Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, II

0.17 MB·English

by  Nanhua Xi

KAZHDAN-LUSZTIG BASIS AND A GEOMETRIC
FILTRATION OF AN AFFINE HECKE ALGEBRA, II
NANHUA XI
Abstract. An affine Hecke algebras can be realized as an equi-
variantK-groupofthe correspondingSteinbergvariety. Thisgives
8
0 rise naturally to some two-sided ideals of the affine Hecke algebra
0 by means of the closures of nilpotent orbits of the corresponding
2 Lie algebra. In this paper we will show that the two-sided ideals

n are in fact the two-sided ideals of the affine Hecke algebra defined
a through two-sided cells of the corresponding affine Weyl group af-
J
ter the two-sided ideals are tensored by Q. This proves a weak

3
form of a conjecture of Ginzburg proposed in 1987.

]
A
Q
.
h
t
a
0. Introduction
m
[ Let H be an affine Hecke algebra over the ring Z[v,v−1] of Lau-

1 rent polynomials in an indeterminate v with integer coefficients. The
v affine Hecke algebra has a Kazhdan-Lusztig basis. The basis has many
2
remarkable properties and play an important role in representation
7
4 theory. Also, Kazhdan-Lusztig and Ginzburg give a geometric realiza-
0
tion of H, which is the key of Kazhdan-Lusztig’s proof for Deligne-
.
1 Langlands conjecture on classification of irreducible modules of affine
0
Hecke algebras over C at non-root of 1. This geometric construction of
8
0 H has some two-sided ideals defined naturally by means of the nilpo-
:
v tentvarietyofthecorrespondingLiealgebra. Thetwo-sidedidealsform
i
X a nice filtration of the affine Hecke algebra. In [G2] Ginzburg conjec-
r tured that the two-sided ideals are in fact the two-sided ideals of the
a
affine Hecke algebra defined through two-sided cells of the correspond-
ing affine Weyl group, see also [L5, T2]. The conjecture is known to be
true for the trivial nilpotent orbit {0} (see Corollary 8.13 in [L5] and
Theorem 7.4 in [X1]) and for type A [TX]. Other evidence is showed
in [L5, Corollary 9.13]. We will prove the two kinds of two-sided ideals
coincide after they are tensored by Q (see Theorem 1.5 in section 1 ).
This proves a weak form of Ginzburg’s conjecture.
2000 AMS Mathematics Subject Classification: Primary 20C08.
N. Xi was partially supported by Natural Sciences Foundation of China (No.
10671193).
1
2 NANHUAXI
1. Affine Hecke algebra
1.1. Let G be a simply connected simple algebraic group over the
complex number field C. The Weyl group W acts naturally on the
0
character group X of a maximal tours of G. The semidirect product
W = W ⋉X with respect to the action is called an (extended) affine
0
Weyl group. Let H be the associated Hecke algebra over the ring
A = Z[v,v−1] (v an indeterminate) with parameter v2. Thus H has an
A-basis {T | w ∈ W} and its multiplication is defined by the relations
w
(T − v2)(T + 1) = 0 if s is a simple reflection and T T = T if
s s w u wu
l(wu) = l(w)+l(u), here l is the length function of W.
1.2. Let g be the Lie algebra of G, N the nilpotent cone of g and
B the variety of all Borel subalgebras of g. The Steinberg variety Z
is the subvariety of N × B × B consisting of all triples (n,b,b′), n ∈
b ∩ b′ ∩ N, b,b′ ∈ B. Let Λ = {(n,b) | n ∈ N ∩ b,b ∈ B} be the
cotangent bundle of B. Clearly Z can be regarded as a subvariety of
Λ × Λ by the imbedding Z → Λ × Λ, (n,b,b′) → (n,b,n,b′). Define
a G × C∗-action on Λ by (g,z) : (n,b) → (z−2ad(g)n,ad(g)b). Let
G×C∗ acts on Λ×Λ diagonally, then Z is a G×C∗-stable subvariety
of Λ×Λ. For 1 ≤ i < j ≤ 3, let p be the projection from Λ×Λ×Λ to
ij
its (i,j)-factor. Note that the restriction of p gives rise to a proper
13
morphism p−1(Z)∩p−1(Z) → Z. Let KG×C∗(Z) = KG×C∗(Λ×Λ;Z) be
12 23
the Grothendieck group of the category of G×C∗-equivariant coherent
sheaves on Λ×Λ with support in Z. We define the convolution product
∗ : KG×C∗(Z)×KG×C∗(Z) → KG×C∗(Z),
F ∗G = (p ) (p∗ F ⊗ p∗ G),
13 ∗ 12 OΛ×Λ×Λ 23
where O is the structure sheaf of Λ×Λ×Λ. This endows with
Λ×Λ×Λ
KG×C∗(Z) an associative algebra structure over the representation ring
RG×C∗ of G × C∗. We shall regard the indeterminate v as the repre-
sentation G × C∗ → C∗, (g,z) → z. Then RG×C∗ is identified with
A⊗Z RG. In particular, KG×C∗(Z) is an A-algebra. Moreover, as A-
algebras, KG×C∗(Z) is isomorphic to the Hecke algebra H, see [G1,
KL2] or [CG, L5]. We shall identify KG×C∗(Z) with H.
1.3. Let C and C′ be two G-orbits in N. We say that C ≤ C′ if C
is in the closure of C′. This defines a partial order on the set of G-
orbits in N. Given a locally closed G-stable subvariety of N, we set
Z = {(n,b,b′) ∈ Z | n ∈ Y}.
Y
If Y is closed, then the inclusion i : Z → Z induces a map
Y Y
(i ) : KG×C∗(Z ) → KG×C∗(Z) (see [G1, KL2]). The image H
Y ∗ Y Y
of (i ) is in fact a two-sided ideal of KG×C∗(Z) (see [L5, Corollary
Y ∗
9.13]), which is generated by G × C∗-equivariant sheaves supported
on Z . It is conjectured that this ideal is spanned by elements in a
Y
Kazhdan-Lusztig basis (see [G2, L5, T2]).
KAZHDAN-LUSZTIG BASIS AND A GEOMETRIC FILTRATION 3
1.4. Let C = v−l(w) P (v2)T , where P are the Kazhdan-
w y≤w y,w y y,w
Lusztig polynomials. TPhen the elements C ( w ∈ W) form an A-basis
w
of H, called a Kazhdan-Lusztig basis of H. Define w ≤ u if a 6= 0
w
LR
in the expression hC h′ = a C (a ∈ A) for some h,h′ in H.
u z∈W z z z
This defines a preorder on WP. The corresponding equivalence classes
are called two-sided cells and the preorder gives rise to a partial order
≤ on the set of two-sided cells of W. (See [KL1].) For an element w
LR
in W and a two-sided cell c of W we shall write w ≤ c if w ≤ u for
LR LR
some (equivalent to for any) u in c.
Lusztig established a bijection between the set of G-orbits in N and
the set of two-sided cells of W (see [L4, Theorem 4.8]). Lusztig’s bijec-
tion preserves the partial orders we have defined, this was conjectured
by Lusztig and verified by Bezrukavnikov (see [B, Theorem 4 (b)]).
Perhaps this bijection is at the heart of the theory of cells in affine
Weyl groups, many deep results are related to it. Now we can state
the main result of this paper.
Theorem 1.5. Let C be a G-orbit in N and c the two-sided cell of
W corresponding to C under Lusztig’s bijection. Then the elements
Cw (w ≤ c) form a Q[v,v−1]-basis of HC¯⊗Z Q, where C¯ denotes the
LR
closure of C and H is the image of the map (i ) : KG×C∗(Z ) →
C¯ C¯ ∗ C¯
KG×C∗(Z) = H.
2. Proof of the theorem
2.1. Before proving the theorem we need recall some results about
representations of an affine Hecke algebra. Let H = C[v,v−1] ⊗ H
A
and for any nonzero complex number q we set Hq = H ⊗C[v,v−1] C,
where C is regarded as a C[v,v−1]-algebra by specializing v to a square
root of q.
ForanyG-stablelocallyclosedsubvarietyY ofN wesetKG×C∗(Z ) =
Y
KG×C∗(Z ) ⊗ C. If Y is closed, then the inclusion i : Z → Z in-
Y Y Y
duces an injective map (i ) : KG×C∗(Z ) ֒→ KG×C∗(Z) = H. If Y is
Y ∗ Y
a closed subset of N, we shall identify KG×C∗(Z ) with the image of
Y
(i ) , which is a two-sided ideal of H. See [KL2, 5.3] or [L5, Corollary
Y ∗
9.13].
Let s be a semisimple element of G and n a nilpotent element in N
such that ad(s)n = qn, where q is in C∗. Let Bs be the subvariety of B
n
consisting of the Borel subalgebras containing n and fixed by s. Then
the component group A(s,n) = C (s,n)/C (s,n)o of the simultaneous
G G
centralizer in G of s and n acts on the total complex homology group
H (Bs). Let ρ be a representation of A(s,n) appearing in the space
∗ n
H (Bs). It is known that if ql(w) 6= 0 then the isomorphism
∗ n w∈W0
classes of irreducible representPations of Hq is one to one corresponding
4 NANHUAXI
to the G-conjugacy classes of all the triples (s,n,ρ), where s ∈ G is
semisimple, n ∈ N satisfying ad(s)n = qn and ρ is an irreducible
representation of A(s,n) appearing in H (Bs). See [KL2, X3].
∗ n
2.2. Fromnowonwe assume thatq isnotarootof1. LetL (s,n,ρ)be
q
an irreducible representation of H corresponding to the triple (s,n,ρ).
q
Kazhdan and Lusztig constructed a standard module M(s,n,q,ρ) of
H such that L (s,n,ρ) is the unique simple quotient of M(s,n,q,ρ)
q q
(see [KL2, 5.12 (b) and Theorem 7.12]). We shall write M (s,n,ρ) for
q
M(s,n,q,ρ). The following simple fact will be needed.
(a) Let C be a G-orbit in N. Then the image H of (i ) acts on
C¯ C¯ ∗
M (s,n,ρ) and L (s,n,ρ) by zero if n is not in C¯.
q q
Proof. Clearly Y = C¯∪(G.n−G.n) is closed. If n is not in C¯, then
the complement in X = C¯∪G.n of Y is G.n. Recall that KG×C∗(ZY′)
is regarded as a two-sided ideal of H for any closed subset Y′ of N (see
2.1). According to[KL2, 5.3(c), (d)and(e)], theinclusions i : Y ֒→ X
and j : G.n ֒→ X induce an exact sequence of H-bimodules
0 → KG×C∗(Z ) → KG×C∗(Z ) → KG×C∗(Z ) → 0.
Y X G.n
Using [KL2, 5.3 (e)] we know the inclusion k : C¯ → Y induces an
injective H-bimodule homomorphism k : KG×C∗(Z ) → KG×C∗(Z ).
∗ C¯ Y
Since M (s,n,ρ) is a quotient module of KG×C∗(Z ) (cf. proof of
q G.n
5.13 in [KL2]), the statement (a) then follows from the exact sequence
above.
2.3. Let J be the based ring of a two-sided cell c of W, which has
c
a Z-basis {t |w ∈ c}. Let D be the set of distinguished involutions
w c
in c. For x,y ∈ W, we write C C = h C , h ∈ A. The
x y z∈W x,y,z z x,y,z
map P
ϕ (C ) = h t , w ∈ W,
c w w,d,u u
X
d∈Dc
u∈W
a(d)=a(u)
defines an A-algebra homomorphism H → Jc ⊗Z A, where a : W → N
is the a-function defined in [L1]. The homomorphism ϕ induces a C-
c
algebra homomorphism ϕc,q : Hq → Jc = Jc⊗ZC. If E is a Jc-module,
through ϕ , E gets an H -module structure, which will be denoted by
c,q q
E . See [L2, L3].
q
Let C be the nilpotent orbit corresponding to c. According to [L4,
Theorems 4.2and4.8], themapE → E defines abijectionbetween the
q
isomorphism classes of simple J -modules and the isomorphism classes
c
of standard modules M (s,n,ρ) with n in C.
q
Now we start to prove Theorem 1.5.
2.4. We first show that H is contained in the two-sided ideal H≤c
C¯
spanned by all C (w ≤ c).
w
LR
KAZHDAN-LUSZTIG BASIS AND A GEOMETRIC FILTRATION 5
Let C = G.n and recall that H stands for the image of (i ) :
C¯ C¯ ∗
KG×C∗(Z ) → KG×C∗(Z) = H. If H was not contained in the A-
C¯ C¯
submodule H≤c of H, we could find x ∈ W such that x 6≤ c and C
x
LR
appearsinH . (WesaythatC appearsinH ifthereexistsanelement
C¯ x C¯
a C (a ∈ A) in H such that a 6= 0.) Choose x ∈ W such
w∈W w w w C¯ x
tPhat Cx appears in HC¯ and x is highest with respect to the preorder
≤ and to H in the following sense: whenever C appears in H , then
C¯ w C¯
LR
either w and x are in the same two-sided cell or x 6≤ w. Let c′ be the
LR
two-sided cell containing x. We then have c′ 6≤ c.
LR
Choose an element h = a C (a ∈ A) in H such that
w∈W w w w C¯
hc′ = w∈c′awCw is nonzeroP. Then we have ϕc′(h) = ϕc′(hc′).
WePclaim that ϕc′(hc′) is nonzero. Let u ∈ c′ be such that au has the
highest degree (as a Laurent polynomial in v) among all a , w ∈ c′.
w
Let d be the distinguished involution such that d and u are in the same
left cell. Then we know that for any distinguished involution d′, the
degree hw,d′,u is less than the degree of hu,d,u if either w 6= u or d′ 6= d
(see [L2, Theorems 1.8 and 1.10]). Thus the degree of awhw,d′,u is less
than the degree of auhu,d,u if either w 6= u or d′ 6= d. Hence ϕc′(h′c) is
nonzero.
Clearly, there are only finitely many q such that ϕc′,q(hc′) is zero
after specializing v to a square root of q. According to Theorem 4 in
[BO], the ring Jc′ is semisimple, that is, its Jacobson radical is zero.
So we can find a nonzero q in C with infinite order and a simple Jc′
module E′ such that ϕc′,q(h) = ϕc′,q(hc′) is nonzero and acts on E′ is
not by zero.
According to [L4, Theorems 4.2 and 4.8], E′ is isomorphic to certain
q
standard module M (s′,n′,ρ) with n′ in the nilpotent orbit C′ corre-
q
sponding to c′. Since c′ 6≤ c, C′ is not in the closure of C (see [B,
LR
Theorem 4 (b)]), by 2.2 (a), the image H of (i ) acts on E′ by zero.
C¯ C¯ ∗ q
This contradicts that ϕc′,q(h) acts on E′ is not by 0. Therefore HC¯ is
contained in the two-sided ideal H≤c.
2.5. In this subsection all tensor products are over Z except other
specifications are given.
Now we show that H≤c⊗Q is equal to H ⊗Q. If C is regular, then
C¯
C¯ is the whole nilpotent cone and the corresponding two-sided cell c
contains the neutral element e, in this case, both H and H≤c are the
C¯
whole Hecke algebra.
We use induction on the partial order ≤ in the set of all two-sided
LR
cells of W. Assume that for all c′ with c ≤ c′ and c′ 6= c, we have
LR
H ⊗Q = H≤c′ ⊗Q, where C′ is the nilpotent orbit corresponding to
C¯′
c′.
6 NANHUAXI
We need to show H ⊗ Q = H≤c ⊗ Q. Let c′ be a two-sided cell
C¯
different from c such that c ≤ c′ but there is no two-sided cell c′′
LR
between c and c′, i.e. no c′′ such that c ≤ c′′ ≤ c′ and c 6= c′′ 6= c′.
LR LR
Let F be an algebraic closure of C(v). We first show that F⊗ H =
A C¯
F ⊗ H≤c. Assume this was not true. Note that F is isomorphic to
A
C (non-canonically), so we can apply the results in [KL2]. By 2.4 and
inductionhypothesis, therewouldexistw ∈ csuch thatC iscontained
w
in F ⊗ H but not in F ⊗ H . By the choice of C′, we know that
A C¯′ A C¯
F⊗ H is the sum of F⊗ H and F ⊗ H for some nilpotent
A C¯′−C′ A C¯ A C¯′′
orbits C′′ with C¯ 6⊆ C¯′′ and C¯′′ 6⊆ C¯ (see [KL2, 5.3(e)]). Since C¯ 6⊆ C¯′′,
by [B, Theorem 4 (b)] we know that C is not in H≤c′′, where c′′ is
w
the two-sided cell corresponding to C′′. By 2.4 we see that F ⊗ H
A C¯′′
does not contain Cw. Thus the image in MC′ = F ⊗A KG×C∗(ZC′) =
F⊗ H /F⊗ H of C is nonzero. According to [KL2, Corollary
A C¯′ A C¯′−C′ w
5.9], each nonzero element in F ⊗A HC¯′ \ F ⊗A HC¯′−C′ acts on MC′ by
nonzero. The argument for [KL2, Proposition 5.13] implies that each
nonzero element in MC′ would have nonzero image in some standard
quotient module of MC′. Thus Cw acts on some standard quotient
module Mv2(s,n′,ρ) of MC′ by nonzero, where n′ ∈ C′.
The homomorphism ϕc′ (see 2.3 for definition) induces a homomor-
phism ϕc′,v2 : F⊗AH → F⊗Jc′. According to [L4, Theorems 4.2 and
4.8], Mv2(s,n′,ρ) is isomorphic to certain Ev2 (defined similar to the Eq
in 2.3) for some simple F⊗Jc′-module E. Note that hw,d,u 6= 0 implies
that u ≤ w. Since C′ is not in the closure of C and w is in the two-sided
LR
c corresponding to C, using [L4, Theorem 4.8] and [B, Theorem 4 (b)],
we see that ϕc′,v2(Cw) = 0. Then Cw acts Mv2(s,n′,ρ) by zero. This
leads to a contradiction. So we have F⊗ H = F⊗ H≤c.
A C¯ A
Thus for each w ∈ c, we can find a nonzero a ∈ F such that aC is
w
in H . Clearly, we must have a ∈ A. Now we show that KG×C∗(Z )
C¯ Y
is a free C[v,v−1]-module for any G-stable locally closed subvariety Y
of N. According to [KL2, 5.3] we may assume that Y is a nilpotent
orbit C. It is enough to show that the completion of KG×C∗(Z ) at
C
any semisimple class in G × C∗ is free over C[v,v−1]. Using [KL2,
5.6] it is enough to show that the right hand side of 5.6(a) in [KL2] is
free. This follows from [KL2, (l3)], the assumption there is satisfied by
[KL2, 4.1]. Using [KL2, 5.3] we know that as free C[v,v−1]-modules
H ⊗C is a direct sum of H ⊗C and KG×C∗(Z ). By assumption,
C¯′ C¯ C¯′−C¯
H ⊗ Q = H≤c′ ⊗ Q, thus H ⊗ Q is a free Q[v,v−1]-module and
C¯′ C¯′
contains C . These imply that if aC ∈ H for some nonzero a ∈ A
w w C¯
then C ∈ H ⊗C. Therefore we can find a nonzero complex number
w C¯
a such that aC is in H . Obviously we have a ∈ Z. Thus H≤c ⊗Q
w C¯
is contained in H ⊗ Q. By 2.4 we then have H≤c ⊗ Q = H ⊗ Q.
C¯ C¯
Theorem 1.5 is proved.
KAZHDAN-LUSZTIG BASIS AND A GEOMETRIC FILTRATION 7
3. Some comments
3.1. If one can show that KG×C∗(Z ) is a free Z-module for any nilpo-
C
tent orbit C, then the argument in 2.5 shows that the image of (i )
C¯ ∗
in H = KG×C∗(Z) contains H≤c, where c is the two-sided cell corre-
sponding to C. Thus Ginzburg’s conjecture would be proved. In fact,
it seems that one can expect more. More precisely, it is likely the
following result is true.
(a)KG×C∗(Z )isafreeA-moduleandKG×C∗(Z )=0forallnilpotent
C 1 C
orbit C. (We refer to [CG, section 5.2] and [Q] for the definition of the
functor KG.)
i
If (a) is true, then we also have
(b) The map (i ) : KG×C∗(Z ) → KG×C∗(Z) is injective.
C¯ ∗ C¯
We explain some evidences for (a) and prove it for G = GL (C),
n
Sp (C) and type G . Let N be a nilpotent element in C and B be
4 2 N
the variety of Borel subalgebras of g containing N. By the Jacobson-
Morozov theorem, there exists a homomorphism ϕ : SL (C) → G
2
0 1 z 0
such that dϕ = N. For z in C∗, let d = . Fol-
(cid:18)0 0(cid:19) z (cid:18)0 z−1(cid:19)
lowing Kazhdan and Lusztig [KL2, 2.4], we define Q = {(g,z) ∈
N
G×C∗ | ad(g)N = z2N}. Then Q is a closed. Let x = (g,z) ∈ Q
N N
act on (G × C∗) × B × B by x(y,b,b′) = (yx−1,ad(g)b,ad(g)b′).
N N
Then Z is isomorphic to the quotient space Q \((G×C∗)×B ×B ).
C N N N
Thus we have KG×C∗(Z ) = KQN(B ×B ) (see [KL2, 5.5] and [Th1,
i C i N N
Prop. 6.2]). It is known that Q = {(g,z) ∈ G × C∗ | gϕ(x)g−1 =
ϕ
ϕ(d xd−1) for all x ∈ SL (C)} is a maximal reductive subgroup of Q
z z 2 N
(see [KL2, 2.4 (d)]). So we have KQN(B ×B ) = KQϕ(B ×B ) (see
i N N i N N
[CG, 5.2.18]).
Let P be the parabolic subgroup of G associated to N (see [DLP,
1.12]). ThenweknowthattheintersectionBN,O ofBN withanyP-orbit
O on B is smooth. The torus D = {ϕ(d ) | z ∈ C∗} is a subgroup of P
z
and acts on BN,O, and BN,O is a vector bundle over the D-fixed point
set BND,O (see [DLP, 3.4 (d)]). Since the action of Qϕ on BN,O commutes
withtheactionofD,accordingto[BB],thisvectorbundleisisomorphic
toaQϕ-stablesubbundleofT(BN,O)|BD ,whereT(BN,O)isthetangent
N,O
bundle of BN,O. Thus the vector bundle is Qϕ-equivariant, so that
the computation of KQϕ(B × B ) is reduced to the computation of
i N N
KQϕ(BD × BD ) for various P-orbits O, O′ on B (see Theorems
i N,O N,O′
2.7 and 4.1 in [Th1], or Theorems 5.4.17 and 5.2.14 in [CG]). Note
that C = {gϕ(d−1)|(g,z) ∈ Q } is a maximal reductive subgroup
ϕ z ϕ
of the centralizer C (N) of N (see [BV, 2.4]) and the map (g,z) →
G
(gϕ(d−1),z) define an isomorphism from Q to C × C∗. Thus we
z ϕ ϕ
have KQϕ(BD × BD ) = KCϕ×C∗(BD × BD ). Now the factor
i N,O N,O′ i N,O N,O′
8 NANHUAXI
C∗ and the group D act on BD × BD trivially, we therefore have
N,O N,O′
KiQϕ(BND,O × BND,O′) = KiCϕ(BND,O × BND,O′) ⊗ RC∗ (see [CG, (5.2.4)],
the argument there works for higher K-groups). Note that we have
identified RC∗ with A = Z[v,v−1]. Thus the statement (a) is equivalent
to the following one.
(c) KCϕ(BD ×BD ) is a free Z-module for i = 0 and is 0 for i = 1.
i N,O N,O′
The statement (c) seems much easier to access. The variety BD
N,O
and its fixed point set Bs,D by any semisimple element s in C are
N,O ϕ
smooth and have good homology properties. See[DLP].
Proposition 3.2. The statement (a) is true for GL (C), Sp (C) and
n 4
type G . In particular, Ginzburg’s conjecture is true in these cases.
2
Proof. We only need to prove statement (c). For G = GL (C),
n
we know that BD has an α-partition into subsets which are affine
N,O
space bundles over the flag variety B′ of C (see Theorem 2.2 and 2.4
ϕ
(a) in [X2]). In this case, (a) is true since we are reduced to compute
KCϕ(B′×B′) (cf. [CG, Lemma 5.5.1]and the argument for [L6, Lemma
i
1.6]). For G = Sp (C) or type G , we know that BD is either empty
4 2 N,O
or the flag variety of C if N is not subregular (see Prop. 4.2 (i)
ϕ
and section 4.4 in [X2]. In this case, we are also reduced to compute
KCϕ(B′ ×B′) (loc.cit), so (a) is true. If N is subregular, then B is a
i N
Dynkincurve andit iseasy tosee thatBD iseither aprojective lineor
N,O
a finite set (see Prop. 4.2 (ii) and section 4.4 in [X2] for a computable
description of B ). The computation for KCϕ(BD ×BD ) is easy, it
N i N,O N,O′
is a free Z-module for i = 0, see 4.3 (b) and 4.4 in [X2], and is 0 for
i = 1 (since this is true for projective line and a finite set). Then we
have the same conclusion for KCϕ(B ×B ) (loc.cit). The proposition
i N N
is proved.
Remark: For GL (C), this proposition also provide an another proof
n
for the main result of [TX], where results in [T1] are used.
Proposition 3.3. Assume that C is connected. Then
ϕ
(a) KCϕ(B ×B ) is a free Z-module.
N N
(b) KQϕ(B ×B ) is a free A-module. That is, KG×C∗(Z ) is a free
N N G.N
A-module.
Proof. Let T be a maximal torus. According to [Th2, (1.11)], we
have split monomorphism KCϕ(B × B ) → KT(B × B ). Similar
N N N N
to the argument for [L6, Lemma 1.13 (d)], we see that KT(B ×B )
N N
is a free R -module. (a) follows.
T
The reasoning for (b) is similar since Q is isomorphic to C ×C∗
ϕ ϕ
and the monomorphism KQϕ(B ×B ) → KT×C∗(B ×B ) is split.
N N N N
The proposition is proved.
KAZHDAN-LUSZTIG BASIS AND A GEOMETRIC FILTRATION 9
Remark: If G = GL (C), then all C are connected and have simply
n ϕ
connected derived group. In this case KQϕ(B × B ) is a free R -
N N Qϕ
module since RQϕ = RCϕ ⊗ A and RT×C∗ is a free RCϕ ⊗ A-module.
Combining this, subsection 2.4 and the argument in subsection 2.5 we
obtain a different proof for the main result in [TX].
3.4. The K-groups KF(B ) and KF(B ×B ) are important in rep-
N N N
resentation theory of affine Hecke algebras for F being Q , C or a
ϕ ϕ
torus of Q (see [KL2, L6]). For the nilpotent element N, in [L4, 10.5]
ϕ
Lusztig conjectures there exists a finite C -set Y which plays a key
ϕ
role in understanding the based ring of the two-sided cell correspond-
ing to G.N. It seems that as R -modules KCϕ(Y) and KCϕ(Y ×Y)
Cϕ
are isomorphic to KCϕ(B ) and KCϕ(B × B ) respectively. Let
N N N
X = B or B × B . In view of [L4, 10.5] one may hope to find a
N N N
canonical Z-basis of KCϕ(X) and a canonical A-basis of KQϕ(X) in
the spirit of [L5, L6]. Moreover, there should exist a natural bijection
between the elements of the canonical basis of KF(B ×B ) (F = C
N N ϕ
or Q ) and the elements of the two-sided cell corresponding to G.N.
ϕ
Acknowledgement: I thank Professor G. Lusztig for very helpful
correspondences and for telling me the argument for the freeness of
KG×C∗(Z ) over C[v,v−1]. I am grateful to Professor T. Tanisaki for
C
a helpful correspondence and to Professor Jianzhong Pan for a helpful
conversation.
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Institute of Mathematics, Chinese Academy of Sciences, Beijing,
100080, China
E-mail address: [email protected]