A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

TL;DR

This paper extends the understanding of Iwahori-spherical representations of non-Archimedean general linear groups by connecting Whittaker functors, Hecke algebra modules, and character formulas through algebraic structures, providing explicit transition matrices.

Contribution

It generalizes Chan-Savin's method to principal degenerate Whittaker functors and links Harish-Chandra--Howe characters with these functors via Zelevinsky's PSH-algebras.

Findings

Explicit unitriangular transition matrix between character coefficients and Whittaker dimensions.

Generalization of Whittaker functor description to principal degenerate cases.

Connection of Murnaghan's formula with Grothendieck group expansions.

Abstract

For Iwahori-spherical representations of non-Archimedean general linear groups, Chan-Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori-Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan's formula for the Harish-Chandra--Howe character as a Grothendieck group expansion of the same module. Comparing the two approaches through the lens of Zelevinsky's PSH-algebras, we obtain an explicit unitriangular transition matrix between coefficients of the character expansion and the principal degenerate Whittaker dimensions.