I will describe a joint work with Uri Bader, Michael Glasner and Roman Sauer in which we study unitary representations of semisimple Lie Groups that are cohomological; namely, such that the continuous cohomology of the group with coefficients in these representations is non-zero. This cohomology is a topological vector space that is not necessarily Hausdorff. The maximal Hausdorff quotient of it is called the reduced cohomology.
An essential work of Vogan and Zuckerman from 84’ solves the question completely for irreducible representations. From Vogan–Zuckerman's work one can also deduce a complete description of representations that have non-zero reduced cohomology: these are precisely representations that admit irreducible cohomological representations as sub-representations.
The question that is left is hence determining unitary representations that admit non-reduced cohomology. Up until now, non-reduced cohomology was mainly studied in the first degree, where its existence is equivalent to not having Kazhdan's property (T): H^1(G,V) is non-reduced if and only if V admits almost invariant vectors (that are not actually invariant). We prove some analoge (but more complicated) results in higher degrees, and settle the question of which representations admit non-reduced cohomology in terms of the Fell topology on the unitary dual of the group.
