Push-pull operators in K-theory

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

Last update: 21 February 2013

Push-pull operators in K-theory

For a positive root α, let sαW be the corresponding reflection and define operators Lα:R(T)R(T) and by Tα:R(T)R(T) by

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

respectively, where ρ=12α>0α. In this section we will show that there is an inductive formula for the classes [𝒪Xw] in terms of the operators Tα and the class [𝒪X1], which was determined in Corollary 2.5.

The operators Tα and Lα have been in the literature for some time, see for example, [Dem1974, §5], [KKu1990], [FLa1994]. Let x,yR(T). 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), Tα is a map of R(G)-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 α be a simple root and let Pα be the minimal parabolic subgroup whose Lie algebra 𝔭α is generated by 𝔟 and the negative root space 𝔤-α. Since Pα/B1, the natural projection

fα:G/BG/Pα (3.2)

is a 1-bundle.

Proposition 3.3. Let α be a simple root. For every xK(G/B),

(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 Tα are the K-theoretic analogues of the BGG operators α (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 sαW be the simple reflection corresponding to a simple root α. Given a Schubert variety XwG/B,

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


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 fα(Xw) and fα-1(fα(Xw)).

For wW let w={w,wsα}. It is convenient to relabel the elements of the set {w,wsα} as w and w where by fiat (w)=(w)=1. Analyzing the Bruhat decomposition of Xw in (1.1) we get

fα(Xw)= fα(Xw) =Xwand fα-1(Xw) =Xw. (3.5)

Since fα:XwXw is an isomorphism of varieties fα:XwXw is birational. This combined with the (deep) fact that Schubert varieties have rational singularities (see the survey [Ram1991] and the references there) implies that

  1. (fα)* (𝒪Xw)= 𝒪Xw, and Rq(fα)* (𝒪Xw)=0 , for q>0.

From the Bruhat decomposition one sees that fα:Xw Xw is the restriction of the ambient 1-bundle fα:G/BG/Pα. Thus

  1. (fα)* (𝒪Xw)= 𝒪Xw, and Rq(fα)* (𝒪Xw)=0 , for q>0.

Finally, from (3.5) we have

  1. (fα)* (𝒪Xw)= 𝒪Xw.

Statements (a) and (b) imply that (fα)! ([𝒪Xw])= [𝒪fα(Xw)] and (c) implies that (fα)! ([𝒪Xw]) =[𝒪Xw].

Corollary 3.6. For each simple root αi let Ti=Tαi. Let w=si1sip be a reduced expression for w and define Tw=Ti1Tip. Then Tw is independent of the choice of the reduced expression of w and

Tw-1 [𝒪X1]= [𝒪Xw].


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 Tw on the elements of the basis { [𝒪v] vW } of K(G/B) is independent of the choice of the reduced word for w. By Proposition 1.5, K(G/B) is a free R(G)-module and thus it follows from (3.1c) that, as an operator on R(T), Tw is independent of the reduced word for w.

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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