  Using Harish-Chandra induction and restriction, we construct a categorical
action of a Kac-Moody algebra on the category of unipotent representations of
finite unitary groups in non-defining characteristic. We show that the
decategorified representation is naturally isomorphic to a direct sum of level
2 Fock spaces. From our construction we deduce that the Harish-Chandra
branching graph coincide with the crystal graph of these Fock spaces, solving a
recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences
between blocks, yielding Brou\'e's abelian defect groups conjecture for
unipotent $\ell$-blocks at linear primes $\ell$.
