Let L be the Lie algebra of smooth sections of a Lie algebra bundle over the manifold X whose typical fibre is a compact semisimple Lie algebra. We endow L with its natural Frechet topology and the ideal L_c of compactly supported smooth sections with its natural direct limit LF-topology . A bounded unitary representation is a continuous homomorphism into the Lie algebra u(H) of bounded skew-hermitian operators on a Hilbert space H.

In this talk we describe a classification of all bounded irreducible unitary representations of L and L_c. Due to the rather coarse topology, for L, the result is rather simple: the irreducible bounded representations are the finite tensor products of evaluation representations (compositions of a representation of a fiber with an evaluation in a point x). For the Lie algebra L_c also infinite tensor products occur, and this lead to a bounded representation theory that is ``wild'' in the sense that factor representations of type II and III occur.

If Γ is a finite group acting on the compact Lie algebra K and freely on the smooth manifold Y and accordingly on the algebra A of smooth functions on Y, then the corresponding equivariant map algebra (K\otimes A)^Γ coincides with the Lie algebra of smooth sections of a Lie algebra bundle over the orbit space.

This is joint work with Bas Janssens.