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Affine Weyl groups in $$K$$-Theory and representation theory. (English) Zbl 1137.14037
Let $$G$$ be a simply connected, simple, complex Lie group, $$B$$ a Borel subgroup of $$G$$ and $$T$$ a maximal torus chosen in such a way that $$T\subset B\subset G.$$ Let $$K_{T}(G/B)$$ be the equivariant $$K$$-theory of a generalized flag variety $$G/B.$$ It is known [cf. B. Kostant and S. Kumar, J. Differ. Geom. 32, No. 2, 549–603 (1990; Zbl 0731.55005)] that $$K_{T}(G/B)$$ is a free module over the representation ring $$R(T)$$ with the basis given by the classes $$[{\mathcal O}_{X_w}]$$ of structure sheaves of Schubert varieties. Here $$w\in W$$ is an element of the corresponding Weyl group. The classical $$K$$-theory Chevalley formula describes the product of the basis element $$[{\mathcal O}_{X_w}]$$ with the class $$[L_{\lambda}]$$ of the line bundle corresponding to the character of $$B$$ determined by $$-{\lambda}$$ [cf. W. Fulton and A. Lascoux, Duke Math. J. 76, No. 3, 711–729 (1994; Zbl 0840.14007); C. Lenart, J. Pure Appl. Algebra 179, No. 1–2, 137–158 (2003; Zbl 1063.14060); H. Pittie and A. Ram, Electron. Res. Announc. Am. Math. Soc. 5, No. 14, 102–107 (1999; Zbl 0947.14025)].
The authors develop the generalization of this for $$K_{T}(G/P),$$ where $$P$$ is a parabolic subgroup of $$G.$$ This Chevalley-type formula allows one to describe a simple combinatorial model for the characters of irreducible representations of $$G$$ and more generally for the Demazure characters. The construction of this model involves a collection of operators which satisfy Yang-Baxter equation. Combinatorially it relies on decompositions in the affine Weyl group and enumeration of saturated chains in the Bruhat order on the nonaffine Weyl group. The authors call this model the alcove path model because it can be viewed as a discrete counterpart of the Littlemann path model [cf. P. Littelmann, Invent. Math. 116, No. 1–3, 329–346 (1994; Zbl 0805.17019), Ann. Math. (2) 142, No. 3, 499–525 (1995; Zbl 0858.17023)]. Based on this model the authors derive simple formula for multiplication of an arbitrary Schubert class by a divisor class and also a dual Chevalley-type formula.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 19E08 $$K$$-theory of schemes
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