Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials

TL;DR

This paper expresses the transition matrix of the Casselman basis of Iwahori vectors in terms of deformations of Kazhdan-Lusztig polynomials, providing new functional equations and a novel proof of a matrix-inverse relation.

Contribution

It introduces a new polynomial deformation of Kazhdan-Lusztig R-polynomials to describe the transition matrix in representation theory of p-adic groups, and proves related functional equations.

Findings

Transition matrix expressed via deformed Kazhdan-Lusztig polynomials

New functional equations under algebraic involution

Alternative proof of matrix-inverse relationship

Abstract

A problem in representation theory of $p$-adic groups is the computation of the \textit{Casselman basis} of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials which are deformations of the Kazhdan-Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.