Decomposition rules for the ring of representations of non-Archimedean $GL_n$

TL;DR

This paper investigates the decomposition of products of irreducible representations in the Grothendieck ring of complex smooth representations of p-adic groups, introducing a width invariant and analyzing ladder representations.

Contribution

It introduces a width invariant to analyze product decompositions and establishes multiplicity-one results for products of ladder representations.

Findings

Width invariant provides necessary conditions for product decomposition.

All irreducible factors appear with multiplicity one in products of two ladder representations.

Proposes and verifies a general rule for the composition series of ladder representation products.

Abstract

Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty$, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in $\mathcal{R}$. We obtain a necessary condition on irreducible factors of a given product by introducing a width invariant. Width $1$ representations form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we propose a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.