Push-pull operators in K-theory

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

Last update: 21 February 2013

Push-pull operators in K-theory

For a positive root $\alpha ,$ let $s_{\alpha }\in W$ be the corresponding reflection and define operators $L_{\alpha }:R\left(T\right)\to R\left(T\right)$ and by $T_{\alpha }:R\left(T\right)\to R\left(T\right)$ by

$$L_{\alpha }\left(x\right)=\frac{x-s_{\alpha }x}{1-e^{-\alpha }}\text{and}{T}_{\alpha }\left(x\right)=e^{-\rho }{L}_{\alpha }\left(e^{\rho }x\right)=\frac{e^{\alpha }x-s_{\alpha }x}{e^{\alpha }-1},$$

respectively, where $\rho =\frac{1}{2}{\sum }_{\alpha >0}\alpha \text{.}$ In this section we will show that there is an inductive formula for the classes $\left[{𝒪}_{X_w}\right]$ in terms of the operators $T_{\alpha }$ and the class $\left[{𝒪}_{X_1}\right],$ which was determined in Corollary 2.5.

The operators $T_{\alpha }$ and $L_{\alpha }$ have been in the literature for some time, see for example, [Dem1974, §5], [KKu1990], [FLa1994]. Let $x,y\in R\left(T\right)\text{.}$ Short direct calculations using the definitions establish the following identities:

$$\begin{array}{cc}\text{(3.1a)}& T_{\alpha }\left(xy\right)=x{T}_{\alpha }\left(y\right),\text{if}\hspace{0.17em}{s}_{\alpha }x=x,\\ \text{(3.1b)}& s_{\alpha }{T}_{\alpha }x=T_{\alpha }x,\\ \text{(3.1c)}& T_{\alpha }{T}_{\alpha }x=T_{\alpha }x,\\ \text{(3.1d)}& e^{\lambda }{T}_{\alpha }x=(T_{\alpha }{e}^{s_{\alpha }\lambda }+\frac{e^{\lambda }-e^{s_{\alpha }\lambda }}{1-e^{-\alpha }})x\text{.}\end{array}$$

Because of (a), $T_{\alpha }$ is a map of $R\left(G\right)\text{-modules}$ and so it descends to an operator on $K(G/B)$ which we shall denote by the same symbol. Moreover, the induced operator on $K(G/B)$ satisfies (3.1a-d).

Let $\alpha$ be a simple root and let $P_{\alpha }$ be the minimal parabolic subgroup whose Lie algebra ${𝔭}_{\alpha }$ is generated by $𝔟$ and the negative root space ${𝔤}_{-\alpha }\text{.}$ Since $P_{\alpha }/B\cong {\mathbb{P}}_1,$ the natural projection

$$\begin{array}{cc}{f}_{\alpha }:G/B\to G/P_{\alpha }& \text{(3.2)}\end{array}$$

is a ${\mathbb{P}}_1\text{-bundle.}$

Proposition 3.3. Let $\alpha$ be a simple root. For every $x\in K(G/B),$

$${\left(f_{\alpha }\right)}^{!}\circ {\left(f_{\alpha }\right)}_{!}\left(x\right)=T_{\alpha }\left(x\right)\text{.}$$

This result is proved in [KKu1990, Prop. 4.11]. In section 5 we shall see that the Grothendieck-Riemann-Roch theorem implies that this fact is equivalent to the corresponding fact in cohomology. This alternate point of view has the advantage that it illustrates why the operators $T_{\alpha }$ are the K-theoretic analogues of the BGG operators ${\partial }_{\alpha }$ (see [BGG1973] and [Dem1974]). The proof of following proposition is a generalization of the argument in [FLa1994, p. 728]. Kostant and Kumar [KKu1990, Lemma 4.12] have also proved the same result.

Proposition 3.4. Let $s_{\alpha }\in W$ be the simple reflection corresponding to a simple root $\alpha \text{.}$ Given a Schubert variety $X_w\subseteq G/B,$

$${\left(f_{\alpha }\right)}^{!}\circ {\left(f_{\alpha }\right)}_{!}\left(\left[{𝒪}_{X_w}\right]\right)=\{\begin{array}{cc}\left[{𝒪}_{X_{w{s}_{\alpha }}}\right],& \text{if}\hspace{0.17em}\ell \left(w{s}_{\alpha }\right)>\ell \left(w\right),\\ \left[{𝒪}_{X_w}\right],& \text{if}\hspace{0.17em}\ell \left(w{s}_{\alpha }\right)<\ell \left(w\right)\text{.}\end{array}$$

		Proof.
	The main idea of the proof is $${\left(f_{\alpha }\right)}^{!}\circ {\left(f_{\alpha }\right)}_{!}\left(\left[{𝒪}_{X_w}\right]\right)={\left(f_{\alpha }\right)}^{!}\left(\left[{𝒪}_{f_{\alpha }\left(X_w\right)}\right]\right)=\left[{𝒪}_{f_{\alpha }^{-1}\left(f_{\alpha }\left(X_w\right)\right)}\right]\text{.}$$ One only has to justify the equalities and identify $f_{\alpha }\left(X_w\right)$ and $f_{\alpha }^{-1}\left(f_{\alpha }\left(X_w\right)\right)\text{.}$ For $w\in W$ let $\bar{w}=\{w,w{s}_{\alpha }\}\text{.}$ It is convenient to relabel the elements of the set $\{w,w{s}_{\alpha }\}$ as $w\prime$ and $w\prime \prime$ where by fiat $\ell \left(w\prime \prime \right)=\ell \left(w\prime \right)=1\text{.}$ Analyzing the Bruhat decomposition of $X_w$ in (1.1) we get $$\begin{array}{cc}{f}_{\alpha }\left(X_{w\prime }\right)=f_{\alpha }\left(X_{w\prime \prime }\right)=X_{\bar{w}}\text{and}{f}_{\alpha }^{-1}\left(X_{\bar{w}}\right)=X_{w\prime \prime }\text{.}& \text{(3.5)}\end{array}$$ Since $f_{\alpha }:X_{w\prime }^{\circ }\to X_{\bar{w}}^{\circ }$ is an isomorphism of varieties $f_{\alpha }:X_{w\prime }\to X_{\bar{w}}$ is birational. This combined with the (deep) fact that Schubert varieties have rational singularities (see the survey [Ram1991] and the references there) implies that ${\left(f_{\alpha }\right)}_{*}\left({𝒪}_{X_{w\prime }}\right)={𝒪}_{X_{\bar{w}}},$ and $R^q{\left(f_{\alpha }\right)}_{*}\left({𝒪}_{X_{w\prime }}\right)=0,$ for $q>0\text{.}$ From the Bruhat decomposition one sees that $f_{\alpha }:X_{w\prime \prime }\to X_{\bar{w}}$ is the restriction of the ambient ${\mathbb{P}}_1\text{-bundle}$ $f_{\alpha }:G/B\to G/P_{\alpha }\text{.}$ Thus ${\left(f_{\alpha }\right)}_{*}\left({𝒪}_{X_{w\prime \prime }}\right)={𝒪}_{X_{\bar{w}}},$ and $R^q{\left(f_{\alpha }\right)}_{*}\left({𝒪}_{X_{w\prime \prime }}\right)=0,$ for $q>0\text{.}$ Finally, from (3.5) we have (fα)* (𝒪Xw‾)= 𝒪Xw′′. Statements (a) and (b) imply that ${\left(f_{\alpha }\right)}_{!}\left(\left[{𝒪}_{X_w}\right]\right)=\left[{𝒪}_{f_{\alpha }\left(X_w\right)}\right]$ and (c) implies that ${\left(f_{\alpha }\right)}^{!}\left(\left[{𝒪}_{X_{\bar{w}}}\right]\right)=\left[{𝒪}_{X_{w\prime \prime }}\right]\text{.}$ $\square$

Corollary 3.6. For each simple root ${\alpha }_i$ let $T_i=T_{{\alpha }_i}\text{.}$ Let $w=s_{i_1}\dots s_{i_p}$ be a reduced expression for $w$ and define $T_w=T_{i_1}\dots T_{i_p}\text{.}$ Then $T_w$ is independent of the choice of the reduced expression of $w$ and

$$T_{w^{-1}}\left[{𝒪}_{X_1}\right]=\left[{𝒪}_{X_w}\right]\text{.}$$

		Proof.
	The formula in the statement follows from Propositions 3.3 and 3.4. These two Propositions, combined with formula (3.1c) also show that the action of $T_w$ on the elements of the basis $\{\left[{𝒪}_v\right]\hspace{0.17em}\mid \hspace{0.17em}v\in W\}$ of $K(G/B)$ is independent of the choice of the reduced word for $w\text{.}$ By Proposition 1.5, $K(G/B)$ is a free $R\left(G\right)\text{-module}$ and thus it follows from (3.1c) that, as an operator on $R\left(T\right),$ $T_w$ is independent of the reduced word for $w\text{.}$ $\square$

Notes and References

This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).

Research supported in part by National Science Foundation grant DMS-9622985.

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