MIT Lie Groups Seminar

Abstract : 

Let G be a real reductive group and let $\widehat{G}$ be the set of irreducible unitary representations of G. The determination of $\widehat{G}$ (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 building blocks of $\widehat{G}$.  Hence, the determination of $\widehat{G}$ 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 ⊂ 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 Unip′(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.