ON SPECIAL REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS

Let F be a non-Archimedean locally compact field, and let G be a split connected reductive group over F. For a parabolic subgroup Q subset of G and a ring L, we consider the G-representation on the L-module C-infinity(G/Q,L)/Sigma(Q'not superset of Q) C-infinity(G/Q', L). (*) Let I subset of G denote an Iwahori subgroup. We define a certain free finite rank-L module m (depending on Q; if Q is a Borel subgroup, then (*) is the Steinberg representation and m is of rank 1) and construct an I-equivariant embedding of(*) into C-infinity (I, m). This allows the computation of the I-invariants in (*). We then prove that if L is a field with characteristic equal to the residue characteristic of F and if G is a classical group, then the G-representation (*) is irreducible. This is the analogue of a theorem of Casselman (which says the same for L = C); it had been conjectured by Vigneras. Herzig (for G = GL(n) (F)) and Abe (for general G) have given classification theorems for irreducible admissible modulo p representations of G in terms of super-singular representations. Some of their arguments rely on the present work.