Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials

0.37 MB·English

by  Masaki Kashiwara

EQUIVARIANT K-THEORY OF AFFINE FLAG MANIFOLDS
AND AFFINE GROTHENDIECK POLYNOMIALS
MASAKI KASHIWARA AND MARK SHIMOZONO
6 Abstract. We study the equivariant K-group of the affine flag manifold with
0 respect to the Borel group action. We prove that the structure sheaf of the
0 (infinite-dimensional) Schubertvarietyin the K-groupis representedby a unique
2
polynomial, which we call the affine Grothendieck polynomial.

n
a
J

4
2 1. Introduction

] Let G be a simply connected semisimple group, B its Borel subgroup, and
G
X = G/B the flag manifold. Its B-orbits are of the form BwB/B for some w
A
in the Weyl group W. Its closure X is called the Schubert variety. It is well-
. w
h known that the equivariant K-group K (X), which is the Grothendieck group
t B
a of the abelian category of coherent B-equivariant O -modules, is a free K (pt)-
m X B
module with {[O ]} as a basis. Note that K (pt) is isomorphic to the group
[ Xw w∈W B
ring Z[P] of the weight lattice P of a maximal torus of B. On the other hand,

2 K (X) ≃ K (G) gives another structure of K (pt)-module on K (X) and we
B B×B B B
v
have a morphism Z[P]⊗Z[P] ≃ K (pt)⊗K (pt) → K (X), which factorsthrough
3 B B B
6 a homomorphism
5
1 (1.1) Z[P]⊗ Z[P] → K (X)
Z[P]W B
0
6 called the equivariant Borel map. Here Z[P]W is the ring of invariants with respect
0
to the action of the Weyl group W. It is also well-known that Z[P]⊗ Z[P] →
/ Z[P]W
h K (X) is an isomorphism. An element in Z[P] ⊗ Z[P] whose image is [O ], is
t B Xw
a
known as a double Grothendieck polynomial when G = SL(n) [9].
m
: The purpose of this paper is to generalize these facts to the affine case. Contrary
v
i to the finite-dimensional case, there are two kinds of flag manifolds; the inductive
X
limit of Schubert varieties BwB/B, each of which is a finite-dimensional projective
r
a variety (see [8] and the references there), and an infinite-dimensional scheme whose
Schubert varieties BwB /B are finite-codimensional subschemes. Here, B is the
− − −
oppositeBorelsubgroup. In[7,8], Kostant-Kumarconsidered thefirst flagmanifold
and studied its equivariant cohomology and K-theory.
In this paper we use the latter flag manifold, which is studied in [3]. We take
the affine flag manifold X = G/B (see §2). It is an infinite-dimensional (not
−
quasi-compact) scheme over C. Its B-orbits are parameterized by the elements of
◦
the Weyl group W. Each B-orbit X is a locally closed subscheme with finite
w
codimension. As a scheme it is isomorphic to A∞ = Spec(C[x ,x ,...]). The flag
1 2
1991 Mathematics Subject Classification. Primary:19L47;Secondary:14M17,17B67, 22E65.
MS is partially supported by NSF DMS-0401012.
1
2 MASAKI KASHIWARAANDMARK SHIMOZONO
manifold X is a union of B-stable quasi-compact open subsets Ω. We define K (X)
B
as the projective limit of K (Ω). Then we have K (X) ∼= K (pt)[O ].
B B w∈W B Xw
Similarly to the finite-dimensional case, we have a homomorphism
Q
(1.2) Z[P]⊗ Z[P] → K (X).
Z[P]W B
In the affine case this morphism is injective but is not surjective; not all [O ] are
Xw
in the image of this morphism. However, as we shall see in this paper, [O ] is in
Xw
the image after a localization.
Let δ be the generator of null roots and let R be the subring of Q(eδ) generated
by e±δ and (enδ −1)−1 (n 6= 0). Tensoring R with (1.2), we have the morphism
(1.3) R ⊗ Z[P] ⊗ Z[P] → R ⊗ K (X).
B
Z[e±δ] Z[P]W Z[e±δ]
Our main result is the following theorem.
Theorem 4.4. For all w ∈ W, [O ] ∈ K (X), considered as an element of
Xw B
R⊗ K (X), is in the image of (1.3).
Z[e±δ] B
Note that we have Z[P]W ≃ Z[PW] in the affine case.
Roughly speaking, we call the element of R⊗ Z[P]⊗ Z[P] corresponding
Z[e±δ] Z[PW]
to [O ] the affine Grothendieck polynomial (see Proposition 4.5).
Xw
In order to prove Theorem 4.4, we use the following vanishing theorem of the
first group cohomology.
Theorem 5.2.
(i) If |I| > 2 then H1(W, R⊗ Z[P]) = 0.
Z[e±δ]
(ii) For any affine Lie algebra g, H1 W,R⊗ ( Zeλ) = 0, where κ∗
Z[e±δ]
is the dual Coxeter number. λ∈P,|hc,λi|<κ∗
(cid:0) L (cid:1)
Here I is the index set of simple roots and c is the canonical central element of g.
The plan of this paper is as follows. In §2 we review the flag manifold of Kac-
Moody Lie algebras. In §3 we study the Demazure operators. We also give a
simple proofofthefact thattheSchubert varieties arenormalandCohen-Macaulay.
This proof seems to be new even in the finite-dimensional case. In §4 we study
the affine flag manifolds. After the preparation in §5, we prove Theorem 5.2 in
§6. As its application, we give in §7 the proof of Theorem 4.4, the existence of
affine Grothendieck polynomials. In §8 we give the character formula of the global
cohomology groups of O (λ) using the affine Grothendieck polynomials. In §9 we
Xw
explain an analogous result for the equivariant cohomology groups of the affine flag
manifolds. In §10 we shall give examples of the affine Grothendieck polynomials.
2. Flag manifolds
Let us recall in this section the definition and properties of the flag manifold of
a symmetrizable Kac-Moody Lie algebra following [3].
Let (a ) be a symmetrizable generalized Cartan matrix, g an associated Kac-
ij i,j∈I
Moody Lie algebra, and t its Cartan subalgebra. Let g = n ⊕ t ⊕ n be the
−
triangular decomposition and g = g the root decomposition. Let ∆ :=
α∈t∗ α
{α ∈ t∗; g 6= 0}\{0} be the set of roots and ∆± the set of positive and negative
α
L
roots.
EQUIVARIANT K-THEORY OF AFFINE FLAG MANIFOLDS 3
Let {α } be the set of simple roots in t∗ and {h } the set of simple co-roots
i i∈I i i∈I
in t. Hence we have hh ,α i = a . Let us take an integral weight lattice P ⊂ t∗.
i j ij
We assume the following conditions:
(i) α ∈ P for all i ∈ I,
i
(ii) {α } is linearly independent,
(2.1)  i i∈I
(iii) h ∈ P∗ for all i ∈ I, where P∗ is the dual lattice Hom(P,Z) ⊂ t,
 i

 (iv) there exists Λ ∈ P such that hh ,Λ i = δ .
i j i ij
Let Tbe the algebraic torus with P as its character lattice. Let W be the Weyl

group. It is the subgroup of Aut(t∗) generated by the simple reflections s (i ∈ I)
i
:s (λ) = λ−hh ,λiα . Let U be the group scheme with n as its Lie algebra. Let
i i i ± ±
B = T ×U be the Borel subgroup, whose Lie algebra is b = t ⊕ n . For any
± ± ± ±
i ∈ I, let us denote by P± the parabolic group whose Lie algebra is b ⊕g . Then
i ± ∓αi
P±/B is isomorphic to the projective line P1.
i ±
Let P+ := {λ ∈ P ; hh ,λi > 0 (i ∈ I)} be the set of dominant integral weights.
i
For λ ∈ P+, let V(λ) (resp. V(−λ)) be the irreducible g-module with highest
weight λ (resp. lowest weight −λ). Then A(g):= V(λ)⊗V(−λ) has an algebra
λ∈P+
structure and we denote Spec(A(g)) by G . TheLscheme G contains a canonical
∞ ∞
point e and is endowed with a left action of P and a right action of P−. The
i i
union of P ···P eP−···P− ⊂ G is an open subset of G and we denote it
i1 im j1 jm ∞ ∞
by G. Then P and P− act freely on G. The flag manifold X is defined as the
i i
quotient G/B . It is a separated (not quasi-compact in general) scheme over C.
−
It is covered by affine open subsets isomorphic to A∞ := Spec(C[x ,x ,...]) (or
1 2
An), and its structure sheaf O is coherent. Let x ∈ X be the image of e ∈ G.
X 0
Then for w ∈ W, wx ∈ X has a sense. The set X has a Bruhat decomposition
0
◦ ◦
X = X , where X is the locally closed subscheme Bwx of X. Let X
w∈W w w 0 w
◦
be the closure of X endowed with the reduced scheme structure. It is called the
F w
Schubert variety. It has codimension ℓ(w), the length of w, and its structure sheaf
◦
O is coherent. As a set we have X = X .
Xw w x>w x
Remark 2.1. (i) For any w, B wB /BF is a finite-dimensional projective sub-
− − −
scheme ofX andits union∪ B wB /B is anind-scheme. This is another
w∈W − − −
flag manifold which we do not use here.
(ii) For a regular dominant integral weight λ, set V(−λ) = V(−λ) where
µ∈P µ
V(−λ) is theweight space ofV(−λ) ofweight µ. Then P(V(−λ)) = (V(−λ)\
µ Q
b
{0})/C× has a scheme structure, and P acts on V(−λ) and P(V(−λ)). Then
i
X is embedded in P(V(−λ)) by x 7→ u , where u isbthe line conbtaining
0 −λ −λ
the lowest weight vector u . b b
−λ
b
Let S be a finite subset of W such that x ∈ S as soon as x 6 y for some y ∈ S.
◦
Then Ω := X is a B-stable quasi-compact open subset which coincides with
S w∈S w
wBx . Conversely, any B-stable quasi-compact open subset is of this form.
w∈S 0S
Let Coh (O ) be the abelian category of coherent B-equivariant O -modules
S B ΩS ΩS
and let K (Ω ) be the Grothendieck group of Coh (O ). For F ∈ Coh (O ),
B S B ΩS B ΩS
let us denote by [F] the corresponding element of K (Ω ). For w ∈ W, the
B S
4 MASAKI KASHIWARAANDMARK SHIMOZONO
sheaf O is a coherent B-equivariant O -module and gives an element [O ] of
Xw X Xw
K (Ω ). The equivariant K-group K (Ω ) is a module over the ring K (pt). For
B S B S B
λ ∈ P, we denote by eλ the element of K (pt) represented by the one-dimensional
B
representationofB givenbyB → T −−e→λ C×. ByP ∋ λ 7→ eλ,K (pt)isisomorphic
B
to the group ring Z[P]. We also denote by eλ the element of Z[P] corresponding
to λ ∈ P.
For w ∈ S, wx ∈ Ω is a T-fixed point. It defines a T-equivariant inclusion
0 S
i : pt → Ω . Since any coherent O -module F has locally a finite resolution by
w S ΩS
locally free modules of finite rank (see Lemma 8.1), the k-th left derived functor
L i∗F vanishes for k ≫ 0. Hence we can define the K (pt)-linear homomorphism
k w T
i∗ : K (Ω ) → K (Ω ) → K (pt) ≃ K (pt)
w B S T S T B
by [F] 7→ ∞ (−1)k[L i∗F]. Note that, similarly to the finite-dimensional case,
k=0 k w
we have
P
(1− eα) if x = w,
(2.2) i∗([O ]) = α∈∆+∩w∆−
x Xw 0 unless x > w.
(
Q
Remark 2.2. The function W → Z[P] given by the equivariant localization x 7→
i∗([O ]), coincides with the function ψw in [7].
x Xw
Lemma 2.3. K (Ω ) is a free K (pt)-module with basis {[O ]} .
B S B Xw w∈S
Proof. Let us argue by induction on the cardinality of S. Let w be a maximal
◦
element of S. Set S′ = S \{w}. Then we have ΩS = ΩS′ ⊔Xw. Hence we have an
exact sequence
◦
KB(Xw) → KB(ΩS) → KB(ΩS′) → 0.
By induction K (Ω ) is a free K (pt)-module with a basis {[O ]} . Also
B S′ B Xx x∈S′
◦
K (X ) is a free K (pt)-module generated by [O ]. Hence K (Ω ) is generated
B w B Xw B S
by {[O ]} . By (2.2) the image of {[O ]} under the map
Xx x∈S Xx x∈S
i∗
K (Ω ) −−−x∈−S−→x K (pt) S
B S B
Q
Q
is linearly independent over K (pt). Here K (pt) S is the product of the copies
B B
of K (pt) parameterized by elements of S. Q (cid:3)
B
Remark 2.4. For ℓ ∈ Z , let n be the direct sum of g where α = m α ∈ ∆+
>0 ℓ α i i i
ranges over positive roots such that m > ℓ. Then n is an ideal of n. Let U be
i i ℓ P ℓ
the normal subgroup of U with n as its Lie algebra. Then for any S as above, U
ℓ ℓ
P
acts on Ω freely for ℓ ≫ 0, and the quotient space U \Ω is a finite-dimensional
S ℓ S
scheme. Hence Ω is the projective limit of {U \Ω } and K (Ω ) is the inductive
S ℓ S ℓ B S
limit of {K (U \Ω )} .
B/Uℓ ℓ S ℓ
We set K (X) = limK (Ω ). Hence we have K (X) = K (pt)[O ].
B ←− B S B B Xw
S w∈W
Forw ∈ W, thehomomorphism i∗ : K (Ω ) → K (pt) indQuces a homomorphism
w B S T
i∗ : K (X) → K (pt).
w B T
EQUIVARIANT K-THEORY OF AFFINE FLAG MANIFOLDS 5
By (2.2) they induce a monomorphism
(2.3) K (X) // // K (pt) W .
B i∗ T
w∈W w Q
Q
For i ∈ I, set X = G/P−. Then there is a canonical projection p : X → X
i i i i
◦ ◦ ◦ ◦ ◦
which is a P1-bundle. We have p (X ) = p (X ) and p−1p (X ) = X ⊔ X
i w i wsi i i w w wsi
◦
for any w ∈ W, and we have the B-orbit decomposition X = ⊔ p (X ).
i i w
w∈W,wsi>w
Similarly to K (X), we define K (X ) as limK (p (Ω )). It is isomorphic to
B B i ←− B i S
S
K (pt)[O ].
B pi(Xw)
w∈W,wsi>w
TQhere exist homomorphisms p : K (X) → K (X ) and p∗: K (X ) → K (X),
i∗ B B i i B i B
defined by [F] 7→ ∞ (−1)k[Rkp F] = [p F] − [R1p F] and [E] 7→ [p∗E],
k=0 i∗ i∗ i∗ i
respectively.
P
Any element λ ∈ P induces a group homomorphism B → T −e→λ C×. Let O (λ)
− X
be the invertible O -module on X = G/B induced by this character of B . Then
X − −
we have ahomomorphismofabeliangroupsZ[P] → K (X)givenby eλ 7→ [O (λ)].
B X
Note that, for λ ∈ P+, we have Γ(X;O (λ)) ≃ V(λ) and Hk(X;O (λ)) = 0 for
X X
k 6= 0 (see [4]).
Similarly to the finite-dimensional case ([1, 8]), we have a commutative diagram
Z[P] // K (X)
B
Di (cid:15)(cid:15) (cid:15)(cid:15) p∗i◦pi∗
Z[P] // K (X).
B
Here D is given by
i
eλ − esiλ−αi
(2.4) D (eλ) = ,
i
1− e−αi
and is called the Demazure operator.
The K (pt)-modulestructure onK (X) induces a K (pt)-linear homomorphism
B B B
K (pt)⊗Z[P] → K (X). Let Z[P]W be the ring of W-invariants of Z[P]. Then
B B
K (pt) ≃ Z[P] are Z[P]W-algebras. The morphism K (pt) ⊗ Z[P] → K (X)
B B B
decomposes as the composition of K (pt)⊗Z[P] → K (pt)⊗ Z[P] and
B B Z[P]W
β: K (pt)⊗ Z[P] → K (X).
B Z[P]W B
It is sometimes called the equivariant Borel map.
Remark 2.5. By [7], the equivariant Borel map K (pt)⊗ Z[P] → K (X) is
B Z[P]W B
an isomorphism if G is a finite-dimensional simply connected semisimple group.
The composition
(2.5) K (pt)⊗ Z[P] −−β→ K (X) −−i∗w→ K (pt) ≃ Z[P]
B Z[P]W B B
is given by a⊗b 7→ a·(wb).
6 MASAKI KASHIWARAANDMARK SHIMOZONO
3. Demazure operators
Let p : X → X be the P1-bundle as in §2. In this section we shall show
i i
[O ] if ws < w,
p∗p ([O ]) = Xwsi i
i i∗ Xw [O ] if ws > w.
( Xw i
Note that if ws > w, then X = p−1p (X ) and p (X ) = p (X ). Moreover,
i w i i w i w i wsi
◦ ◦ ◦
X → p (X ) = p (X ) is an isomorphism.
wsi i wsi i w
Lemma 3.1. We have R1p O = 0 for all w ∈ W.
i∗ Xw
Proof. Assume that ws < w. Since X = p−1p (X ), we have O = p∗O ,
i wsi i i w Xwsi i pi(Xw)
which implies that R1p O = 0. Applying the right exact functor R1p to the
i∗ Xwsi i∗
exact sequence O → O → 0, we obtain R1p O = 0. (cid:3)
Xwsi Xw i∗ Xw
As shown in [3], for any point p ∈ X , there exist an open neighborhood Ω of p
w
and a closed subset S of An for some n such that we have a commutative diagram
∼
X ∩Ω S ×A∞
w
(cid:127)_ (cid:127)_
(cid:15)(cid:15) (cid:15)(cid:15)
Ω ∼ An ×A∞.
Hence various properties of X (such as normality) make sense.
w
Proposition 3.2. For any w ∈ W, we have
(i) X is normal.
w
(ii) p O ≃ O .
i∗ Xw pi(Xw)
Proof. Let us show (i) and (ii) by induction on the length ℓ(w). Note that when
ws > w, (ii) follows from X = p−1p (X ).
i w i i w
When w = e, (i) and (ii) are obvious. Assume that ℓ(w) > 0.
Let us first show (ii). We may assume ws < w. We have a monomorphism
i
◦
j: O = p O ֌ p O , which is an isomorphism on p (X ). By the
pi(Xw) i∗ Xwsi i∗ Xw i w
induction hypothesis, X as well as p (X ) = p (X ) is normal. Hence j is
wsi i w i wsi
globally an isomorphism since p O is a coherent O -module.
i∗ Xw Xi
Next let us show (i). Let O be the normalization of O . Then O is a
Xw Xw Xw
coherent O -module. Let S be the support of O /O . Then S is a B-stable
X Xw Xw
g◦ g
closed subset contained in X \X . We shall show that S is an empty set.
w w
g
Otherwise let x ∈ W be a minimal element such that X ⊂ S. Then x > w. Let
x
us take i ∈ I such that xs < x. Assume first ws > w. Then O = p∗O .
i i Xw i pi(Xw)
HenceO istheinverseimageofthenormalizationofO . HenceS = p−1p (S).
Xw pi(Xw) i i
This contradicts X 6⊂ S. Hence we have ws < w.
xsi i
We hgave monomorphisms O ֌ p O ֌ p O . By the induction hy-
pi(Xw) i∗ Xw i∗ Xw
pothesis, X as well as p (X ) = p (X ) is normal. Since p O is a coherent
wsi i w i wsi i∗ Xw
g ◦
O -module and O → p O → p O are isomorphisms on p (X ), the
pi(Xw) pi(Xw) i∗ Xw i∗ Xw i w
g
g
EQUIVARIANT K-THEORY OF AFFINE FLAG MANIFOLDS 7
normality ofp (X )implies thatO ֌ p O isanisomorphism onX . Hence
i w pi(Xw) i∗ Xw i
we have isomorphisms O −∼→p O −∼→p O .
pi(Xw) i∗ Xw i∗ Xw
We have an exact sequence p O → p Og→ p (O /O ) → R1p O .
i∗ Xw i∗ Xw i∗ Xw Xw i∗ Xw
Since R1p O = 0 by Lemma 3.1, we obtain pg(O /O ) = 0. On the other
i∗ Xw i∗ Xw Xw
hand, since S → X is an isomorphism on pg(X◦ ), the sugpport of p (O /O )
i i x i∗ Xw Xw
◦ g
contains p (X ), which is a contradiction. (cid:3)
i x
g
Corollary 3.3. For any w ∈ W, we have
[O ] if ws < w,
p∗p ([O ]) = Xwsi i
i i∗ Xw [O ] if ws > w.
( Xw i
Proposition 3.4. The module O is Cohen-Macaulay for any w ∈ W.
Xw
Proof. Since X has codimension ℓ(w), Extk (O ,O ) = 0 for k < ℓ(w). Hence
w OX Xw X
it is enough to show that Extk (O ,O ) = 0 for k > ℓ(w). We shall prove it by
OX Xw X
induction on ℓ(w). When w = e, it is obvious. Assume that ℓ(w) > 0.
Set
(3.1) F = τ>ℓ(w)RHom (O ,O ),
OX Xw X
where τ>ℓ(w) is the truncation functor (see e.g. [6]). Let us set S = Supp(F).
Then S is a B-stable closed subset of X . Let x ∈ W be a minimal element of
w
{x ∈ W ; X ⊂ S}. Let us take i ∈ I such that xs < x. If ws > w, then we have
x i i
F ≃ p ∗τ>ℓ(w)RHom (O ,O ),
i OXi pi(Xw) Xi
which implies that p−1p (S) = S. Hence X ⊂ S, which is a contradiction. Hence
i i xsi
we have ws < w.
i
Let Ω be the relative canonical sheaf, which is isomorphic to O (−α ). Then
X/Xi X i
the Grothendieck-Serre duality theorem says that
(3.2) Rp RHom (O ,Ω [1]) ≃ RHom (Rp O ,O ).
i∗ OX Xw X/Xi OXi i∗ Xw Xi
Since Rp O ≃ O by Lemma 3.1andProposition3.2andsince theinduction
i∗ Xw pi(Xw)
hypothesis implies that p (X ) = p (X ) is Cohen-Macaulay, we have
i w i wsi
RHom (Rp O ,O ) ≃ Extℓ(w)−1(O ,O )[1−ℓ(w)].
OXi i∗ Xw Xi OXi pi(Xw) Xi
Hence (3.2) implies that
Rp RHom (O ,Ω ) ≃ Extℓ(w)−1(O ,O )[−ℓ(w)].
i∗ OX Xw X/Xi OXi pi(Xw) Xi
Applying Rp to the distinguished triangle
i∗
Extℓ(w)(O ,Ω )[−ℓ(w)] → RHom (O ,Ω ) → F ⊗Ω −+−→1 ,
OX Xw X/Xi OX Xw X/Xi X/Xi
we obtain a distinguished triangle
Rp Extℓ(w)(O ,Ω )[−ℓ(w)] → Extℓ(w)−1(O ,O )[−ℓ(w)]
i∗ OX Xw X/Xi OXi pi(Xw) Xi
→ Rp (F ⊗Ω ) −+−1→ .
i∗ X/Xi
Hence, taking cohomology groups, we obtain
Rkp (F ⊗Ω )−∼→Rk−ℓ(w)+1p Extℓ(w)(O ,Ω ) ≃ 0
i∗ X/Xi i∗ OX Xw X/Xi
8 MASAKI KASHIWARAANDMARK SHIMOZONO
◦
for k > ℓ(w). On a neighborhood of p (X ), S → X is an embedding, and we have
i x i
Rkp (F ⊗Ω ) ≃ p Hk(F ⊗Ω ). Hence Hk(F ⊗Ω ) = 0 for k > ℓ(w)
i∗ X/Xi i∗ X/Xi X/Xi
◦
on a neighborhood of X , which is a contradiction. (cid:3)
x
4. Affine flag manifolds
In this section we shall consider affine flag manifolds. Let (a ) be an affine
ij ij∈I
Cartan matrix. Let Q = Zα and L = ZΛ . We take L⊕ Q as the integral
i i
i∈I i∈I
weight lattice P. In the LCartan subalgebrLa t = Hom(P,C), we give the simple
coroots h ∈ t by
i
hh ,α i = a , and hh ,Λ i = δ .
i j ij i j ij
Remark 4.1. (i) We have taken L⊕Q as the integral weight lattice P. Let B
denotetheassociatedBorelsubgroup. LetP′ beanotherintegralweight lattice
satisfying (2.1) and let B′ be its associated Borel subgroup. Then there is a
map P → P′ and therefore a morphism B′ → B. Hence we have morphisms
∼
K (pt) → K (pt) and K (pt)⊗ K (X)−→K (X). In this sense, our
B B′ B′ KB(pt) B B′
choice of P is universal.
(ii) However our choice of P can often be realized as a direct sum P′ ⊕S, where
S ⊂ PW and Q ⊂ P′. Then we have KB(X) ≃ Z[S]⊗Z KB′(X).
The Weyl group W acts on P and Q, and we have an exact sequence of W-
modules
0 → Q → P → L → 0,
where L is endowed with the trivial action of W.
Let us set Q = Z α and let δ ∈ Q be the imaginary root such that
+ i∈I >0 i +
{α ∈ Q; hh ,αi = 0 for all i} = Zδ.
P i
Similarly, let us choose c ∈ Z h such that
i∈I >0 i
h ∈ PZh ; hh,α i = 0 for all i = Zc.
i i
i∈I
n o
P
We write
δ = a α and c = a∨h .
i i i i
i∈I i∈I
X X
Then there exists a unique symmetric bilinear form ( , ): P ×P → Q such that
(λ,λ′) = 0 for any λ,λ′ ∈ L,
2(α ,λ)
i
(4.1) hh ,λi = for any λ ∈ P,
i
(α ,α )
i i
hc,λi = (δ,λ) for any λ ∈ P.
Set Q = Q/Zδ. Then W acts on Q . Let us choose 0 ∈ I such that the image
cl cl
W of W in Aut(Q ) is generated by the image of {s } and a = 1. These
cl cl i i∈I\{0} 0
conditions are equivalent to saying that a = 1 and δ−α is a constant multiple of
0 0
a root. Such a 0 exists uniquely up to a Dynkin diagram automorphism. Note that
2 if g 6≃ A(2), (α ,α )
(α ,α ) = 2n and θ:=δ −α ∈ 0 0 ∆+.
0 0 (4 if g ≃ A(2), 0 2
2n
EQUIVARIANT K-THEORY OF AFFINE FLAG MANIFOLDS 9
Let PW denote the space of W-invariant integral weights. We have in the affine
case
Z[PW]−∼→Z[P]W,
since, for any λ ∈ P, either λ ∈ PW or Wλ is an infinite set (see Remark 11.2).
We see easily that PW = {λ ∈ P ; hh ,λi = 0 for all i} is isomorphic to Q by the
i
∼
map η: Q−→PW given by
(4.2) η(β) = β − hh ,βiΛ .
i i
i∈I
X
Hence we have P = L PW and Z[P] ∼= Z[L]⊗Z Z[PW], which implies
KB(pt)⊗Z[P]W Z[P]L∼= Z[P]⊗Z[PW] Z[P] ∼= Z[P]⊗Z Z[L] ∼= Z[L]⊗Z Z[P].
For any w ∈ W, let i∗ ◦β: Z[P]⊗ Z[P] → Z[P] be as in (2.5).
w Z[PW]
Lemma 4.2. The homomorphism induced by the i∗ ◦β’s
w
Z[P]⊗ Z[P] → Z[P] W
Z[PW]
Q
is injective. Here Z[P] W is the product of the copies of Z[P] parameterized by the
elements of W. Q
We shall give the proof of this lemma in §11.
As a corollary (together with (2.3)), we have
Corollary 4.3. The homomorphism
β
K (pt)⊗ Z[P] −→ K (X)
B Z[PW] B
is injective.
Let R be the subring of Q(eδ) generated by e±δ and (enδ −1)−1 (n > 0). Then
we have an injective homomorphism
(4.3) β: R⊗ K (pt)⊗ Z[P] ֌ R⊗ K (X).
Z[e±δ] B Z[PW] Z[e±δ] B
Our main result is the following theorem.
Theorem 4.4. For all w ∈ W, [O ] ∈ K (X), considered as an element of
Xw B
R⊗ K (X), is in the image of the map β in (4.3).
Z[e±δ] B
Hence, K (pt)[O ]mayberegardedasasubmoduleofR ⊗ K (pt)⊗
B Xw B Z[PW]
w∈W Z[e±δ]
Z[P]. L
We shall prove a slightly more precise result.
Let ξ: Z[P] → KB(pt)⊗Z[PW]Z[P] ≃ Z[L]⊗Z Z[P] be the homomorphism given
by
(4.4) ξ(eλ+β) = e−λ ⊗ eλ+β for λ ∈ L and β ∈ Q.
We extend this to
ξ: R⊗ Z[P] ֌ R⊗ K (pt)⊗ Z[P].
Z[e±δ] Z[e±δ] B Z[PW]
Set ρ = Λ . Then κ∗ :=hc,ρi = a∨ is called the dual Coxeter number
i∈I i i∈I i
of g. Let us introduce the W-submodule of Z[P]:
P P
(4.5) Z[P] := Zeλ.
[0,−κ∗)
λ∈P,0>hc,λi>−κ∗
L
10 MASAKI KASHIWARAANDMARK SHIMOZONO
Proposition 4.5. For any w ∈ W, there exists a unique
G ∈ R⊗ Z[P]
w Z[e±δ] [0,−κ∗)
such that β ◦ξ(G ) = [O ].
w Xw
We call G the affine Grothendieck polynomial. The proof of this proposition is
w
given in §7 as an application of Theorem 5.2 below.
Remark 4.6. In the finite-dimensional case, Z[P]W is much bigger than Z[PW],
and the choice of Grothendieck polynomials is not unique. However, in the affine
case G is uniquely determined.
w
Remark 4.7. Let ψ: Z[P] → Z[P] be the homomorphism given by
ψ(eλ+α) = eλ−η(α) for λ ∈ L and α ∈ Q.
Then we have
(4.6) G = ψ(G ) for any w ∈ W.
w−1 w
Indeed, let ϕ: g → g be the Lie algebra homomorphism such that ϕ(e ) = f ,
i i
ϕ(f ) = e and ϕ(h) = −h for h ∈ t. It induces group scheme morphisms ϕ: B →
i i
B and ϕ: B → B. Note that ϕ induces an isomorphism
− −
∼
Z[P] ≃ K (pt) ≃ K (pt)−→K (pt) ≃ K (pt) ≃ Z[P],
T B ϕ∗ B− T
which is given by eλ 7→ e−λ. There exists a unique scheme isomorphism a: G → G
such that a(e) = e and a(bgb−1) = ϕ(b )a(g)ϕ(b)−1 for b ∈ B, b ∈ B and g ∈ G.
− − − −
We have K (X) ≃ K (G), and a: G → G induces a commutative diagram
B B×B−
Z[P] ξ // K (pt)⊗K (pt) β // K (G) ∼ K (X)
B B− B×B− B
ψ k a ψ˜
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Z[P] ξ // K (pt)⊗K (pt) β // K (G) ∼ K (X).
B B− B×B− B
Here k(eλ ⊗ eµ) = e−µ ⊗ e−λ. Indeed, kξ(eλ+α) = k(e−λ ⊗ eλ+α) = e−λ−α ⊗ eλ =
e−λ−α+η(α) ⊗ eλ−η(α) = ξ(eλ−η(α)).
Since a(BwB ) = Bw−1B , we have ψ˜([O ]) = [O ]. which implies (4.6).
− − Xw Xw−1
5. Vanishing of the Weyl group cohomology
Let (W,S) be a Coxeter group, where S is a system of generators. For a subset
S′ of S, let us denote by W the subgroup of W generated by S′.
S′
Lemma 5.1. For any W-module V, we have
{{v } ; v ∈ V satisfy conditions (i) and (ii) below}
H1(W,V) = s s∈S s ,
{{v } ; there exists v ∈ V such that v = (1−s)v}
s s∈S s
(i) (1+s)v = 0,
s
(ii) for any pair of distinct elements s,t ∈ S such that W is a finite group,
{s,t}
(−1)ℓ(w)wv = (−1)ℓ(w)wv .
s t
w∈W{Xs,t},ws>w w∈W{Xs,t},wt>w
Here ℓ: W → Z is the length function.
>0