Pn,2, wedefineaninteger zλ byzλ = 2k′+k′′zλ′zλ′′. Wenowdefineafunction
z (t) ∈ Q(t) by
λ
(1) (2)
(1.3.4) z (t) = z z (t)z (t).
λ λ λ λ
Let Ξ[t] = Z[t]⊗ Ξ be the free Z[t]-module, and Ξ (t) = Q(t)⊗ Ξ be the Q(t)-
Z Q Z
space. Then {p (x) | λ ∈ P } gives a basis of Ξn(t). We define a scalar product
λ n,2 Q
on Ξ (t) by
Q
hp ,p i= δ z (t).
λ µ λ,µ λ
4 LIU ANDSHOJI
We express a double partitionλ = (λ′,λ′′) as λ′ = (λ′,...,λ′),λ′′ = (λ′′,...,λ′′)
1 k 1 k
with some k, by allowing zero on parts λ′,λ′′. We define a composition c(λ) of n by
i i
c(λ) = (λ′,λ′′,λ′,λ′′,...,λ′,λ′′).
1 1 2 2 k k
We define a partial order λ ≥ µ on P by the the condition c(λ) ≥ c(µ), where
n,2
≥ is the dominance order on the set of compositions of n defined in a similar way
as in the case of partitions.
The following fact is known.
Proposition 1.4 ([S1, S2]). There exists a unique function P (x;t) ∈ Ξ [t] satis-
λ Q
fying the following properties.
(i) P is expressed as a linear combination of Schur functions s as
λ µ
P (x;t) = s (x)+ u (t)s (x)
λ λ λ,µ µ
µ<λ
X
with u (t) ∈ Q(t).
λ,µ
(ii) hP ,P i= 0 unless λ = µ.
λ µ
Remark 1.5. P is called the Hall-Littlewood function associated to a double
λ
partiton λ. More generally, Hall-Littlewood functions associated to r-partitions of
n was introduced in [S1]. However the arguments in [S1] is based on a fixed total
orderwhich iscompatible withthepartialorder≥onP even inthecaseofdouble
n,2
partitions. In [S2, Theorem 2.8], the closed formula for P is given in the case of
λ
double partitions. This implies that P is independent of the choice of the total
λ
order, and is deterimned uniquely as in the above proposition. (The uniqueness of
P also follows from the result of Achar-Henderson, see Theorem 2.4.)
λ
1.6. By Proposition 1.4, {P | λ ∈ P } gives a basis of Ξ (t). For λ,µ ∈
λ n,2 Q
P , we define a function K (t) ∈ Q(t) by the formula
n,2 λ,µ
s (x) = K (t)P (x;t).
λ λ,µ µ
µX∈Pn,2
K (t) are called the Kostka functions associated to double partitions. For each
λ,µ
λ = (λ′,λ′′) ∈ P , put n(λ) = n(λ′ +λ′′) = n(λ′)+n(λ′′). We define an integer
n,2
a(λ) by
(1.6.1) a(λ) = 2n(λ)+|λ′′|.
The following result was proved in [S2, Prop. 3.3].
Proposition 1.7. K (t) ∈ Z[t]. K (t) = 0 unless λ ≥ µ. If λ ≥ µ, K (t) is
λ,µ λ,µ λ,µ
a monic of degree a(µ)−a(λ), hence K (t) = 1. In particular, P (x;t) ∈ Ξn[t],
λ,λ λ
and u (t) ∈ Z[t].
λ,µ
1.8. Since K (t) is a polynomial in t associated to double partitions, we call
λ,µ
it the double Kostka polynomial. Put K (t) = ta(µ)K (t−1). By Proposition 1.7,
λ,µ λ,µ
e
DOUBLEKOSTKAPOLYNOMIALS 5
K (t) is again contained in Z[t], which we call the modified double Kostka polyno-
λ,µ
mial. In the case of Kostka polynomial K (t), we also put K (t) = tn(µ)K (t−1).
λ,µ λ,µ λ,µ
Bey 1.2, K (t) is a polynomial in Z[t], which is called the modified Kostka polyno-
λ,µ
mial. e
Folloewing [S1, S2], we give a combinatorial characterization of K (t) and
λ,µ
K (t). In order to discuss both cases simultaneously, we introduce some nota-
λ,µ
tion. For r = 1,2, put W = S ⋉ (Z/2Z)n. Hence W is the symmeetric group
n,r n n,r
Se of degree n if r = 1, and is the Weyl group of type C if r = 2. For a (not
n n
necessarily irreducible) character χ of W , we define the fake degree R(χ) by
n,r
n (tir −1) ε(w)χ(w)
(1.8.1) R(χ) = i=1 ,
|W | det (t−w)
Q n,r w∈XWn,r V0
where ε is the sign character of W , and V is the reflection representation of W
n,r 0 n,r
if r = 2 (i.e., dimV = n), and its restriction on S if r = 1. Let R(W ) = N R
0 n n,r i=1 i
be the coinvariant algebra over Q associated to W , where N is the number of
n,r
L
positive roots of the root system of type C (resp. type A ) if r = 2 (resp. r = 1).
n n−1
Then R(W ) is a graded W -module, and we have
n,r n,r
N
(1.8.2) R(χ) = hχ,R i ti,
i Wn,r
i=1
X
whereh, i is the inner product of characters of W . It follows that R(χ) ∈ Z[t].
Wn,r n,r
It is known that irreducible characters of W are parametrized by P (we use the
n,r n,r
convention that P = P ). We denote by χλ the irreducible character of W
n,1 n n,r
corresponding to λ ∈ P . (Here we use the parametrization such that the identity
n,r
character corresponds to λ = ((n),−) if r = 2, and λ = (n) if r = 1.) We define a
square matrix Ω = (ω ) by
λ,µ λ,µ
(1.8.3) ω = tNR(χλ ⊗χµ ⊗ε).
λ,µ
We have the following result. Note that Theorem 5.4 in [S1] is stated for a fixed
total order on P . But in our case, it can be replaced by the partial order (see
n,2
Remark 1.5).
Proposition 1.9 ([S1, Thm. 5.4]). Assume that r = 2. There exist unique matrices
P = (p ), Λ = (ξ ) over Q[t] satisfying the equation
λ,µ λ,µ
PΛtP = Ω,
subject to the condition that Λ is a diagonal matrix and that
0 unless µ ≤ λ ,
p =
λ,µ ta(λ) if λ = µ .
(
6 LIU ANDSHOJI
Then the entry p of the matrix P coincides with K (t).
λ,µ λ,µ
A simialr result holds for the case r = 1 by replacing λ,µ ∈ P by λ,µ ∈ P ,
n,2 n
and by replacing a(λ) by n(λ). e
1.10. Assume that λ = (−,λ′′) ∈ P . If µ ≤ λ, then µ is of the form
n,2
µ = (−,µ′′) with µ′′ ≤ λ′′. Thus K (t) = 0 unless µ satisfies this condition. The
λ,µ
following result was showm by Achar-Henderson [AH] by a geometric method (see
Proposition 2.5 (ii)). We give beloew an alternate proof based on Proposition 1.9.
Proposition 1.11. Assume that λ = (−,λ′′),µ = (−,µ′′) ∈ P . Then
n,2
(1.11.1) Kλ,µ(t) = tnKλ′′,µ′′(t2).
In particular, we have e e
(1.11.2) Kλ,µ(t) = Kλ′′,µ′′(t2).
Proof. (1.11.2) follows from (1.11.1). We show (1.11.1). We shall compute ω =
λ,µ
tNR(χλ ⊗ χµ ⊗ ε) for λ = (−,λ′′),µ = (−.µ′′). χλ corresponds to the irreducible
representation of S with character χλ′′, extended by the action of (Z/2Z)n such
n
that any factor Z/2Z acts non-trivially. This is the same for χµ. Hence χλ ⊗ χµ
corresponds to the representation of S with character χλ′′ ⊗χµ′′, extended by the
n
trivial action of (Z/2Z)n. Thus χλ⊗χµ⊗ε corresponds to the representation of S
n
with character χλ′′⊗χµ′′⊗ε′, extended by the actionof (Z/2Z)n such that any factor
Z/2Z acts non-trivially, where ε′ denote the sign character of S . Let {s ,...,s }
n 1 n
be the set of simple reflections of W . We identify the symmetric algebra S(V∗)
n 0
of V with the polynomial ring R[y ,...,y ] with the natural W action, where s
0 1 n n i
permutes y and y (1 ≤ i ≤ n − 1), and s maps y to −y . Then (Z/2Z)n-
i i+1 n n n
invariant subalgebra of R[y ,...,y ] coincides with R[y2,...,y2]. It follows that
1 n 1 n
the (Z/2Z)n-invariant subalgerba R(W )(Z/2Z)n of R(W ) is isomorphic to R(S )
n n n
as graded algebras, where the degree 2i-part of R(W )(Z/2Z)n corresponds to the
n
degree i part of R(S ). Let X be the subspace of R(W ) consisitng of vectors on
n n
which (Z/2Z)n acts in such a way that each factor Z/2Z acts non-trivially. Then
X = y ···y R(W )(Z/2Z)n. It follows that
1 n n
R(χλ ⊗χµ ⊗ε)(t) = tnR(χλ′′ ⊗χµ′′ ⊗ε′)(t2).
Since N = n2 for W -case, and N = n(n−1)/2 for S -case, this implies that
n n
(1.11.3) ωλ,µ(t) = t2nωλ′′,µ′′(t2)
We consider the embedding P ֒→ P by λ′′ 7→ (−,λ′′). This embedding is
n n,2
compatible with the partial order of P and P , and in fact, P is identifed with
n n,2 n
the subset {µ ∈ P | µ ≤ (−,(n))} of P . We consider the matrix equation
n,2 n,2
PΛtP = Ω as in Proposition 1.9 for r = 2. Let P ,Λ ,Ω be the submatrices of
0 0 0
P,Λ,Ω obtained by restricting the indices from P to P . Then these matrices
n,2 n
satisfy the relation P Λ tP = Ω . By (1.11.3) Ω coincides with t2nΩ′(t2), where
0 0 0 0 0
DOUBLEKOSTKAPOLYNOMIALS 7
Ω′ denotes the matrix Ω in the case r = 1. If we put P′ = t−nP ,Λ′ = Λ , we have
0 0
a matrix equation P′Λ′tP′ = Ω′(t2). Note that the (λ′′,λ′′)-entry of P′ coincides
with t−nta(λ) = t2n(λ′′). Hence P′,Λ′,Ω′ satisfy all the requirements in Proposition
1.9 for the case r = 1, by replacing t by t2. Now by Proposition 1.9, we have
t−nKλ,µ(t) = Kλ′′,µ′′(t2) as asserted. (cid:3)
As a corollary, we have
e e
Corollary 1.12. Assume that λ = (−,λ′′). Then Pλ(x;t) = Pλ′′(x(2);t2).
Proof. Since λ = (−,λ′′), we have sλ(x) = sλ′′(x(2)). By (1.11.2), we have
sλ′′(x(2)) = Kλ′′,µ′′(t2)Pµ(x;t).
µX′′∈Pn
We have also
sλ′′(x(2)) = Kλ′′,µ′′(t2)Pµ′′(x(2);t2).
µX′′∈Pn
Since (Kλ′′,µ′′(t2)) is a non-singular matrix indexed by Pn, the assertion follows. (cid:3)
2. Geometric interpretation of double Kostka polynomials
2.1. In [L1], Lusztig gave a geometric interpretation of Kostka polynomials
in terms of the intersection cohomology complex associated to the nilpotent orbits
of gl . Let V be an n-dimensional vector space over an algebraically closed field k,
n
and put G = GL(V). Let g be the Lie algebra of G, and g the nilpotent cone of
nil
g. G acts on g by the adjoint action, and the set of G-orbits in g is in bijective
nil nil
correspondence with P via the Jordan normal form of nilpotent elements. We
n
denote by O the G-orbit corresponding to λ ∈ P . Let O be the closure of O in
λ n λ λ
g . Then we have O = O , where µ ≤ λ is the dominance order of P . Let
nil λ µ≤λ µ n
A = IC(O ,Q¯ ) be the intersection cohomology complex of Q¯ -sheaves, and H iA
λ λ l ` l x λ
the stalk at x ∈ O of the i-th cohomology sheaf H iA . Lusztig’s result is stated
λ λ
as follows.
Theorem 2.2 ([L1, Thm. 2]). H iA = 0 for odd i. For each x ∈ O ⊂ O ,
λ µ λ
K (t) = tn(λ) (dimH 2iA )ti.
λ,µ x λ
i≥0
X
e
2.3. The geometric interpretation of double Kostka polynomials analogous to
Theorem 2.2 was established by Achar-Henderson [AH]. We follow the setting in 2.1.
Consider the direct product X = g×V, on which G acts as g : (x,v) 7→ (gx,gv),
where gv is the natural action of G on V. Put X = g × V. X is a G-
nil nil nil
stable subset of X , and is called the enhanced nilpotent cone. It is known by
Achar-Henderson [AH] and by Travkin [T] that the set of G-orbits in X is in
nil
bijective correspondence with P . The correspondence is given as follows; take
n,2
8 LIU ANDSHOJI
(x,v) ∈ X . Put Ex = {g ∈ End(V) | gx = xg}. Then W = Exv is an x-
nil
stable subspace of V. Let λ′ be the Jordan type of x| , and λ′′ the Jordan type of
W
x| . Then λ = (λ′,λ′′) ∈ P , and the assignment (x,v) 7→ λ gives the required
V/W n,2
correspondence. We denote by O the G-orbit corresponding to λ ∈ P . The
λ n,2
closure relation for O was described by [AH, Thm. 3.9] as follows;
λ
(2.3.1) O = O ,
λ µ
µ≤λ
a
where the partial order µ ≤ λ is the one defined in 1.3. We consider the intersec-
tion cohomology complex A = IC(O ,Q¯ ) on X associated to λ ∈ P . The
λ λ l nil n,2
following result was proved by Achar-Henderson.
Theorem 2.4 ([AH, Thm. 5.2]). Assume that A is attached to the enhanced
λ
nilpotent cone. Then H iA = 0 for odd i. For z ∈ O ⊂ O ,
λ µ λ
K (t) = ta(λ) (dimH 2iA )t2i.
λ,µ z λ
i≥0
X
e
Note that H 2i corresponds to t2i in the enhanced case, which is different from
the correspondence H 2i ↔ ti in the g case. As a corollary, we have
nil
Proposition 2.5 ([AH, Cor. 5.3]). (i) K (t) ∈ Z [t]. Moreover, only pow-
λ,µ ≥0
ers of t congruent to a(λ) modulo 2 occur in the polynomial.
(ii) Assume that λ = (−,λ′′),µ = (−,µ′′)e. Then Kλ,µ(t) = tnKλ′′,µ′′(t2).
(iii) Assume that λ = (λ′,−) and µ = (µ′,µ′′). Then Kλ,µ(t) = Kλ′,µ′+µ′′(t2).
e e
Proof. For the sake of completeness, we give the proof here. (i) is clear from the
theorem. For (ii), take λ = (−,λ′′). Then by the correespondenceegiven in 2.3, if
(x,v) ∈ Oλ, then v = 0, and x ∈ Oλ′′. It follows that Oλ = Oλ′′ and that Aλ ≃ Aλ′′.
z ∈ Oµ is also wtitten as z = (x,0) with x ∈ Oµ′′. Then (ii) follows by comparing
Theorem 2.2 and Theorem 2.4. For (iii), it was proved in [AH, Lemma 3.1] that
Oλ = Oλ′×V for λ = (λ′,−). Thus IC(Oλ,Q¯l) ≃ IC(Oλ′,Q¯l)⊠(Q¯l)V, where (Q¯l)V
is the constant sheaf Q¯l on V. It follows that Hz2iAλ = Hx2iAλ′ for z = (x,v) ∈ Oµ.
Since x ∈ Oµ′+µ′′, (iii) follows from Theorem 2.2 (note that a(λ) = 2n(λ′)). (cid:3)
Remark 2.6. Proposition 2.5 (ii) was also proved in Proposition 1.11 by a com-
binatorial method. We don’t know whether (iii) is proved in a combinatorial way.
However if we admit that K (t) depends only on µ′ + µ′′ for λ = (λ′,−) (this is
λ,µ
a consequence of (iii)), a similar argument as in the proof of Proposition 1.1 can be
applied. e
Proposition 2.5 (iii) implies the following.
Corollary 2.7. For ν ∈ P , we have
n
Pν(x(1);t2) = t|µ′′|P(µ′,µ′′)(x;t).
ν=µ′+µ′′
X
DOUBLEKOSTKAPOLYNOMIALS 9
Proof. It follows from Proposition 2.5 (iii) that Kλ,µ(t) = t|µ′′|Kλ′,µ′+µ′′(t2) for λ =
(λ′,−). Since sλ(x) = sλ′(x(1)), we have
sλ′(x(1)) = Kλ,µ(t)Pµ(x;t)
µX∈Pn,2
= Kλ′,µ′+µ′′(t2)t|µ′′|Pµ(x;t)
µX∈Pn,2
= Kλ′,ν(t2) t|µ′′|P(µ′,µ′′)(x;t).
νX∈Pn ν=Xµ′+µ′′
On the other hand, we have
sλ′(x(1)) = Kλ′,ν(t2)Pν(x(1);t2).
νX∈Pn
Since (Kλ′,ν(t2)) is a non-singular matrix, we obtain the required formula. (cid:3)
Remark 2.8. The formula in Corollary 2.7 suggests that the behaviour of P (x;t)
µ
at t = 1 is different from that of ordinally Hall-Littlewood functions given in (1.2.1).
In fact, by Corollary 1.12, P (x;t) = P (x(2);t2). Hence P (x;1) = m (x(2))
(−,ν) ν (−,ν) ν
by (1.2.1). Also by (1.2.1) P (x(1);1) = m (x(1)). Then by Corollary 2.7, we have
ν ν
mν(x(1)) = mν(x(2))+ P(µ′,µ′′)(x;1).
ν=µ′+µ′′,µ′6=∅
X
This formula shows that a certain cancelation occurs in the expression of P (x;1)
µ
as a sum of monomials. Concerning this, we will have a related result later in
Proposition 3.23.
2.9. Thereexists a geometricrealizationofdoubleKostkapolynomialsinterms
of the exotic nilpotent cone instead of the enhanced nilpotent cone. Let V be a 2n-
dimensional vector space over an algebraically closed field k of odd characteristic.
Let G = GL(V) and θ an involutive automorphism of G such that Gθ = Sp(V).
Put H = Gθ. Let g be the Lie algebra of G. θ induces a linear automorphism of
order 2 on g, which we denote also by θ. g is decomposed as g = gθ ⊕g−θ, where
g±θ is the eigenspace of θ with eigenvalue ±1. Thus g±θ are H-invariant subspaces
in g. We consider the direct product X = g−θ × V, on which H acts diagonally.
Put g−θ = g−θ ∩g . Then g−θ is H-stable, and we consider X = g−θ ×V. X
nil nil nil nil nil nil
is an H-invariant subset of X , and is called the exotic nilpotent cone. It is known
by Kato [K1] that the set of H-orbits in X is in bijective correspondence with
nil
P . We denote by O the H-orbit corresponding to λ ∈ P . It is also known
n,2 λ n,2
by [AH] that the closure relations for O are given by the partial order ≤ on P .
λ n,2
We consider the intersection cohomology complex A = IC(O ,Q¯ ) on X . The
λ λ l nil
following result was proved by Kato [K2], and [SS2], independently.
10 LIU ANDSHOJI
Theorem 2.10. Assume that A is attached to the exotic nilpotent cone. Then
λ
H iA = 0 unless i ≡ 0 (mod 4). For z ∈ O ⊂ O , we have
λ µ λ
K (t) = ta(λ) (dimH 4iA )t2i.
λ,µ z λ
i≥0
X
e
2.11. Let W be the Weyl group of type C . The advantage of the use of
n n
the exotic nilpotent cone relies on the fact that it has a good relationship with
representations of Weyl groups, as explained below. Let B be a θ-stable Borel
subgroup of G. Then Bθ is a Borel subgrouop of H, and we denote by B the flag
variety H/Bθ of H. Let 0 = M ⊂ M ⊂ ··· ⊂ M ⊂ V be the (full) isotropic flag
0 1 n
fixed by Bθ. Hence M is a maximal isotropic subspace of V. Put
n
X = {(x,v,gBθ) ∈ g−θ ×V ×B | g−1x ∈ LieB,g−1v ∈ M },
nil nil n
and definefa map π : X → X by (x,v,gBθ) 7→ (x,v). Then X is smooth,
1 nil nil nil
irreducible and π is proper surjective. Let V be the irreducible representation
1 λ
of W corresponidng tofχλ (λ ∈ P ). We consider the direct iamfge (π ) Q¯ of
n n,2 1 ∗ l
the constant sheaf Q¯ on X . The following result is an analogue of the Springer
l nil
correspondene for reductive groups, and was proved by Kato [K1], and [SS1], inde-
pendently. f
Theorem 2.12. (π ) Q¯ [dimX ] is a semisimple perverse sheaf on X , equipped
1 ∗ l nil nil
with W -action, and is decomposed as
n
(2.12.1) (π ) Q¯ [dimX ] ≃ V ⊗A [dimO ],
1 ∗ l nil λ λ λ
λ∈MPn,2
where A [dimO ] is a simple perverse sheaf on X .
λ λ nil
2.13. For each z = (x,v) ∈ X , put
nil
B = {gBθ ∈ B | g−1x ∈ LieB,g−1v ∈ M }.
z n
B is isomorphic to π−1(z), and is called the Springer fibre. Since H i((π ) Q¯ ) ≃
z 1 z 1 ∗ l
Hi(B ,Q¯ ), Hi(B ,Q¯ ) has a structure of W -module, which we call the Springer
z l z l n
representation of W . Put K = (π ) Q¯ . By taking the stalk at z ∈ X of the i-th
n 1 ∗ l nil
cohomology of both sides in (2.12.1), we have an isomorphism of W -modules,
n
H iK ≃ Hi(B ,Q¯ ) ≃ V ⊗H i+dimOλ−dimXnilA .
z z l λ z λ
λ∈MPn,2
Since dimX − dimO = 2a(λ) (see [SS2, (5.7.1)], this together with Theorem
nil λ
2.10 imply the following result.