  An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$
is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M.
Boyarchenko asserts that, in the case where $F$ is a non-archimedean local
field, every irreducible smooth representation of $G$ is admissible and
smoothly induced by a one-dimensional smooth representation of some algebra
subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution
$\sigma:J\to J$, then $\sigma$ naturally defines a group automorphism of $G$,
and we may consider the fixed point subgroup $C_{G}(\sigma)$. Assuming that $F$
has characteristic different from $2$, we extend Boyarchenko's result and show
that every irreducible smooth representation of $C_{G}(\sigma)$ is admissible
and smoothly induced by a one-dimensional smooth representation of a subgroup
of the form $C_{H}(\sigma)$ where $H$ is an $\sigma$-invariant algebra subgroup
of $G$. As a particular case, the result holds for maximal unipotent subgroups
of the classical Chevalley groups defined over $F$.
