Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions

Let G be a unitary, symplectic or orthogonal group over a non-archimedean local field of residual characteristic different from 2, considered as the fixed point subgroup in a general linear group G of an involution. Following [7] and [13], we generalize the notion of a semisimple character for G and for G. In particular, following the formalism of [4], we show that these semisimple characters have certain functorial properties. Finally, we show that any positive level supercuspidal representation of G contains a semisimple character.

Let normalF be a non‐Archimedean locally compact field of residue characteristic p, let normalD be a finite‐dimensional central division normalF‐algebra and let ℓ be a prime number different from p. We develop a theory of ℓ‐modular types for the group GLm(normalD), m⩾1, in preparation of the study of the ℓ‐modular smooth representations of this group.

Soit F un corps commutatif localement compact non archimédien, et soit D une algèbrè a division de centre F. Nous prouvons que toute représentation irréductible supercuspidale du groupe GLm(D), de niveau non nul, est l'induite compacte d'une représentation d'un sous-groupe ouvert compact modulo le centre de GLm(D). Plus précisément, nous prouvons que de telles représentations contiennent un type simple maximal au sens de Bushnell et Kutzko.Abstract 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 GLm(D) is compactly induced from a representation of a compact mod centre open subgroup of GLm(D). More precisely, we prove that such representations contain a maximal simple type in the sense of Bushnell and Kutzko.Mots clés : groupe réductif p-adique ; représentation lisse supercuspidale ; type ; paire couvrante ; foncteur de Jacquet ; immeuble de Bruhat-Tits

Let G be a unitary group over a nonarchimedean local field of odd residual characteristic. This paper concerns the study of the "wild part" of the irreducible smooth representations of G, encoded in a so-called "semisimple character". We prove two fundamental results concerning them, which are crucial steps towards a classification of the cuspidal representations of G. First we introduce a geometric combinatoric condition under which we prove an "intertwining implies conjugacy" theorem for semisimple characters, both in G and in the ambient general linear group. Second, we prove a Skolem-Noether theorem for the action of G on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of G which have the same characteristic polynomial must be conjugate under an element of G if there are corresponding semisimple strata which are intertwined by an element of G.It turns out that this condition is also sufficient to obtain an "intertwining implies conjugacy" result:Theorem (Theorem 10.2). Let θ P CpΛ, m, βq and θ 1 P CpΛ 1 , m, β 1 q be semisimple characters which intertwine, let ζ : I Ñ I 1 be the matching given by Theorem 10.1, and suppose that the condition (1.1) holds. Then θ is conjugate to θ 1 by an element ofŨpΛq. Now we turn to our results for classical groups, so we assume that our underlying strata rΛ, q, r, βs are self-dual -that is, β is in the Lie algebra of G and Λ is in the building of the centralizer in G of β (see [BS09]). Our first main result here is a Skolem-Noether theorem for semisimple strata, which is crucial in the sequel [KSS16]:Theorem (Theorem 7.12). Let rΛ, q, r, βs and rΛ 1 , q, r, β 1 s be two self-dual semisimple strata which intertwine in G, and suppose that β and β 1 have the same characteristic polynomial. Then, there is an element g P G such that gβg´1 " β 1 .In order to prove this statement, in Section 4 we analyse the Witt groups W˚pEq of finite field extensions E of F and trace-like maps from W˚pEq and W˚pF q .Given a self-dual semisimple stratum rΛ, q, r, βs, the set C´pΛ, m, βq of semisimple characters for G is obtained by restricting the semisimple characters in CpΛ, m, βq. (Equivalently, one may just restrict those senisimple characters which are invariant under the involution defining G.) Our final result is an "intertwining implies conjugacy" theorem for semisimple characters for G.Theorem (Theorem 10.3). Let θ´P C´pΛ, m, βq and θ 1 P C´pΛ, m, β 1 q be two semisimple characters of G, which intertwine over G, and assume that their matching satisfies (1.1). Then, θ´and θ 1 are conjugate under UpΛq "ŨpΛq X G.