On intertwining implies conjugacy for classical groups

Abstract

Let G be a unitary group of a signed-Hermitian form h given over a non-Archimedian local field k of residue characteristic not two. Let V be the vector space on which h is defined. We consider minimal skew-strata, more precisely pairs (b,a) consisting of a Lie algebra element b and a hereditary order $a$ stable under the adjoint involution of h, such that b generates a field whose multiplicative group is a subset of the normalizer of $a$, and some more conditions. We prove that if two minimal skew-strata (b_i,a), i=1,2 interwine by an element of G, then they are conjugate under G, and we give a natural generalization for minimal semisimple skew-strata.