Unipotent Representations of Complex Reductive Groups

Unipotent representations are a mysterious class of representations of a reductive group over the real or complex numbers, which are conjectured to form the `building blocks' of the unitary dual. In 1985, Barbasch and Vogan defined a class of representations of a complex reductive group called `special unipotent representations.' These representations have proven to be fundamental objects in the study of unitary representations, but they constitute only a fraction of all unipotent representations (for example, the metaplectic representations are excluded). In this talk, I will propose a more general definition of 'unipotent,' inspired by the Orbit Method. I will catalog the properties of our unipotent representations (including their classification) and describe an intriguing relationship between our representations and those of Barbasch-Vogan, which is related to symplectic duality. This talk is based on joint work with Ivan Losev and Dmitryo Matvieievskyi.