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Let  $G$ be an orthogonal, symplectic or unitary group over a non-archimedean local field of odd residual characteristic. This paper concerns the study of the “wild part” of an irreducible smooth representation of  $G$ , encoded in its “semisimple character”. We prove two fundamental results concerning them, which are crucial steps toward a complete classification of the cuspidal representations of  $G$ . First we introduce a geometric combinatorial 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  $G$ .



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