Cuspidal representations of reductive p-adic groups are relatively injective and projective

TL;DR

This paper proves that cuspidal representations of reductive p-adic groups are relatively injective and projective, revealing their structural properties and categorical decompositions in the context of smooth representations over suitable fields.

Contribution

It establishes the relative injectivity and projectivity of cuspidal representations and describes the categorical structure of smooth representations over fields with characteristic not dividing the pro-order of G.

Findings

Cuspidal representations are relatively injective and projective.

The category decomposes into cuspidal and parabolically induced subrepresentations.

Provides structural insights into smooth representations over specific fields.

Abstract

Cuspidal representations of a reductive p-adic group G over a field of characteristic different from p are relatively injective and projective with respect to extensions that split by a U-equivariant linear map for any subgroup U that is compact modulo the centre. The category of smooth representations over a field whose characteristic does not divide the pro-order of G is the product of the subcategories of cuspidal representations and of subrepresentations of direct sums of parabolically induced representations.