Last update: 21 February 2013
For a positive root let be the corresponding reflection and define operators and by by
respectively, where In this section we will show that there is an inductive formula for the classes in terms of the operators and the class which was determined in Corollary 2.5.
The operators and have been in the literature for some time, see for example, [Dem1974, §5], [KKu1990], [FLa1994]. Let Short direct calculations using the definitions establish the following identities:
Because of (a), is a map of and so it descends to an operator on which we shall denote by the same symbol. Moreover, the induced operator on satisfies (3.1a-d).
Let be a simple root and let be the minimal parabolic subgroup whose Lie algebra is generated by and the negative root space Since the natural projection
Proposition 3.3. Let be a simple root. For every
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 are the K-theoretic analogues of the BGG operators (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 be the simple reflection corresponding to a simple root Given a Schubert variety
The main idea of the proof is
One only has to justify the equalities and identify and
For let It is convenient to relabel the elements of the set as and where by fiat Analyzing the Bruhat decomposition of in (1.1) we get
Since is an isomorphism of varieties is birational. This combined with the (deep) fact that Schubert varieties have rational singularities (see the survey [Ram1991] and the references there) implies that
From the Bruhat decomposition one sees that is the restriction of the ambient Thus
Finally, from (3.5) we have
Statements (a) and (b) imply that and (c) implies that
Corollary 3.6. For each simple root let Let be a reduced expression for and define Then is independent of the choice of the reduced expression of and
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 on the elements of the basis of is independent of the choice of the reduced word for By Proposition 1.5, is a free and thus it follows from (3.1c) that, as an operator on is independent of the reduced word for
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.