In this paper we verify the local Langlands correspondence for pure inner forms of unramified p-adic groups and tame Langlands parameters in "general position". For each such parameter, we explicitly construct, in a natural way, a finite set ("L-packet") of depth-zero supercuspidal representations of the appropriate p-adic group, and we verify some expected properties of this L-packet. In particular, we prove, with some conditions on the base field, that the appropriate sum of characters of the representations in our L-packet is stable; no proper subset of our L-packets can form a stable combination. Our L-packets are also consistent with the conjectures of B. Gross and D. Prasad on restriction from SO 2n+1 to SO 2n [24].These L-packets are, in general, quite large. For example, Sp 2n has an L-packet containing 2 n representations, of which exactly two are generic. In fact, on a quasi-split form, each L-packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced BruhatTits building. When the group has connected center, every depth-zero generic supercuspidal representation appears in one of these L-packets.We emphasize that there is nothing new about the representations we construct. They are induced from Deligne-Lusztig representations on subgroups of finite index in maximal compact mod-center subgroups, see [42], [44], [61]. The point here is to assemble these representations into L-packets in a natural and explicit way and to verify that these L-packets have the required properties.
Let k denote a complete nonarchimedean local field with finite residue field. Let G be the group of k-rational points of a connected reductive linear algebraic group defined over k. Subject to some conditions, we establish a range of validity for the Harish-Chandra-Howe local expansion for characters of admissible irreducible representations of G. Subject to some restrictions, we also verify two analogues of this result. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. -Soit k un corps local non archimédien de corps résiduel fini. Soit G le groupe des points k-rationnels d'un groupe algébrique linéaire réductif connexe défini sur k. Sous certaines conditions, nous établissons le domaine de validité pour le développement local de Harish-Chandra-Howe pour les caractères des représentations irréductibles admissibles de G. Nous vérifions également deux analogues de ce résultat. 2002 Éditions scientifiques et médicales Elsevier SAS Homogeneity for charactersSuppose that (π, V ) is an irreducible admissible representation of G. We first recall some facts about the character of (π, V ).Let C ∞ c (G) denote the space of complex-valued, locally constant, compactly supported functions on G. Let dg denote a Haar measure on G.dg ✩ Supported by National Science Foundation Postdoctoral Fellowship 98-04375. ANNALES SCIENTIFIQUES DE L'ÉCOLE NORMALE SUPÉRIEURE 0012-9593/02/03/ 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved 392 S. DEBACKERfor v ∈ V . In [14] Harish-Chandra shows that the character distributionis represented on G reg , the set of regular semisimple elements in G, by a function which is locally constant on G reg . We abuse notation and denote by Θ π both the character distribution and the function which represents it. Thus we have, for allSuppose that either G = GL n (k) or k has characteristic zero. Suppose we have a reasonable map ϕ from g to G in a neighborhood of zero. (If k has characteristic zero, then the exponential map will do.) Howe [15] and Harish-Chandra [13] showed that in a sufficiently small neighborhood of zero we have a local expansion for the character Θ π . That is, for all regular semisimple X in g sufficiently close to zero, we have the asymptotic expansionThis expression is referred to as the Harish-Chandra-Howe local expansion. Here O(0) is the set of nilpotent orbits, the c O (π) are complex-valued constants which depend on our orbit O and our representation (π, V ), and the µ O s are functions on the Lie algebra which are determined by O. In other words, sufficiently near the identity, a character is given by a linear combination of functions on g, and these functions are independent of our representation.Let G again denote the group of k-rational points of an arbitrary connected reductive group defined over k. The conjecture of Hales, Moy, and Prasad defines a region on which the Harish-Chandra-Howe local expansion ought to be valid. This predicted region depends on a rational number which Moy and Prasad [21] call ρ(π), the depth of (π, V ). For...
Let k denote a field with nontrivial discrete valuation. We assume that k is complete with perfect residue field. Let G be the group of k-rational points of a reductive, linear algebraic group defined over k. Let g denote the Lie algebra of G. Fix r ∈ R. Subject to some restrictions, we show that the set of distinguished degenerate Moy-Prasad cosets of depth r (up to an equivalence relation) parametrizes the nilpotent orbits in g.
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