A filtration on rings of representations of non-Archimedean $GL_n$

TL;DR

This paper introduces a new width-based filtration on the Grothendieck ring of smooth representations of GL_n over p-adic fields, revealing structural properties and multiplicity phenomena for ladder representations.

Contribution

It defines a width invariant that induces a filtration on the representation ring and characterizes ladder representations, providing new insights into their product structure.

Findings

Width invariant induces an increasing filtration on the ring

Ladder representations are exactly those of width 1

Product of ladder representations exhibits multiplicity-one phenomenon

Abstract

Let $F$ be a $p$-adic field. Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\{GL_n(F)\}_{n=0}^\infty$ taken together, with multiplication defined in the sense of parabolic induction. We introduce a width invariant for elements of $\mathcal{R}$ and show that it gives an increasing filtration on the ring. Irreducible representations of width $1$ are precisely those known as ladder representations. We thus obtain a necessary condition on irreducible factors of a product of two ladder representations. For such a product we further establish a multiplicity-one phenomenon, which was previously observed in special cases.