## Push-pull operators in K-theory

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

$Lα(x)= x-sαx 1-e-α and Tα(x)= e-ρLα (eρx)= eαx-sαx eα-1 ,$

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:

$(3.1a) Tα(xy)=x Tα(y),if sαx=x, (3.1b) sαTαx= Tαx, (3.1c) TαTαx=Tαx, (3.1d) eλTαx= ( Tαesαλ+ eλ-esαλ 1-e-α ) x.$

Because of (a), ${T}_{\alpha }$ is a map of $R\left(G\right)\text{-modules}$ and so it descends to an operator on $K\left(G/B\right)$ which we shall denote by the same symbol. Moreover, the induced operator on $K\left(G/B\right)$ 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 ${P}_{\alpha }/B\cong {ℙ}_{1},$ the natural projection

$fα:G/B→G/Pα (3.2)$

is a ${ℙ}_{1}\text{-bundle.}$

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

$(fα)!∘ (fα)!(x) =Tα(x).$

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 ${X}_{w}\subseteq G/B,$

$(fα)!∘ (fα)! ([𝒪Xw])= { [𝒪Xwsα], if ℓ(wsα) >ℓ(w), [𝒪Xw], if ℓ(wsα) <ℓ(w).$

 Proof. The main idea of the proof is $(fα)!∘ (fα)! ([𝒪Xw])= (fα)! ([𝒪fα(Xw)])= [𝒪fα-1(fα(Xw))].$ 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 $\stackrel{‾}{w}=\left\{w,w{s}_{\alpha }\right\}\text{.}$ It is convenient to relabel the elements of the set $\left\{w,w{s}_{\alpha }\right\}$ 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 $fα(Xw′)= fα(Xw′′) =Xw‾and fα-1(Xw‾) =Xw′′. (3.5)$ Since ${f}_{\alpha }:{X}_{w\prime }^{\circ }\to {X}_{\stackrel{‾}{w}}^{\circ }$ is an isomorphism of varieties ${f}_{\alpha }:{X}_{w\prime }\to {X}_{\stackrel{‾}{w}}$ is birational. This combined with the (deep) fact that Schubert varieties have rational singularities (see the survey [Ram1991] and the references there) implies that ${\left({f}_{\alpha }\right)}_{*}\left({𝒪}_{{X}_{w\prime }}\right)={𝒪}_{{X}_{\stackrel{‾}{w}}},$ 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 ${f}_{\alpha }:{X}_{w\prime \prime }\to {X}_{\stackrel{‾}{w}}$ is the restriction of the ambient ${ℙ}_{1}\text{-bundle}$ ${f}_{\alpha }:G/B\to G/{P}_{\alpha }\text{.}$ Thus ${\left({f}_{\alpha }\right)}_{*}\left({𝒪}_{{X}_{w\prime \prime }}\right)={𝒪}_{{X}_{\stackrel{‾}{w}}},$ 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 ${\left({f}_{\alpha }\right)}^{!}\left(\left[{𝒪}_{{X}_{\stackrel{‾}{w}}}\right]\right)=\left[{𝒪}_{{X}_{w\prime \prime }}\right]\text{.}$ $\square$

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

$Tw-1 [𝒪X1]= [𝒪Xw].$

 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 $\left\{\left[{𝒪}_{v}\right] \mid v\in W\right\}$ of $K\left(G/B\right)$ is independent of the choice of the reduced word for $w\text{.}$ By Proposition 1.5, $K\left(G/B\right)$ 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.