On Unitarizability in the Case of Classical p-Adic Groups

Abstract

In [78] we propose an approach to the unitarizability problem in the case of classical groups over a p-adic field of characteristic zero based on cuspidal reducibility points. The unitarizability for these groups is reduced to the case of so called weakly real representations in [73]. Following C. Jantzen, to an irreducible weakly real representation π of a classical group one can attach a sequence (π 1 , . . . , π k ) of irreducible representations of classical groups, each of them supported by a line of cuspidal representations X ρ of general linear groups containing a selfcontragredient representation ρ, and an irreducible cuspidal representation σ of a classical group ([25]). The first question is if π is unitarizable if and only if all π i are unitarizable.Further, the pair ρ, σ determines the non-negative reducibility exponent α ρ,σ ∈ 1 2 Z among ρ and α. The following question is if the unitarizability of irreducible representations supported by X ρ ∪ σ can be described solely in terms of the reducibility point α ρ,σ (see [78] for precise statement). If the answer to the above two questions is positive, then the unitarizability problem for classical p-adic groups would be reduced to a problem of a systems of real numbers.Following the above proposed strategy, in this paper we solve the unitarizability problem for irreducible subquotients of representations Ind G P (τ ), where G is a classical group over a p-adic field of characteristic zero, P is a parabolic subgroup of G of the generalized rank (at most) 3 and τ is an irreducible cuspidal representation of a Levi factor M of P . As a consequence, this gives also a solution of the unitarizability problem for classical p-adic groups of the split rank (at most) three. This paper also provides some very limited support for the possibility of the above approach to the unitarizability could work in general.

Unitary dual of p-adic group SO(7) with support on minimal parabolic subgroup

“…Given the assumption of the theorem, all of the remaining irreducible unitarizable subquotients of the induced representation (4.2) we get using the second part of [22,Theorem 8.13.]. This finishes the proof.…”

“…2 and a 2 = 5 2 , is the irreducible unitarizable subquotient of the induced representation (4.2). Also, using [22,Theorem 8.13. ] we can conclude that the subquotient…”

“…In order to get the complete description of the unitary dual of the p-adic group SO (7) with the support on the minimal parabolic subgroup, we identify unitarizable classes. That was achieved using primarily Tadić's result about the unitarizability of representations of classical p-adic groups of corank three from [22], as well as unitary duals of p-adic general linear groups from [18] and unitary dual of p-adic group SO(5) from [9]. This paper is organized as follows.…”

Uluslararasi Hukukta İnsanliğa Karşi Suçlar

“…"Herhangi bir" ifadesi Tadic davasına göre,insanlığa karşı suçların, failler ile aynı ulusta olanlara karşı yahut farklı ulusta olanlara veya uyruksuzlara karşı işlenmesidir. 38 Diğer bir unsur ise sivilunsurudur. Sivil unsuru kamu düzenini korumak ile görevli olan silahlı personel dışındaki herkesi kapsamaktadır 39 .Sivil nüfus uluslararası ya da uluslararası olmayan çatışmalarda yer almayan, çatışan tarafların yanında yer almayan insanları kapsamaktadır.…”

On representations of reductive p-adic groups over ℚ-algebras

“…In this paper we define and study certain category of smooth modules for reductive p-adic groups analogous to the usual smooth complex representations ( [1,2,3,4,8]). Nowadays there is an active current research in the field of complex representation theory as one can observe from the review articles [18,19]. Representations in positive characteristic are also well understood thanks to the recent works of Henniart, Vignéras and others (see [12]).…”