arXiv subjects
Publications and source records attributed to Vincent Secherre.
Let F be a non-Archimedean local field of characteristic 0, and let D be a finite dimensional central division algebra over F. We prove that any unitary irreducible representation of a Levi subgroup of GL(m,D), with m a positive integer, induces irreducibly to GL(m,D). This ends the classification of the unitary dual of GL(m,D) initiated by Tadic.
Let F be a non Archimedean locally compact field of residue characteristic different from 2, let G be a connected reductive group defined over F, let s be an involutive F-automorphism of G and H an open F-subgroup of the fixed points group of s. We denote by G(F) (resp. H(F)) the group of F-points of G (resp. H). In this paper, we obtain an analogue of the Cartan decomposition for the reductive symmetric space G(F)/H(F). More precisely, we obtain a decomposition of G(F) as a union of H(F)-cosets which is related to the H(F)-conjugacy classes of maximal s-anti-invariant F-split tori in G. When G is F-split, we get a more precise result, involving the stabilizer of a special point of the Bruhat-Tits building of G over F.
Let F be a non Archimedean locally compact field and let D be a central F-division algebra. We prove that any positive level supercuspidal irreducible representation of the group GL(m,D) is compactly induced from a representation of a compact mod center open subgroup of GL(m,D). More precisely, we prove that such representations contain a maximal simple type.