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An irreducibility result in non-zero characteristic. (Un résultat d’irréductibilité en caractéristique non nulle.) (French) Zbl 1064.22003
The purpose of this paper is to prove that if \(F\) is a non-archimedean local field and \(G\) is an inner twist of \(GL_n/F,\) then a representation of \(G(F)\) parabolically induced from a square-integrable representation is irreducible. In the case of zero characteristic this was proved by J. N. Bernstein, P. Deligne, D. A. Kazhdan and M.-F. Vignéras [Représentations des groupes réductifs sur un corps local (1984; Zbl 0583.22009), Theorem B.2.d].
The present paper gives a new proof in the characteristic zero case and a proof in the case of positive characteristic. Novel aspects of the proofs are the reduction of the question of irreducibility to one about orbital integrals using the Paley-Wiener theorem of J. Bernstein, P. Deligne and D. A. Kazhdan [J. Anal. Math. 47, 180–192 (1986; Zbl 0634.22011)] and Kazhdan’s (and Krasner’s) theory of close local fields described in an article of P. Deligne in the book referred to above.

MSC:
22E35 Analysis on \(p\)-adic Lie groups
20G05 Representation theory for linear algebraic groups
20G25 Linear algebraic groups over local fields and their integers
20G30 Linear algebraic groups over global fields and their integers
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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