On local character expansions for principal series representations of general linear groups

TL;DR

This paper provides explicit formulas for the local character expansion of irreducible p-adic general linear group representations, using degenerate Whittaker models and Kazhdan-Lusztig polynomials, advancing understanding of their structure.

Contribution

It introduces two new explicit formulas for the local character expansion of irreducible representations in principal blocks of p-adic general linear groups, generalizing previous work.

Findings

Explicit formulas in terms of degenerate Whittaker models

Closed expressions involving Kazhdan-Lusztig polynomial values

Extension of previous Iwahori-spherical case results

Abstract

We obtain two explicit formulas for the full local character expansion of any irreducible representation of a p-adic general linear group in principal blocks. The first, generalizing previous work of the author on the Iwahori-spherical case, expresses the expansion in terms of dimensions of degenerate Whittaker models. The second gives a closed expression in terms of values of Kazhdan-Lusztig polynomials of a suitable permutation group.