WHY ANY UNITARY PRINCIPAL SERIES REPRESENTATION OF SLn OVER A p-ADIC FIELD DECOMPOSES SIMPLY

Recently A. Knapp [3] announced that every unitary principal series representation of a semisimple lie group decomposes simply; that is, no two distinct irreducible components of a given unitary principal series representation are equivalent. In proving this result Knapp analyzed in detail the structure of the spaces of intertwining operators for principal series representations. His analysis used a detailed description, due to Harish-Chandra, of the Fourier transform on semisimple Lie groups. In this note we want to prove the analogue of Knapp's result for SLn over a p-adic field. Our proof will be conceptually quite different from Knapp's. Although the case we consider is admittedly quite special, there is hope that this approach to the problem will generalize. Let F be a nonarchimedean local field and let G = GLn(F). Let D be the diagonal subgroup of G and U the upper unipotent matrices. Let B = D U be the group of all nonsingular upper triangular matrices and write 8 for the modular function of B. (If djb is a left Haar measure for B, then d(b)djb is a right Haar measure.) Let A" be a maximal compact subgroup of G and recall that G = KB. Let x be any (unitary) character of D and regard it as a character of B. The induced representation TÏX = Ind^XxS ^) is called the (unitary) principal series representation attached to xTo describe the unitary representation 7rx explicitly we let Hx denote the Hubert space of all complex-valued measurable functions h on G such that h(gb) = X (^)5" / (^)^(^) (g€G,bGB) and such that fK\h(k)\ dk < °°. Then TTX is just left translation in f/x: (*x(x)h) Or) = h(x~ g) (h e Hx; g,xe G). We use the subscript " 1 " to denote the subgroup of Gx = SLn(F) ob-