The endomorphism ring of projectives and the Bernstein centre

Abstract

Let F be a local non-archimedean field and 𝒪 F its ring of integers. Let Ω be a Bernstein component of the category of smooth representations of GL n (F), let (J,λ) be a Bushnell–Kutzko Ω-type, and let ℨ Ω be the centre of the Bernstein component Ω. Let σ be a direct summand of Ind J GL n (𝒪 F ) λ. We will begin by computing c--Ind GL n (𝒪 F ) GL n (F) σ⊗ ℨ Ω κ(𝔪), where κ(𝔪) is the residue field at maximal ideal 𝔪 of ℨ Ω , and the maximal ideal 𝔪 belongs to a Zariski-dense set in Specℨ Ω .