## Converting to ${H}_{T}^{*}\left(G/B\right)$

Last update: 24 February 2013

## Converting to ${H}_{T}^{*}\left(G/B\right)$

The graded nil-Hecke algebra is the algebra ${H}_{\text{gr}}$ given by generators ${t}_{1},\dots ,{t}_{n}$ and ${x}_{\lambda },$ $\lambda \in P,$ with relations

$ti2=2, titjti… ⏟mijfactors = tjtitj… ⏟mijfactors ,xλ+μ= xλ+xμ,and xλti= tixsiλ+ ⟨λ,αi∨⟩. (4.1)$

The subalgebra of ${H}_{\text{gr}}$ generated by the ${x}_{\lambda }$ is the polynomial ring $ℤ\left[{x}_{1},\dots ,{x}_{n}\right],$ where ${x}_{i}={x}_{{\omega }_{i}},$ and $W$ acts on $ℤ\left[{x}_{1},\dots ,{x}_{n}\right]$ by

$wxλ=xwλ and w(fg)= (wf)(wg), for w∈W,λ∈P, f,g∈ℤ [x1,…,xn].$

Then the last formula in (4.1) generalizes to

$fti=ti(sif) +f-sifαi, for f∈ℤ [x1,…,xn].$

Let ${t}_{w}={t}_{{i}_{1}}\dots {t}_{{i}_{p}}$ for a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ and let $ℤ{W}_{\text{gr}}$ be the subalgebra of ${H}_{\text{gr}}$ spanned by the ${t}_{w},$ $w\in W\text{.}$ Then

${ x1m1… xnmntw ∣ w∈W, mi∈ℤ≥0 } and { twx1m1… xnmn ∣ w∈W,mi∈ ℤ≥0 }$

are bases of ${H}_{\text{gr}}\text{.}$

Let $S=ℤ\left[{y}_{1},\dots ,{y}_{n}\right]$ and extend coefficients to $S$ so that ${H}_{\text{gr,}S}=S{\otimes }_{ℤ}{H}_{\text{gr}}$ and $S\left[{x}_{1},\dots ,{x}_{n}\right]=S{\otimes }_{ℤ}ℤ\left[{x}_{1},\dots ,{x}_{n}\right]$ are $S\text{-algebras.}$ Define ${H}_{T}^{*}\left(G/B\right)$ to be the ${H}_{\text{gr,}S}$ module

$HT*(G/B)=S -span {[Xw] ∣ w∈W}, (4.2)$

so that the $\left[{X}_{w}\right],$ $w\in W,$ are an $S\text{-basis}$ of ${K}_{T}\left(G/B\right),$ with ${H}_{\text{gr,}S}\text{-action}$ given by

$xi[X1]= yi[X1], andti[Xw]= { [Xwsi], if wsi>w, 0, if wsi

Let $y$ be the $S\text{-algebra}$ homomorphism given by

$y: S[x1,…,xn] ⟶ S xi ⟼ yi$

so that ${H}_{T}^{*}\left(G/B\right)\cong {H}_{\text{gr,}S}{\otimes }_{S\left[{x}_{1},\dots ,{x}_{n}\right]}y$ as ${H}_{\text{gr,}S}\text{-modules.}$ Then, using analogous methods to the ${K}_{T}\left(G/B\right)$ case proves the following theorem, which gives the ring structure of ${H}_{T}^{*}\left(G/B\right)$ (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

$ker Φ= ⟨ f-y(f) ∣ f∈S [x1,…,xn]W, ⟩$

the ideal of the ring $S\left[{x}_{1},\dots ,{x}_{n}\right]$ generated by the elements $f-y\left(f\right)$ for $f\in S{\left[{x}_{1},\dots ,{x}_{n}\right]}^{W}\text{.}$ Hence

$HT*(G/B)≅ ℤ [ y1,…,yn, x1,…,xn ] ⟨ f-y(f) ∣ f∈S [x1,…,xn]W ⟩$

has the structure of a ring.

As a vector space ${H}_{\text{gr}}=ℤ\left[{x}_{1},\dots ,{x}_{n}\right]\otimes ℤ{W}_{\text{gr.}}$ Let $\stackrel{^}{{H}_{\text{gr}}}=ℚ\left[\left[{x}_{1},\dots ,{x}_{n}\right]\right]\otimes ℚ{W}_{\text{gr}}$ with multiplication determined by the relations in (4.1). Then $\stackrel{^}{{H}_{\text{gr}}}$ is a completion of ${H}_{\text{gr}}$ (this simply allows us to write infinite sums) and the elements of $\stackrel{^}{{H}_{\text{gr}}}$ given by

$ch(Xλ)= ∑r≥0 1r!xλr andch(Ti) =ti· xαi 1-ch(Xαi) (4.5)$

satisfy the relations of $\stackrel{\sim }{H}$ and thus ch extends to a ring homomorphism $\text{ch} : \stackrel{\sim }{H}⟶\stackrel{^}{{H}_{\text{gr}}}\text{.}$ It is this fact that really makes possible the transfer from $K\text{-theory}$ to cohomology possible. Though it is not difficult to check that the elements in (3.5) satisfy the defining relations of $\stackrel{\sim }{H}$ it is helpful to realize that these formulas come from geometry. As explained in [PRa1998], the action of ${T}_{i}$ on ${K}_{T}\left(G/B\right)$ and the action of ${t}_{i}$ on ${H}_{T}^{*}\left(G/B\right)$ are, respectively, the push-pull operators ${\pi }_{i}^{*}{\left({\pi }_{i}\right)}_{!}$ and ${\pi }_{i}^{*}{\left({\pi }_{i}\right)}_{*},$ where if ${P}_{i}$ is a minimal parabolic subgroup of $G$ then ${\pi }_{i}:G/{P}_{i}\to G/B$ is the natural surjection. Then the first formula in (3.5) is the definition of the Chern character, and the second formula is the Grothedieckâ€“Riemannâ€“Roch theorem applied to the map ${\pi }_{i}\text{.}$ The factor $X{\alpha }_{i}/\left(1-\text{ch}\left({X}^{{\alpha }_{i}}\right)\right)$ is the Todd class of the bundle of tangents along the fibers of ${\pi }_{i}$ (see [Hir1995, p. 91]).

Then $\stackrel{^}{{H}_{T}^{*}}{\left(G/B\right)}_{ℚ}=ℚ\left[\left[{y}_{1},\dots ,{y}_{n}\right]\right]{\otimes }_{ℤ\left[{y}_{1},\dots ,{y}_{n}\right]}{H}_{T}^{*}\left(G/B\right)$ is the appropriate completion of ${H}_{T}^{*}\left(G/B\right)$ to use to transfer the ring homomorphism $\text{ch} : {\stackrel{\sim }{H}}_{R}\to \stackrel{^}{{H}_{\text{gr}}}$ to a ring homomorphism

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

for $h\in {\stackrel{\sim }{H}}_{R}\text{.}$ The ring $\stackrel{^}{{H}_{T}^{*}}{\left(G/B\right)}_{ℚ}$ is a graded ring with

$deg(yi)=1and deg([Xw])= ℓ(w0)-ℓ(w), (4.7) and,for w∈W, ch([𝒪Xw])= [Xw]+higher degree terms. (4.8)$

In summary, if ${e}_{i}={e}^{{\omega }_{i}},$ ${X}_{i}={X}^{{\omega }_{i}},$ ${y}_{i}={y}_{{\omega }_{i}},$ ${x}_{i}={x}_{{\omega }_{i}},$

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

then there is a commutative diagram of ring homomorphisms

$KT(G/B)= R[X] ⟨ f-e(f) ∣ f∈R[X]W ⟩ ⟶ch HT* (G/B)ℚ= S^[x1,…,xn] ⟨ f-y(f) ∣ f∈S^ [x1,…,xn]W ⟩ ↓ ei=1 ↓yi=0 K(G/B)= ℤ[X] ⟨ f-f(1) ∣ f∈ℤ[X]W ⟩ ⟶ch H* (G/B)ℚ= ℚ[x1,…,xn] ⟨ f-f(0) ∣ f∈ℚ [x1,…,xn]W ⟩ .$

## 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.