The K_+-fixed vectors of Iwahori-spherical GL_n-representations: connections with Zelevinsky's segments

TL;DR

This paper analyzes the fixed vectors of Iwahori-spherical representations of GL_n over non-archimedean fields, linking their decomposition to partitions and Kostka numbers, and provides algorithms for explicit determination.

Contribution

It introduces a combinatorial framework connecting fixed vector decompositions with partitions, Kostka numbers, and algorithms for explicit module occurrence determination.

Findings

Decomposition controlled by partitions dominated by a main partition.

Multiplicity of modules given by Kostka numbers.

Algorithm to determine module occurrences and multiplicities.

Abstract

We study the space of K_+-fixed vectors of Iwahori-spherical representations of GL_n over a non-archimedean local field. For a generic Iwahori-spherical representation, we show that its decomposition into irreducible modules of the finite Lie group K/K_+ is controlled by a partition determined by the representation: an irreducible module occurs only if its partition is dominated by that partition, and when it occurs the multiplicity is a Kostka number. For an arbitrary irreducible Iwahori-spherical representation, we attach a partition from its data and prove a necessary condition: any occurring module must correspond to a partition dominated by this one, and the module attached to the partition itself occurs exactly once. We also give a combinatorial algorithm which, by further computation, determines precisely which modules actually occur and with what multiplicities. This answers a question of Prasad.