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SUMMARY:Lucas Mason-Brown (MIT)
DTSTART:20200429T203000Z
DTEND:20200429T213000Z
DTSTAMP:20260423T021040Z
UID:MITLie/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITLie/4/">U
 nipotent representations of real reductive groups</a>\nby Lucas Mason-Brow
 n (MIT) as part of MIT Lie groups seminar\n\n\nAbstract\nLet $G$ be a real
  reductive group and let ${\\widehat G}$ be the set of\nirreducible unitar
 y representations of $G$. The determination of $\\widehat G$ (for\narbitra
 ry $G$) is one of the fundamental unsolved problems in\nrepresentation the
 ory. In the early 1980s\, Arthur introduced a finite\nset Unip($G$) of (co
 njecturally unitary) irreducible representations of\n$G$ called {\\it unip
 otent representations}. In a certain sense\, these\nrepresentations form t
 he building blocks of $\\widehat G$. Hence\, the\ndetermination of $\\wide
 hat G$ requires as a crucial ingredient the determination\nof Unip($G$). I
 n this thesis\, we prove three results on unipotent\nrepresentations.  Fir
 st\, we study unipotent representations by\nrestriction to $K\\subset G$\,
  a maximal compact subgroup. We deduce a formula\nfor this restriction in 
 a wide range of cases\, proving (in these\ncases) a long-standing conjectu
 re of Vogan. Next\, we study the\nunipotent representations attached to in
 duced nilpotent orbits. We\nfind that Unip($G$) is ‘generated’ by an e
 ven smaller set $\\hbox{Unip}'(G)$\nconsisting of representations attached
  to rigid nilpotent\norbits. Finally\, we study the unipotent representati
 ons attached to\nthe principal nilpotent orbit. We provide a complete clas
 sification of\nsuch representations\, including a formula for their $K$-ty
 pes.\n
LOCATION:https://researchseminars.org/talk/MITLie/4/
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