Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T, and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class by a homogeneous line bundle (dominant or anti-dominant) and for products of a Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley formulas are given in terms of the combinatorics of the Littelmann path model. We give explicit computations of products of Schubert classes for the rank two cases and this data allows us to make a "positivity conjecture" generalizing the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B), respectively

Systems of parameters and holonomicity of A-hypergeometric systems

The main result is an elementary proof of holonomicity for A-hypergeometric systems, with no requirements on the behavior of their singularities, originally due to Adolphson [Ado94] after the regular singular case by Gelfand and Gelfand [GG86]. Our method yields a direct de novo proof that A-hypergeometric systems form holonomic families over their parameter spaces, as shown by Matusevich, Miller, and Walther [MMW05]