Converting to HT*(G/B)

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

Last update: 24 February 2013

Converting to HT*(G/B)

The graded nil-Hecke algebra is the algebra Hgr given by generators t1,,tn and xλ, λP, with relations

ti2=2, titjti mijfactors = tjtitj mijfactors ,xλ+μ= xλ+xμ,and xλti= tixsiλ+ λ,αi. (4.1)

The subalgebra of Hgr generated by the xλ is the polynomial ring [x1,,xn], where xi=xωi, and W acts on [x1,,xn] by

wxλ=xwλ andw(fg)= (wf)(wg), forwW,λP, f,g [x1,,xn].

Then the last formula in (4.1) generalizes to

fti=ti(sif) +f-sifαi, forf [x1,,xn].

Let tw=ti1tip for a reduced word w=si1sip and let Wgr be the subalgebra of Hgr spanned by the tw, wW. Then

{ x1m1 xnmntw wW, mi0 } and { twx1m1 xnmn wW,mi 0 }

are bases of Hgr.

Let S=[y1,,yn] and extend coefficients to S so that Hgr,S=SHgr and S[x1,,xn]= S [x1,,xn] are S-algebras. Define HT*(G/B) to be the Hgr,S module

HT*(G/B)=S -span {[Xw]wW}, (4.2)

so that the [Xw], wW, are an S-basis of KT(G/B), with Hgr,S-action given by

xi[X1]= yi[X1], andti[Xw]= { [Xwsi], ifwsi>w, 0, ifwsi<w. (4.3)

Let y be the S-algebra homomorphism given by

y: S[x1,,xn] S xi yi

so that HT*(G/B) Hgr,S S[x1,,xn] y as Hgr,S-modules. Then, using analogous methods to the KT(G/B) case proves the following theorem, which gives the ring structure of HT*(G/B) (see also the proof of [KRa2002, Prop. 2.9] for the same argument with (non-nil) graded Hecke algebras).

Theorem 4.4. The composite map

Φ: S[x1,,xn] Hgr,Stw0 Hgr,S HT*(G/B) f ftw0 h h[X1]

is surjective with kernel

kerΦ= f-y(f) fS [x1,,xn]W,

the ideal of the ring S[x1,,xn] generated by the elements f-y(f) for fS[x1,,xn]W. Hence

HT*(G/B) [ y1,,yn, x1,,xn ] f-y(f) fS [x1,,xn]W

has the structure of a ring.

As a vector space Hgr= [x1,,xn] Wgr. Let Hgr^= [[x1,,xn]] Wgr with multiplication determined by the relations in (4.1). Then Hgr^ is a completion of Hgr (this simply allows us to write infinite sums) and the elements of Hgr^ given by

ch(Xλ)= r0 1r!xλr andch(Ti) =ti· xαi 1-ch(Xαi) (4.5)

satisfy the relations of H and thus ch extends to a ring homomorphism ch:HHgr^. It is this fact that really makes possible the transfer from K-theory to cohomology possible. Though it is not difficult to check that the elements in (3.5) satisfy the defining relations of H it is helpful to realize that these formulas come from geometry. As explained in [PRa1998], the action of Ti on KT(G/B) and the action of ti on HT*(G/B) are, respectively, the push-pull operators πi*(πi)! and πi*(πi)*, where if Pi is a minimal parabolic subgroup of G then πi:G/PiG/B is the natural surjection. Then the first formula in (3.5) is the definition of the Chern character, and the second formula is the Grothedieck–Riemann–Roch theorem applied to the map πi. The factor Xαi/(1-ch(Xαi)) is the Todd class of the bundle of tangents along the fibers of πi (see [Hir1995, p. 91]).

Then HT*^ (G/B)= [[y1,,yn]] [y1,,yn] HT*(G/B) is the appropriate completion of HT*(G/B) to use to transfer the ring homomorphism ch:HRHgr^ to a ring homomorphism

ch:KT(G/B) HT*^ (G/B) by setting ch(h[OX1]) =ch(h)[X1], (4.6)

for hHR. The ring HT*^(G/B) is a graded ring with

deg(yi)=1and deg([Xw])= (w0)-(w), (4.7) and,forwW, ch([𝒪Xw])= [Xw]+higher degree terms. (4.8)

In summary, if ei=eωi, Xi=Xωi, yi=yωi, xi=xωi,

R[X]= [ e1±1,, en±1, X1±1,, Xn±1 ] ,[X]= [ X1±1,, Xn±1 ] , and S^ [x1,,xn]= [[y1,,yn]] [x1,,xn],

then there is a commutative diagram of ring homomorphisms

KT(G/B)= R[X] f-e(f) fR[X]W ch HT* (G/B)= S^[x1,,xn] f-y(f) fS^ [x1,,xn]W ei=1 yi=0 K(G/B)= [X] f-f(1) f[X]W ch H* (G/B)= [x1,,xn] f-f(0) f [x1,,xn]W .

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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