Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. In this talk we discuss various aspects of representations of Hilbert--Lie groups, in particular of the group U_2(H) of unitary Hilbert--Schmidt perturbations of the identity.
We us covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half-Lie groups that reduces matters to the case where a ``maximal torus'' is fixed, so that compatible weight decompositions can be studied.
We also consider projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from ``bounded extremal weights'', and for these the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists. This leads to projective representations of restricted unitary groups.