  In this paper, we construct a categorical double quantum Heisenberg action on
the representation category of finite classical groups $\mathrm{O}_{2n+1}(q)$,
$\mathrm{Sp}_{2n}(q)$ and $\mathrm{O}^{\pm}_{2n}(q)$ with $q$ odd. Over a field
of characteristic zero or characteristic $\ell$ with $\ell\nmid q(q-1)$, we
deduce a categorical action of a Kac-Moody algebra
$\mathfrak{s}\mathfrak{l}'_{I_+}\oplus\mathfrak{s}\mathfrak{l}'_{I_-}$ on the
representation category of finite classical groups. We show that the colored
weight functions $\mathbb{O}^+(u)(\bullet)$, $\mathbb{O}^-(v)(\bullet)$ and
uniform projection can distinguish all irreducible characters of finite
classical groups. In particular, the colored weight functions are complete
invariants of quadratic unipotent characters. We also show that using the theta
correspondence and extra symmetries of categorical double quantum Heisenberg
action, the Kac-Moody action on the Grothendieck group of the whole category
can be determined explicitly.
