Elliptic Springer theory

We introduce an elliptic version of the Grothendieck–Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the resolution of degree-zero, semistable $G$-bundles by degree-zero $B$-bundles over an elliptic curve $E$. From a representation theory perspective, they produce a full embedding of representations of the elliptic or double affine Weyl group into perverse sheaves with nilpotent characteristic variety on the moduli of $G$-bundles over $E$. The resulting objects are principal series examples of elliptic character sheaves, objects expected to play the role of character sheaves for loop groups.

INTEGRABLE FUNCTIONS FOR BERNOULLI MEASURES OF RANK 1 — II

In [Integrable functions for Bernoulli measures of rank 1, Ann. Math. Blaise Pascal17 (2010) 349–363; Mesures p-adiques et suites classiques de nombres. Thèse de Doctorat, Université de Bamako, (2011)] we have begun the study of non-Archimedean integrable functions with respect to the normalized Bernoulli measures of rank 1, denoted by μ1,α. The integrability is within the theory of non-Archimedean integration due to Monna and Springer [Intégration non-archimédienne. II, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math.25 (1963) 643–653; Intégration non-archimédienne. I, Nederl. Akad. Wetensch. Proc. Ser. A 66 = Indag. Math.25 (1963) 634–642], which we will call the Monna–Springer theory. In this paper which contains strictly the results of [Integrable functions for Bernoulli measures of rank 1, Ann. Math. Blaise Pascal17 (2010) 349–363; Mesures p-adiques et suites classiques de nombres. Thèse de Doctorat, Université de Bamako (2011)], we cover the integrability with respect to all normalized Bernoulli measures of rank 1. We show that, the integrable functions are reduced to the continuous functions when α is a p-adic unit different from 1 and -1, and all p-adic functions are integrable for α ∈ {-1, 1}. Again following the Monna–Springer theory, we also show that for the measure μ(5) = ∑α4=1 μ1,α, the space of integrable functions is equal to the space of continuous functions.

A GRAPHICAL DESCRIPTION OF (Dn,An−1) KAZHDAN–LUSZTIG POLYNOMIALS

We give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.