Regular Bernstein blocks

For a connected reductive group G defined over a non-archimedean local field F, we consider the Bernstein blocks in the category of smooth representations of G ⁢ ( F ) {G(F)} . Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called regular Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of F is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of G ⁢ ( F ) {G(F)} is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of G 0 ⁢ ( F ) {G^{0}(F)} , where G 0 {G^{0}} is a certain twisted Levi subgroup of G. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.

A Künneth Theorem for p-Adic Groups

Let G1 and G2 be p-adic groups. We describe a decomposition of Ext-groups in the category of smooth representations of G1 × G2 in terms of Ext-groups for G1 and G2. We comment on for a supercuspidal representation π of a p-adic group G. We also consider an example of identifying the class, in a suitable Ext1, of a Jacquet module of certain representations of p-adic GL2n.

Canonical Extensions of Harish-Chandra Modules to Representations of G

Let G be the group of R-rational points on a reductive, Zariskiconnected, algebraic group defined over R, let K be a maximal compact subgroup, and let g be the corresponding complexified Lie algebra of G. It is a curious fault of the current representation theory of G that for technical reasons one very rarely works with representations of G itself, but rather with a certain category of simultaneous representations of g and K. The reasons for this are, roughly speaking, that for a given (g,K)-module of finite length there are clearly any number of overlying rather distinct continuous G-representations, whose ‘essence’ is captured by the (g, K)-module alone. At any rate, this paper will propose a remedy for this inconvenience, and define a category of smooth representations of G of finite length which will, I hope, turn out to be as easy to work with as representations of (g, K) and occasionally much more convenient. It is to be considered a report on what has been to a great extent joint work with Nolan Wallach, and is essentially a sequel to [38].