The ring KT(G/B)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 23 February 2013

The ring KT(G/B)

Let H and [X] be as in (1.5). The trivial representation of H is defined by the homomorphism 1:H given by 1(Ti)=1. The first of the maps

[X] HTw0 HH1 f fTw0 f1

is an H-module isomorphism if the action of H on [X] is given by

Ti·f= Xαif-sif Xαi-1 ,forf[X]. (2.1)

The group algebra of P is

R=-span {eλλP} witheλeμ= eλ+μ, (2.2)

for λ,μP. Extend coefficients to R so that HR=RH and R[X]=R[X] are R-algebras. Define KT(G/B) to be the HR-module

KT(G/B)=R -span {[𝒪Xw]wW}, (2.3)

so that the [𝒪Xw], wW, are an R-basis of KT(G/B), with HR-action given by

Xλ[𝒪X1]= eλ[𝒪X1], andTi [𝒪Xw]= { [𝒪Xwsi], ifwsi>w, [𝒪Xw], ifwsi<w. (2.4)

If R is an R[X]-module via the R-algebra homomorphism given by

e: R[X] R Xλ eλ (2.5)

then as HR-modules, KT(G/B) HRR[X] Re, where Re is the R-rank 1 R[X]-module determined by the homomorphism e.

Let Q be the field of fractions of R and let Q be the algebraic closure of Q. For wW let

bwinQ RKT(G/B) be determined byXλbw= ewλbw,for λP. (2.6)

If the bw exist, then they are a Q-basis of QRKT(G/B) since they are eigenvectors with distinct eigenvalues. If τi, 1in, are the operators on QRKT(G/B) given by

τi=Ti- 11-X-αi, thenb1= [𝒪X1]and τibw=bwsi, forwsi>w, (2.7)

because, a direct computation with relation (1.3) gives that Xλτibw =τiXsiλbw =τiewsiλbw =ewsiλbwsi. Thus the bw,wW, exist and the form of the τ-operators shows that, in fact, they form a Q-basis of QRKT(G/B) (it was not really necessary to extend coefficients all the way to Q). Eqs. (2.6) and (2.7) force

τiτjτi mijfactors = τjτiτj mijfactors ,and the equalityτi2= 1 (Xαi-1) (X-αi-1)

is checked by direct computation using (1.3). Let τw=τi1τip for a reduced word w=si1sip. Then, for wW,

bw=τw-1b1 ,[𝒪Xw]= Tw-1[𝒪X1] and we define[Xw] =εw-1[𝒪X1], (2.8)

where εw is as in (1.11). In terms of geometry, [𝒪Xw] is the class of the structure sheaf of the Schubert variety Xw in G/B and, up to a sign, [Xw] is class of the sheaf Xw determined by the exact sequence 0Xw𝒪Xw𝒪Xw0, where Xw=v<wBvB (see [Mat2000, Theorem 2.1(ii)] and [LSe2003, Eq. (4)]). We are not aware of a good geometric characterization of the basis {[X-λw]wW} of KT(G/B) which appears in the following theorem.

Theorem 2.9. Let λw, wW, be as defined in Theorem 1.7 and let [Xλ] =Xλ[𝒪Xw0] =XλTw0[𝒪X1] for λP. Then the [X-λw],wW, form an R-basis of KT(G/B).


Up to constant multiples, [𝒪Xw0]= Tw0[𝒪X1] is determined by the property

Ti[𝒪Xw0] =Tw0[OX1] ,for all1in. (2.10)

If constants cwQ are given by

[𝒪Xw0]= wWcwbw,

then comparing coefficients of bwsi, for wsi>w, on each side of (2.10) yields a recurrence relation for the cw,

cw=cwsi (11-e-wαi) forwsi>w,

which implies

cw0v-1= αR(v) 11-ew0α (2.11)

via (1.1) and the fact that cw0=1. Thus,

[X-λv]= X-λv [𝒪Xw0]= wWcw e-wλvbw,

and if C,M and A are the W×W matrices given by

C=diag(cw),M= (e-wλv), andA=(azw), wherebw=zW azw[𝒪Xz],

then the transition matrix between the X-λv and the [𝒪Xz] is the product ACM. By (2.8) and the definition of the τi, the matrix A has determinant 1. Using the method of Steinberg [Ste1975] and subtracting row e-sαwλv from row e-wλv in the matrix M allows one to conclude that det(M) is divisible by

αR+ (1-e-α)W/2 and identifyingwW e-wλw= i=1n siw<w e-ωi= (e-ρ)W/2

as the lowest degree term determines det(M) exactly. Thus,

det(ACM)=1· ( wW αR(w) 11-e-α ) ( eραR+ (1-e-α) ) W/2 =(eρ)W/2.

Since this is a unit in R, the transition matrix between the [𝒪Xw] and the X-λv is invertible.

Theorem 2.12. The composite map

Φ: R[X] HRTw0 HR KT(G/B) f fTw0 h h[OX1]

is surjective with kernel

kerΦ= f-e(f) fR[X]W ,

the ideal of the ring R[X] generated by the elements f-e(f) for fR[X]W. Hence

KT(G/B) R[X] f-e(f) fR[X]W

has the structure of a ring.


Since Φ(Xλ)= XλTw0 [OX1]=Xλ [𝒪Xw0], it follows from Theorem 2.9 that Φ is surjective. Thus KT(G/B) R[X]/kerΦ. Let I= f-e(f) fR[X]W . If fR[X]W then, for all λP,

Φ(xλ(f-e(f))) = Xλ(f-e(f)) Tw0[OX1] =XλTw0 (f-e(f)) [OX1] = XλTw0 (e(f)-e(f)) [OX1]=0,

since f-e(f)Z(HR). Thus IkerΦ. The ring KT(G/B)= R[X]/kerΦ is a free R-module of rank W and, by Theorem 1.7, so is R[X]/I. Thus kerΦ=I.

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and the Schubert calculus authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.

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