Motivated by topological quantum field theory, we investigate the geometric
aspects of unitary 2-representations of finite groups on 2-Hilbert spaces,
and their 2-characters. We show how the basic ideas of geometric quantization are
‘categorified’ in this context: just as representations of groups correspond to equivariant
line bundles, 2-representations of groups correspond to equivariant gerbes. We
also show how the 2-character of a 2-representation can be made functorial with
respect tomorphisms of 2-representations.Under the geometric correspondence, the
2-character of a 2-representation corresponds to the geometric character of its associated
equivariant gerbe. This enables us to show that the complexified 2-character
is a unitarily fully faithful functor from the complexified Grothendieck category of
unitary 2-representations to the category of unitary conjugation equivariant vector
bundles over the group.