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

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.