Unipotent representations from a geometric point of view

For a real reductive group G, the set of unipotent representations is a finite set of irreducible representations that satisfy certain desired properties and are supposed to be “building blocks” for all unitary representations of G. In 1982 Arthur proposed a definition of a special unipotent representation, which have been studied extensively in past years. However, this notion does not include some interesting unitary representations, such as a metaplectic representation. We restrict our attention to the case of a complex semisimple group G. In an ongoing project with Ivan Losev and Lucas Mason-Brown we propose a new definition of unipotent representations that extends the set of special unipotent ones. Using this definition we were able to classify unipotent representations and prove their key properties including some that were not known for special unipotent representations before. In this talk, I will explain our construction of unipotent representations, and time permitting sketch proofs of the main results.