Reductive P-adic Groups Research Articles

On certain Iwahori invariants in the unramified principal series

An affine Hecke algebra is additively the tensor product of a finite dimensional Hecke algebra with the coordinate ring Θ of a complex torus. In this paper we give explicit formulas for eigenvectors of Θ in unramified principal series representations of the reductive p-adic group G associated to H. This leads to new information about intertwining operators, Jacquet modules and submodules of principal series representations

Induced representations of locally profinite groups

Let G be a separable, locally prolinite (i.e., locally compact and totally disconnected) group with centre Z, and L an open subgroup of G containing Z and such that L/Z is compact. This note contains a couple of results on the structure of (smooth) representations of G induced from irreducible (therefore finite-dimensional) representations of L. The first demonstrates the equivalence of various “finiteness” conditions on the induced representation, while the second gives a finiteness property in the general case. An immediate corollary of the second is the fact (well known for p-adic reductive groups) that a given irreducible representation of L can occur in at most finitely many irreducible supercuspidal representations of G. In the case, for example, where G is a general linear group over a nonArchimedean local field and L is a maximal compact mod centre subgroup, it has long been appreciated that an irreducible representation of G induced from L is automatically supercuspidal. Indeed, this observation is the starting point of the current theory of classification of supercuspidals, see, for example, [6]. Our results answer the frequently encountered question as to what happens when the induced representation is not irreducible. As the extreme generality of our hypotheses indicates, the results here are straightforward, even formal, in nature. It is this essential simplicity, as much as the results themselves, which prompts me to write this rather easy little note.

A Proof of Langlands' Conjecture on Plancherel Measures; Complementary Series of -adic groups

analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at least when the group is quasi-split and the inducing representation is generic. Assuming two natural conjectures in harmonic analysis of p-adic groups, we also prove the validity of the conjecture in general (Theorem 9.5). As our second result we obtain all the complementary series and special representations of quasi-split p-adic groups coming from rank-one parabolic subgroups and generic supercuspidal represen

Reduced C∗-algebra for reductive p-adic groups

Reduced C∗-algebra for reductive p-adic groups

Characters of Non-Connected, Reductive p-Abic Groups

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

P-adic measures for spherical representations of reductive P-adic groups

P-adic measures for spherical representations of reductive P-adic groups

Asymptotics and integrability properties for matrix coefficients of admissible representations of reductive p-adic groups

We obtain the necessary and sufficient conditions for the pth power integrability of a matrix coefficient of an admissible representation of a reductive p-adic group in terms of the exponents of the matrix coefficient.

On cuspidal representations of 𝑝-adic reductive groups

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.

All reductive p-adic groups are tame

All reductive p-adic groups are tame