On cuspidal representations of 𝑝-adic reductive groups

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Let k be a p-adic field, and G a reductive connected algebraic group over k. Fix a maximal torus T of G which splits in an unramified extension of k, and which has the same split rank as the center of G. For each character 6 of T(k), satisfying some conditions, there is a cuspidal representation y$ of G{k) which is a sum of a finite number of irreducible representations; the correspondence 0 |-*JQ is one-to-one on the orbits of such characters by the little Weyl group of T; furthermore, the formulas for the formal degree of JQ and its character for sufficiently regular elements of T(k) are given: they are formally the same as is the discrete series for real reductive groups. 1. Unramified maximal tori. Let k be a p-adic field, that is a finite extension of Q or a field of formal series over a finite extension of F p . We denote by k the residue field of order q. Let G be a reductive connected algebraic group defined over k, the derived group Gde r of which is simply connected. A maximal torus of G defined over k is called minisotropic if it normalizes no (proper) horocyclic subgroup of G defined over k. LEMMA. Suppose there exists a minisotropic maximal torus T of G which splits in a finite unramified extension L of G. Then the Galois group r of L over k has a unique fixed point v in the apartment of T in the building ofGder(L) [2] ; moreover, the face ofv is minimal amongst the faces in this apartment which are invariant by T. 2. Characters. We conserve notations and hypotheses of §1 and the Lemma. Let 0 be a continuous character of 7\k). For each X £ ^(T), the lattice of rational one-parameter subgroups of T, we define a character 6X of Lby AMS (MOS) subject classifications (1970). Primary 22E50, 20G25; Secondary 20C15.