The Bernstein decomposition and the Bernstein centre

What is the Bernstein decomposition? In rough terms, it expresses the category R(G) of smooth complex representations of a reductive p-adic group G as the product of certain indecomposable full subcategories, often called the components of R(G). The primary reference is [Ber84] where the decomposition forms a crucial step in a description of the centre of the abelian category R(G). This is the theory of the Bernstein centre which we also discuss below. The notes [BR92] present a somewhat different approach to both the Bernstein decomposition and the theory of the Bernstein centre. These have very strongly influenced the account presented here. In particular, we directly follow [BR92] in expressing each component of R(G) as a module category via the construction of suitable progenerators. (The necessary elementary categorical algebra is reviewed in the body of the chapter.) This breaks into two cases, the cuspidal and the non-cuspidal. The cuspidal case is elementary. The non-cuspidal case, however, is far from elementary. It relies principally on a deep result of Bernstein, often referred to as the Second Adjoint Theorem. I have omitted the long and non-trivial proof of this theorem. A full account, following the approach of [BR92], is to appear in a forthcoming book by David Renard [Ren]. (Note we cannot appeal to Bushnell’s alternative proof of the Second Adjoint Theorem [Bus01] because of its dependence on [Ber84] — see Remark 1.8.1.4 below.) Working within the associated module categories in the two cases, cuspidal and non-cuspidal, we recover Bernstein’s description of the centre of R(G). Here the treatment of the cuspidal case roughly parallels [BR92] but the precise route we have taken was prompted by [BH03, Section 8]. In the non-cuspidal case, our treatment diverges from [BR92] and is closer in spirit to [Ber84]. The introductions to the individual sections give an overview of the stages (and detours) on the way to the final decomposition and the description of the Bernstein centre. In Section 1.2 and in parts of Sections 1.3 and 1.4, I have borrowed heavily from my notes from a beautiful set of lectures given by Robert Kottwitz at the University of Chicago in the early nineties. Finally, I am grateful to David Renard for alerting me to a gap in a preliminary version of this article.