Converting to $H_T^{*}(G/B)$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 24 February 2013

Converting to $H_T^{*}(G/B)$

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

$$\begin{array}{cc}{t}_i^2=2,\underset{\underset{m_{ij}\text{factors}}{⏟}}{t_i{t}_j{t}_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{t_j{t}_i{t}_j\dots },x_{\lambda +\mu }=x_{\lambda }+x_{\mu },\text{and}\hspace{0.17em}{x}^{\lambda }{t}_i=t_i{x}_{s_i\lambda }+⟨\lambda ,{\alpha }_i^{\vee }⟩\text{.}& \text{(4.1)}\end{array}$$

The subalgebra of $H_{\text{gr}}$ generated by the $x_{\lambda }$ is the polynomial ring $\mathbb{Z}[x_1,\dots ,x_n],$ where $x_i=x_{{\omega }_i},$ and $W$ acts on $\mathbb{Z}[x_1,\dots ,x_n]$ by

$$w{x}_{\lambda }=x_{w\lambda }\hspace{0.17em}\text{and}\hspace{0.17em}w\left(fg\right)=\left(wf\right)\left(wg\right),\text{for}\hspace{0.17em}w\in W,\lambda \in P,f,g\in \mathbb{Z}[x_1,\dots ,x_n]\text{.}$$

Then the last formula in (4.1) generalizes to

$$f{t}_i=t_i\left(s_{i}f\right)+\frac{f-s_{i}f}{{\alpha }_i},\text{for}\hspace{0.17em}f\in \mathbb{Z}[x_1,\dots ,x_n]\text{.}$$

Let $t_w=t_{i_1}\dots t_{i_p}$ for a reduced word $w=s_{i_1}\dots s_{i_p}$ and let $\mathbb{Z}{W}_{\text{gr}}$ be the subalgebra of $H_{\text{gr}}$ spanned by the $t_w,$ $w\in W\text{.}$ Then

$$\{x_1^{m_1}\dots x_n^{m_n}{t}_w\hspace{0.17em}\mid \hspace{0.17em}w\in W,m_i\in {\mathbb{Z}}_{\ge 0}\}\text{and}\{t_w{x}_1^{m_1}\dots x_n^{m_n}\hspace{0.17em}\mid \hspace{0.17em}w\in W,m_i\in {\mathbb{Z}}_{\ge 0}\}$$

are bases of $H_{\text{gr}}\text{.}$

Let $S=\mathbb{Z}[y_1,\dots ,y_n]$ and extend coefficients to $S$ so that $H_{\text{gr,}S}=S{\otimes }_{\mathbb{Z}}{H}_{\text{gr}}$ and $S[x_1,\dots ,x_n]=S{\otimes }_{\mathbb{Z}}\mathbb{Z}[x_1,\dots ,x_n]$ are $S\text{-algebras.}$ Define $H_T^{*}(G/B)$ to be the $H_{\text{gr,}S}$ module

$$\begin{array}{cc}{H}_T^{*}(G/B)=S\text{-span}\hspace{0.17em}\{\left[X_w\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\},& \text{(4.2)}\end{array}$$

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

$$\begin{array}{cc}{x}_i\left[X_1\right]=y_i\left[X_1\right],\text{and}{t}_i\left[X_w\right]=\{\begin{array}{cc}\left[X_{w{s}_i}\right],& \text{if}\hspace{0.17em}w{s}_i>w,\\ 0,& \text{if}\hspace{0.17em}w{s}_i<w\text{.}\end{array}& \text{(4.3)}\end{array}$$

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

$$\begin{array}{cccc}y:& S[x_1,\dots ,x_n]& ⟶& S\\ & x_i& ⟼& y_i\end{array}$$

so that $H_T^{*}(G/B)\cong H_{\text{gr,}S}{\otimes }_{S[x_1,\dots ,x_n]}y$ as $H_{\text{gr,}S}\text{-modules.}$ Then, using analogous methods to the $K_T(G/B)$ case proves the following theorem, which gives the ring structure of $H_T^{*}(G/B)$ (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

$$\begin{array}{cccccccc}\Phi :& S[x_1,\dots ,x_n]& ⟶& H_{\text{gr,}S}{t}_{w_0}& \hookrightarrow & H_{\text{gr,}S}& ⟶& H_T^{*}(G/B)\\ & f& ⟼& f{t}_{w_0}& & h& ⟼& h\left[X_1\right]\end{array}$$

is surjective with kernel

$$\text{ker}\hspace{0.17em}\Phi =⟨f-y\left(f\right)\hspace{0.17em}\mid \hspace{0.17em}f\in S{[x_1,\dots ,x_n]}^W,⟩$$

the ideal of the ring $S[x_1,\dots ,x_n]$ generated by the elements $f-y\left(f\right)$ for $f\in S{[x_1,\dots ,x_n]}^W\text{.}$ Hence

$$H_T^{*}(G/B)\cong \frac{\mathbb{Z}[y_1,\dots ,y_n,x_1,\dots ,x_n]}{⟨f-y\left(f\right)\hspace{0.17em}\mid \hspace{0.17em}f\in S{[x_1,\dots ,x_n]}^W⟩}$$

has the structure of a ring.

As a vector space $H_{\text{gr}}=\mathbb{Z}[x_1,\dots ,x_n]\otimes \mathbb{Z}{W}_{\text{gr.}}$ Let $\widehat{H_{\text{gr}}}=\mathbb{Q}\left[[x_1,\dots ,x_n]\right]\otimes \mathbb{Q}{W}_{\text{gr}}$ with multiplication determined by the relations in (4.1). Then $\widehat{H_{\text{gr}}}$ is a completion of $H_{\text{gr}}$ (this simply allows us to write infinite sums) and the elements of $\widehat{H_{\text{gr}}}$ given by

$$\begin{array}{cc}\text{ch}\left(X^{\lambda }\right)=\sum _{r\ge 0}\frac{1}{r!}{x}_{\lambda }^r\text{and}\text{ch}\left(T_i\right)=t_i·\frac{x_{{\alpha }_i}}{1-\text{ch}\left(X^{{\alpha }_i}\right)}& \text{(4.5)}\end{array}$$

satisfy the relations of $\stackrel{\sim }{H}$ and thus ch extends to a ring homomorphism $\text{ch}\hspace{0.17em}:\hspace{0.17em}\stackrel{\sim }{H}⟶\widehat{H_{\text{gr}}}\text{.}$ 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 $K_T(G/B)$ and the action of $t_i$ on $H_T^{*}(G/B)$ 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 Grothedieck–Riemann–Roch theorem applied to the map ${\pi }_i\text{.}$ The factor $X{\alpha }_i/(1-\text{ch}\left(X^{{\alpha }_i}\right))$ is the Todd class of the bundle of tangents along the fibers of ${\pi }_i$ (see [Hir1995, p. 91]).

Then $\widehat{H_T^{*}}{(G/B)}_{\mathbb{Q}}=\mathbb{Q}\left[[y_1,\dots ,y_n]\right]{\otimes }_{\mathbb{Z}[y_1,\dots ,y_n]}{H}_T^{*}(G/B)$ is the appropriate completion of $H_T^{*}(G/B)$ to use to transfer the ring homomorphism $\text{ch}\hspace{0.17em}:\hspace{0.17em}{\stackrel{\sim }{H}}_R\to \widehat{H_{\text{gr}}}$ to a ring homomorphism

$$\begin{array}{cc}\text{ch}:\hspace{0.17em}{K}_T(G/B)⟶\widehat{H_T^{*}}{(G/B)}_{\mathbb{Q}}\text{by setting ch}\left(h\left[O_{X_1}\right]\right)=\text{ch}\left(h\right)\left[X_1\right],& \text{(4.6)}\end{array}$$

for $h\in {\stackrel{\sim }{H}}_R\text{.}$ The ring $\widehat{H_T^{*}}{(G/B)}_{\mathbb{Q}}$ is a graded ring with

$$\begin{array}{cc}\text{deg}\left(y_i\right)=1\text{and}\text{deg}\left(\left[X_w\right]\right)=\ell \left(w_0\right)-\ell \left(w\right),& \text{(4.7)}\\ \text{and},\text{for}\hspace{0.17em}w\in W,\text{ch}\left(\left[{𝒪}_{X_w}\right]\right)=\left[X_w\right]+\text{higher degree terms.}& \text{(4.8)}\end{array}$$

In summary, if $e_i=e^{{\omega }_i},$ $X_i=X^{{\omega }_i},$ $y_i=y_{{\omega }_i},$ $x_i=x_{{\omega }_i},$

$$\begin{array}{c}R\left[X\right]=\mathbb{Z}[e_1^{\pm 1},\dots ,e_n^{\pm 1},X_1^{\pm 1},\dots ,X_n^{\pm 1}],\mathbb{Z}\left[X\right]=\mathbb{Z}[X_1^{\pm 1},\dots ,X_n^{\pm 1}],\\ \text{and}\hat{S}[x_1,\dots ,x_n]=\mathbb{Q}\left[[y_1,\dots ,y_n]\right][x_1,\dots ,x_n],\end{array}$$

then there is a commutative diagram of ring homomorphisms

$$\begin{array}{ccc}{K}_T(G/B)=\frac{R\left[X\right]}{⟨f-e\left(f\right)\hspace{0.17em}\mid \hspace{0.17em}f\in R{\left[X\right]}^W⟩}& \stackrel{\text{ch}}{⟶}& H_T^{*}{(G/B)}_{\mathbb{Q}}=\frac{\hat{S}[x_1,\dots ,x_n]}{⟨f-y\left(f\right)\hspace{0.17em}\mid \hspace{0.17em}f\in \hat{S}{[x_1,\dots ,x_n]}^W⟩}\\ \downarrow e_i=1& & \downarrow y_i=0\\ K(G/B)=\frac{\mathbb{Z}\left[X\right]}{⟨f-f\left(1\right)\hspace{0.17em}\mid \hspace{0.17em}f\in \mathbb{Z}{\left[X\right]}^W⟩}& \stackrel{\text{ch}}{⟶}& H^{*}{(G/B)}_{\mathbb{Q}}=\frac{\mathbb{Q}[x_1,\dots ,x_n]}{⟨f-f\left(0\right)\hspace{0.17em}\mid \hspace{0.17em}f\in \mathbb{Q}{[x_1,\dots ,x_n]}^W⟩}\text{.}\end{array}$$

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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