2026-09-21T18:08:31Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1269312026-06-17T14:44:40Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
David Vogan.
Mason-Brown, Lucas(Lucas David)
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2020-09-03T16:41:29Z
2020-09-03T16:41:29Z
2020
2020
https://hdl.handle.net/1721.1/126931
1191267388
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 207-210).
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.
by Lucas Mason-Brown.
Ph.D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
210 pages
eng
Massachusetts Institute of Technology
MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Unipotent representations of real reductive groups
Thesis
application/pdf
2020-09-03T16:41:28Z
Publication
Doctoral
Math
unknown
unknown
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<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.</Abstract>
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