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

Abstract

Cuspidal representations of a reductive p p -adic group G G over a field of characteristic different from p p are relatively injective and projective with respect to extensions that split by a U U -equivariant linear map for any subgroup U 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 G is the product of the subcategories of cuspidal representations and of subrepresentations of direct sums of parabolically induced representations.