On $p$-adic Speh representations

Abstract

This note contains simple proofs of some known results (unitarity, character formula) on Speh representations of a group GL n (D) where D is a local non Archimedean division algebra of any characteristic. Introduction. In this note I give simple proofs of some known results on Speh representations of groups GL n (D) where D is a central division algebra of finite dimension over a local non archimedean field F of any characteristic. The new idea is to use the Moeglin-Waldspurger algorithm (MWA) for computing the dual of an irreducible representation. For the unitarizability of Speh representations , previous proofs were based either on the trace formula and the close fields theory, or on deep results of type theory. The proof here is combinatoric , independent of D, the characteristic , and Bernstein's (also called U0) theorem. Short general proofs using MWA are also given for other known facts. The proof is always the same: one wants to prove a relation (R) involving some representations (for example an induced representation π is irreducible). One starts by writing the naive relation (R') between these representations, known from standard theory, but not as strong as (R) (for example the semi simplification of π is a sum with non negative coefficients of some irreducible representations). Usually (R') has more terms that (R), because it is weaker, and one wants to prove that some terms which are supposed to appear in (R') are actually not there (for example all the subquotients of π except of the expected one have coefficient zero). The method then is to consider also the dual relation (R ) to (R'), and to play with the MWA, in order to show that the mild constraints one has on (R') and (R ) are enough to show that the extra terms are not there and (R') reduces actually to (R). All the results in this paper are already known, and here I only give new short proofs. So I kindly ask the reader, when using one of these facts, to quote, at least in first place, the original reference (see the historical notice at the end of the paper). Beside the Zelevinsky and Tadi´cTadi´c classification of the admissible dual ([Ze], [Ta2]), the proofs here rely on [Au] (dual of an irreducible representation is irreducible), [MW] and [BR2] (algorithm for computing the dual) and some easy tricks from [Ta1], [Ba3] and [CR], and do not involve any complicated technique. The idea of searching for simple proof for classification of unitary representations, together with a list of basic tricks to use, are due to Marko Tadi´cTadi´c ([Ta3] for example). He also was the first to formulate some properties of Speh representations (formula for ends of complimentary series, character formula, Speh representations are prime elements in the ring of representations, dual of a Speh representation is Speh) and to prove them when D = F. The starting point of my proof here of the assertion Speh representations are unitary is also due to Tadi´cTadi´c who found the simple but brilliant trick reducing the problem of unitarity to a problem of irreducibility. I would like to thank Guy Henniart who read the paper and made useful observations.