Unicity of types and local Jacquet--Langlands correspondence

TL;DR

This paper investigates the relationship between the uniqueness of types in representations of inner forms of GL(N) over non-archimedean fields and the local Jacquet--Langlands correspondence, revealing new insights into their interplay.

Contribution

It establishes a connection between the strong unicity property of types and the Jacquet--Langlands correspondence for inner forms of GL(N).

Findings

Demonstrates conditions under which types are unique for representations.

Shows how the unicity of types relates to the Jacquet--Langlands correspondence.

Provides new criteria linking types and correspondence in non-archimedean local fields.

Abstract

Let $F$ be a non-archimedean local field. For any irreducible representation $\pi$ of an inner form $G'=\mathrm{GL}_{m}(D)$ of $G=\mathrm{GL}_{N}(F)$, there exists an irredubile representation of a maximal compact open subgroup in $G'$ which is also a type for $\pi$. Then we can consider the problem whether these types are unique or not in some sense. If such types for $\pi$ are unique, we say $\pi$ has the strong unicity property of types. On the other hand, there exists a correspondence connecting irreducible representations of $G'$ and $G$, called the Jacquet--Langland correspondence. In this paper, we study the ralation between the strong unicity of types and the Jacquet--Langlands correspondence.