Wednesday 28 September

We will explain an explicit relationship between the specializations of symmetric and nonsymmetric Macdonald polynomials at t = 0 and t = infinity with the graded characters of level-zero Demazure modules. Here level-zero Demazure modules are Demazure-type submodules of an extremal weight module of a level-zero extremal weight over a quantum affine algebra. We will also give a combinatorial description of these specializations in terms of semi-infinite Lakshmibai-Seshadri paths, or equivalently, in terms of quantum Lakshmibai-Seshadri paths.

In this talk, I propose a class of eigenfunctions for the elliptic van Diejen operators

(Ruijsenaars operators of type BC) which are represented by elliptic hypergeometric

integrals of Selberg type.

They are constructed from simple seed eigenfunctions by integral transformations,

thanks to gauge symmetries and kernel function identities of the van Diejen operators.

This talk is based on a collaboration with Farrokh Atai (University of Leeds, UK).

S. Helgason has a paper entitled “Harish-Chandra’s c-function. A Mathematical Jewel”. In his work on spherical functions on p-adic group Macdonald pointed to an analogue of the c-function for p-adic groups. In Lusztig's work on affine Hecke algebras this version of the c-function for p-adic groups appears in the formula for the action of the Demazure-Lusztig operators. In this talk we will explain how the c-function enters into (and simplifies) formulas for Macdonald polynomials: expansions, principal specializations, and norm formulas.

This talk will be about (dual) (p)-canonical bases of quantum groups. Associated to a quantised enveloping algebra is a canonical basis, whose dual basis was discovered to have remarkable multiplicative structures, in what was one of the motivations for the development of the theory of cluster algebras. I will discuss this structure and its categorification, through the lenses of KLR algebras and folding.