  We review the categorical representation of a Kac-Moody algebra on unipotent
representations of finite unitary groups in non-defining characteristic given
by the authors. Then, we extend this construction to finite reductive groups of
types B or C, in non-defining characteristic. We show that the decategorified
representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce
that the Harish-Chandra branching graph coincides with the crystal graph of
these Fock spaces. We also obtain derived equivalences between blocks, yielding
Broue's abelian defect group conjecture for unipotent l-blocks at linear
primes.
