Unipotent representations of real reductive groups

Abstract

Let G be a real reductive group and let Ĝ be the set of irreducible unitary representations of G. The determination of Ĝ (for arbitrary G) is one of the fundamental unsolved problems in representation theory. In the early 1980s, Arthur introduced a finite set Unip(G) of (conjecturally unitary) irreducible representations of G called unipotent representations. In a certain sense, these representations form the build-ing blocks of Ĝ. Hence, the determination of Ĝ requires as a crucial ingredient the determination of Unip(G). In this thesis, we prove three results on unipotent representations. First, we study unipotent representations by restriction to K [subset symbol] G, a maximal compact subgroup. We deduce a formula for this restriction in a wide range of cases, proving (in these cases) a long-standing conjecture of Vogan. Next, we study the unipotent representations attached to induced nilpotent orbits. We find that Unip(G) is 'generated' by an even smaller set Unip2(G) consisting of representations attached to rigid nilpotent orbits. Finally, we study the unipotent representations attached to the principal nilpotent orbit. We provide a complete classification of such representations, including a formula for their K-types.