\(R\)-groups and elliptic representations for \(SL_ n\)
The author continues his investigation of representations of classical \(p\)-adic groups that are parabolically induced from the discrete series. In this paper the group is \(\text{SL}_n (F)\) where \(F\) is a locally compact, non-discrete, non-archimedean field of characteristic zero. The reducibility and number of irreducible components is determined for any representation of \(\text{SL}_n (F)\) induced from a discrete series representation of a Levi subgroup. For this the theory of \(R\)-groups is used, and the \(R\)-groups are computed through restriction from \(\text{GL}_n (F)\), extending results of Gelbart and Knapp. The results are an explicit description of the elliptic tempered representations of \(\text{SL}_n (F)\) and, also, a description of those tempered representations that are not irreducibly induced from elliptic representations.
scientific article; zbMATH DE number 695065