Casselman's basis of Iwahori vectors and the Bruhat order

TL;DR

This paper investigates the Casselman basis of Iwahori vectors in p-adic group representations, revealing simplified expressions for certain matrix elements and their inverses, and proposing conjectures related to the Bruhat order.

Contribution

It provides explicit formulas for parts of the matrix associated with the Casselman basis and explores their implications for the Bruhat order, advancing understanding of Iwahori vectors.

Findings

Certain matrix elements have explicit, simplified expressions.

The inverse matrix also admits a nice expression.

Conjectures are proposed relating to the Bruhat order.

Abstract

The Casselman basis of Iwahori fixed vectors in a principal series representation of a p-adic group G is dual to the standard intertwining operators. To compute it one must compute a matrix m(u,v) indexed by pairs of Weyl group elements. This matrix is upper triangular with respect to the Bruhat order. In general this matrix is difficult to compute but it is shown that certain elements have a nice expression. This is also true of the inverse matrix to m(u,v). This leads to interesting conjectures regarding the Bruhat order.