Abstract
AbstractWe prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a$\mathbb {Z}\left [q^{\pm 1}\right ]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars$e^{\lambda }$, where$\lambda $is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type$E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.
Highlights
Let Qrat be the semi-infinite flag manifold
We distinguish the semi-infinite Schubert variety Q := Q ( ) ⊂ Qrat associated to the identity element e of the affine Weyl group, and call it the semi-infinite flag manifold
The purpose of this paper is to prove a completely explicit, combinatorial inverse Chevalley formula in the equivariant K-group ×C∗ (Q ) in the case of a laced group G and a minuscule weight
Summary
Let Qrat be the semi-infinite flag manifold. This is a reduced ind-scheme whose set of C-valued points is (C (( )))/( (C) · (C (( )))) (see [11] for details), where G is a connected simple algebraic group over C, = ⊂ is a Borel subgroup, H is a maximal torus and N is the unipotent radical of B. The Chevalley formulas of [13, 24, 19] provide the complete analogue for semi-infinite flag manifolds of their previously well-understood K-theory counterparts for the standard Kac–Moody flag varieties [26, 22, 7, 20, 21] In all such formulas, the objective is to expand the tensor product of a Schubert class with an equivariant line bundle, as a linear combination of Schubert classes with equivariant scalar coefficients. One approach to understanding equations (1.1) and (1.2) is that they relate two actions of the group algebra Z[ ] on ×C∗ Qrat , one given by equivariant scalar multiplication (the left-hand side of equation (1.2)) and the other by the tensor product with equivariant line bundles (the left-hand side of equation (1.1)). In the sequel [16] to this paper, we will use Kato’s isomorphism to derive a corresponding inverse Chevalley formula in ( / )
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