Model Theory of Hilbert Spaces with a Discrete Group Action

In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is existentially closed, has quantifier elimination, is ${\mathrm{\aleph }}_0$-categorical, ${\mathrm{\aleph }}_0$-stable and SFB. On the other hand, when the group involved is countably infinite, the theory of the Hilbert space expanded by the representation of this group is ${\mathrm{\aleph }}_0$-categorical up to perturbations. Additionally, when the expansion is model complete, we prove that it is ${\mathrm{\aleph }}_0$-stable up to perturbations.