  Let~$G$ be a unitary group of an~$\epsilon$-hermitian form~$h$ given over a
nonarchimedean local field~$F_0$ of odd residue characteristic. We introduce a
geometric combinatoric condition under which we prove "Intertwining implies
Conjugacy" for semisimple characters of~$G$ and the general linear group of the
ambient vector space of~$G$. Further we prove a Skolem-Noether result for the
action of~$G$ on its Lie algebra, more precisely two Lie algebra elements
of~$G$ which have the same characteristic polynomial over~$F$ must be conjugate
under an element of~$G$ if there are corresponding semisimple characters which
intertwine over an element of~$G$ } 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~$G$.
