Winter School 2015 on Representation Theory of Real Reductive Groups

Date:	January 24 (Sat)-26 (Mon), 2015
Venue:	Graduate School of Mathematical Sciences, The University of Tokyo, Japan [ Access ]

January 24 (Sat) 13:00-14:00, Room 126
Peter Trapa
"Unitary representations of reductive Lie groups I"

14:30-15:30, Room 126
Raul Gomez
"The Tor and Ext functors for smooth representations of real algebraic groups"

16:30-17:30, Room 126
Benjamin Harris
"The Geometry of Tempered Characters"

January 25 (Sun) 9:00-10:00, Room 126
Raul Gomez
"Generalized and degenerate Whittaker models associated to nilpotent orbits"

10:30-11:30, Room 126
Benjamin Harris
"The Geometry of Harmonic Analysis"

12:00-13:00, Room 126
Peter Trapa
"Unitary representations of reductive Lie groups II"

January 26 (Mon) 9:30-10:30, Room 122
Benjamin Harris
"The Geometry of Nontempered Characters"

11:00-12:00, Room 122
Raul Gomez
"Local Theta lifting of generalized Whittaker models"

12:30-13:30, Room 128
Peter Trapa
"Unitary representations of reductive Lie groups III"

Speaker Benjamin Harris (Oklahoma State University) [ pdf ]

Title The Geometry of Tempered Characters

Abstract In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).

Title The Geometry of Harmonic Analysis

Abstract In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.

Title The Geometry of Nontempered Characters

Abstract In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.

Speaker Peter Trapa (University of Utah) [ pdf ]

Title Unitary representations of reductive Lie groups I, II, III

Abstract Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form. This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.

Speaker Raul Gomez (Cornel University) [ pdf ]

Title 1. The Tor and Ext functors for smooth representations of real algebraic groups

Abstract Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.

Title 2. Generalized and degenerate Whittaker models associated to nilpotent orbits

Abstract In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications. This is joint work with Dmitry Gourevitch and Siddhartha Sahi.

Title 3. Local Theta lifting of generalized Whittaker models

Abstract In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.

Organizers: T. Kobayashi, T. Kubo, H. Matumoto, H. Sekiguchi

Home

January 24 (Sat)	13:00-14:00, Room 126 Peter Trapa "Unitary representations of reductive Lie groups I"
	14:30-15:30, Room 126 Raul Gomez "The Tor and Ext functors for smooth representations of real algebraic groups"
	16:30-17:30, Room 126 Benjamin Harris "The Geometry of Tempered Characters"
January 25 (Sun)	9:00-10:00, Room 126 Raul Gomez "Generalized and degenerate Whittaker models associated to nilpotent orbits"
	10:30-11:30, Room 126 Benjamin Harris "The Geometry of Harmonic Analysis"
	12:00-13:00, Room 126 Peter Trapa "Unitary representations of reductive Lie groups II"
January 26 (Mon)	9:30-10:30, Room 122 Benjamin Harris "The Geometry of Nontempered Characters"
	11:00-12:00, Room 122 Raul Gomez "Local Theta lifting of generalized Whittaker models"
	12:30-13:30, Room 128 Peter Trapa "Unitary representations of reductive Lie groups III"
Speaker	Benjamin Harris (Oklahoma State University) [ pdf ]
Title	The Geometry of Tempered Characters
Abstract	In this introductory talk, we will briefly recall parts of Harish-Chandra's theory of characters for reductive groups and the geometric formula of Rossmann and Duflo for tempered characters of reductive groups. Examples will be given in the case G=SL(2,R).
Title	The Geometry of Harmonic Analysis
Abstract	In this talk, we will present recent joint work with Tobias Weich. When G is a real, reductive algebraic group and X is a homogeneous space for G with an invariant measure, we will completely describe the regular, semisimple asymptotics of the support of the Plancherel measure for L^2(X). We will give concrete examples of this theorem, describing what can and cannot be deduced from this result.
Title	The Geometry of Nontempered Characters
Abstract	In this talk, we will survey the results of Rossmann and Schmid-Vilonen on geometric formulas for nontempered characters of reductive groups, and we will mention an old result of Barbasch-Vogan on the special case A_q(lambda). We will discuss what nontempered character formulas would be necessary to generalize the main formula of the second talk, and we will make conjectures.
Speaker	Peter Trapa (University of Utah) [ pdf ]
Title	Unitary representations of reductive Lie groups I, II, III
Abstract	Let G be a real reductive group. I will describe an algorithm to determine the unitary dual of G. More precisely, I will describe an algorithm to determine if an irreducible (g,K) module (specified in the Langlands classification) is unitary in the sense that it admits a positive definite invariant Hermitian form. This is joint work with Jeffrey Adams, Marc van Leeuwen, and David Vogan.
Speaker	Raul Gomez (Cornel University) [ pdf ]
Title	1. The Tor and Ext functors for smooth representations of real algebraic groups
Abstract	Inspired by the recent work of Dipendra Prasad in the $p$-adic setting, we define the Tor and Ext functors for an appropriate category of smooth representations of a real algebraic group $G$, and give some applications. This is joint work with Birgit Speh.
Title	2. Generalized and degenerate Whittaker models associated to nilpotent orbits
Abstract	In this talk, we examine the relation between the different spaces of Whittaker models that can be attached to a nilpotent orbit. We will also explore their relation to other nilpotent invariants (like the wave front set) and show some examples and applications. This is joint work with Dmitry Gourevitch and Siddhartha Sahi.
Title	3. Local Theta lifting of generalized Whittaker models
Abstract	In this talk, we describe the behavior of the space of generalized Whittaker models attached to a nilpotent orbit under the local theta correspondence. This description is a generalization of a result of Moeglin in the p-adic setting. This is joint work with Chengbo Zhu.
	Organizers: T. Kobayashi, T. Kubo, H. Matumoto, H. Sekiguchi