Notes on representations of non-Archimedean SL(n)

Abstract

Let F be a non-archimedean local field. In this paper the relation between irreducible representations of GL(n, F) and SL(«, F) is studied. Using the results on GL(n, F) a parametrization of (var- ious classes of) irreducible representations of SL(n, F) by parame- ters expressed in terms of cuspidal representations of GL(n, F) is obtained. Introduction. Before we give a more detailed description of the con- tent of this paper, a few historical remarks on SL(n, F) are needed. Gelfand and Naimark gave in (8) proof of the irreducibility of unitary principal series representations of SL(n, C). The same proof gives the irreducibility of unitary principal series for GL(n) over any local field. Using the fact that the unitary principal series have non-trivial Whittaker models for GL(n), and the uniqueness of the model proved by Rodier ((18)), Howe and Silberger proved in (10) that the unitary principal series of GL(n, F) restricted to SL(n, F) are multiplicity free. The same idea appears in Labesse and Langlands paper (14). In this way, Howe and Silberger obtained that unitary principal series representations of SL(n, F) are multiplicity free. Shahidi observed in (20) that one can prove, using the same idea of Whittaker models, that any irreducible tempered representation of GL(n, F) restricted to SL(n , F) is multiplicity free. In this way one obtains that the parabolically induced representation of SL(«, F) by irreducible tem- pered representation is multiplicity free. A general approach to the reducibility and the multiplicities was done by Keys. The structure of the commuting algebras of unitary principal series representations for Chevalley groups was described by him in (11) and it turned out the multiplicities are not always one. This was also shown earlier by Knapp and Zuckerman in (12). Gelbart and Knapp gave in (5) a de- scription of irreducible constituents of the restriction to SL(w, F) of the unitary principal series representations of GL(n, F). Their pa- per (6) is based on two working hypotheses, the second of them is the multiplicity one of the restriction to SL(«,/ Γ) of irreducible represen- tations of GL(/ι, F). Bernstein showed in (1) that any parabolically