Kazhdan s orthogonality conjecture for real reductive groups

Size: px
Start display at page:

Download "Kazhdan s orthogonality conjecture for real reductive groups"

Transcription

1 Kazhdan s orthogonality conjecture for real reductive groups Binyong Sun (Joint with Jing-Song Huang) Academy of Mathematics and Systems Science Chinese Academy of Sciences IMS, NUS March 28, 2016

2 Contents 1. Euler-Poincaré pairing 2. Global characters 3. Elliptic pairings 4. The elliptic Grothendieck group 5. Kazhdan s orthogonality conjecture 6. The real case 7. The proof

3 1. Euler-Poincaré pairing Let G be a connected reductive algebraic group over Q p. Let G V 1, V 2 be smooth representations. Assume that they have finite length.

4 Definition: The Euler-Poincaré pairing EP G (V 1, V 2 ) := i Z( 1) i dim Ext i G (V 1, V 2 ).

5 Example 1 If G has a non-compact center, then EP G (V 1, V 2 ) = 0.

6 Example 2 If G has a compact center, and either V 1 or V 2 is supercuspidal, then EP G (V 1, V 2 ) = dim Hom G (V 1, V 2 ).

7 2. Global characters The space of generalized functions: C (G) := Hom C (C 0 (G) dg, C). The space of locally integrable functions: } L 1 loc {f (G) := : f (g)ϕ(g) dg < for all ϕ /. G

8 Put G rs := {regular semisimple element in G}. Inclusions and restrictions: C (G) L 1 loc (G) C (G) injective injective C (G rs ) L 1 loc (G rs) C (G rs )

9 Let G V be a smooth representation of finite length. Its global character Θ V C (G) is defined to be ( ϕ dg the trace of V V, v G ) ϕ(g)g.v dg.

10 Clearly, Θ V C (G) G. Harish-Chandra s regularity theorem: Θ V L 1 loc (G) and Θ V Grs C (G rs ).

11 3. Elliptic pairings Definition: An element x G is elliptic if its centralizer Z G (x) in G has a compact center. Put G re := {regular elliptic element in G} and C := G re /G.

12 Remarks: elliptic semisimple. C is non-empty G has a compact center. We are concerned with Θ V C C (C).

13 The Weyl measure dc on C: f (g) dg = for all f C 0 (G re ) C 0 (G). G C G f (gcg 1 ) dg dc, Lemma (Kazhdan, 1986) Θ V C L 2 (C; dc).

14 Definition: The elliptic pairing (V 1, V 2 ) ell := Θ V1 (c 1 )Θ V2 (c) dc. Remarks: (V 1, V 2 ) ell = (V 2, V 1 ) ell. Θ V (c 1 ) = Θ V (c). C

15 Example 1 If G has a non-compact center, then (V 1, V 2 ) ell = 0.

16 Example 2 If either V 1 or V 2 is properly induced, then (V 1, V 2 ) ell = 0.

17 4. The elliptic Grothendieck group Let R(G) denote the Grothendieck group (with C-coefficients) of the category of finite length smooth representations of G. The elliptic Grothendieck group is R(G) := R(G) the span of all properly induced representations.

18 Lemma (Harish-Chandra) R(G) C (G), V Θ V. Lemma (Kazhdan) Assume that G has a compact center. R(G) L 2 (C; dc), V Θ V C. Remark: Assume that G has a compact center. The elliptic pairing induces an inner product on R(G).

19 5. Kazhdan s orthogonality conjecture Theorem (Kazhdan s orthogonality conjecture) (Schneider-Stuhler, 1997 and Bezrukavnikov, 1998) EP G (V 1, V 2 ) = (V 1, V 2 ) ell. The same holds in the case of real groups.

20 6. The real case Let G be a real reductive group in Harish-Chandra s class: (a) it has only finitely many connected components, and its Lie algebra g 0 is reductive; (b) it has a connected closed subgroup with finite center whose Lie algebra equals [g 0, g 0 ]; (c) for every g G, the adjoint action Ad(g) : g g is an inner automorphism of g, where g denotes the complexification of g 0.

21 Let K be a maximal compact subgroup of G. Let (g, K) V 1, V 2 be two modules. Assume that they have finite length. The Euler-Poincaré pairing EP g,k (V 1, V 2 ) := i Z( 1) i dim Ext i g,k (V 1, V 2 ).

22 Define C := G re /G and dc as in the p-adic case. Remark: C is non-empty rankg = rankk.

23 For every (g, K) V of finite length, the global character Θ V C (G) is defined; Θ V L 1 loc (G); Θ V Grs C (G rs ); Θ V C L 2 (C).

24 The elliptic pairing: (V 1, V 2 ) ell := C Θ V1 (c 1 )Θ V2 (c) dc. Remark: The elliptic pairing (V 1, V 2 ) ell only depends on V 1 K and V 2 K.

25 Theorem (Huang-Sun, arxiv 2015) EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell.

26 Corollary If rankg rankk, then EP g,k (V 1, V 2 ) = 0. Example: G = SL n (R), n 3.

27 Corollary The Euler-Poincaré pairing EP g,k is symmetric and non-negative. Assume that G has a compact center. It defines an inner product on R(g, K) R := the K-group, with R-coef., of f.l. (g, K)-modules. the span of properly induced representations

28 Remarks: If G is connected and rankg = rankk, then the Dirac pairing (V 1, V 2 ) Dir Z is defined by using the Dirac cohomologies. It only depends on V 1 K and V 2 K. It is easy to verify that (V 1, V 2 ) Dir = (V 1, V 2 ) ell. This is carried out by Huang and Renard.

29 7. The proof Step 1: If G has a non-compact center, then EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell = 0.

30 Lemma Let a be an abelian finite dimensional complex Lie algebra. If a 0, then ( 1) i dim H i (a; V ) = 0 i Z for all finite dimensional representations V of a. Proof. ( ) n n i=0( 1) i = 0, for n 1. i

31 Step 2: If V 2 is properly induced, then EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell = 0. Proof. Using Frobenius reciprocity, this is reduced to Step 1. Every proper Levi subgroup has a non-compact center.

32 then Step 3: If V 2 is in discrete series or limit of discrete series, EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell. Proof. Assume that rankg = rankk, and let T K be a Cartan subgroup of G. Then V 2 is cohomologically induced from T V 0, using a Borel subalgebra b = t n. Then dim K/T ( 1) 2 EP g,k (V 1, V 2 ) = j Z( 1) j dim Hom T (H j (n; V 1 ), V 0 ).

33 For all t T G rs, Vogan: Θ V1 (t) = i Z ( 1)i Θ Hi (n,v )(t). i Z ( 1)i Θ i n(t) Harish-Chandra: dim K/T ( 1) 2 Θ V2 (t) = w W(T,G) Θ V0 (w.t) i Z ( 1)i Θ i n(w.t) ;

34 Step 4: The general case. Proof. Assume that G has a compact center. Then R(G) is generated by properly induced representations, when rankg rankk; discrete series representations, limit of discrete series representations, and properly induced representations, when rankg = rankk.

35 Thank you for attention!

Dirac Cohomology, Orbit Method and Unipotent Representations

Dirac Cohomology, Orbit Method and Unipotent Representations Dirac Cohomology, Orbit Method and Unipotent Representations Dedicated to Bert Kostant with great admiration Jing-Song Huang, HKUST Kostant Conference MIT, May 28 June 1, 2018 coadjoint orbits of reductive

More information

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the

More information

Topics in Representation Theory: Roots and Weights

Topics in Representation Theory: Roots and Weights Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our

More information

Representations Are Everywhere

Representations Are Everywhere Representations Are Everywhere Nanghua Xi Member of Chinese Academy of Sciences 1 What is Representation theory Representation is reappearance of some properties or structures of one object on another.

More information

0 A. ... A j GL nj (F q ), 1 j r

0 A. ... A j GL nj (F q ), 1 j r CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected

More information

On the Harish-Chandra Embedding

On the Harish-Chandra Embedding On the Harish-Chandra Embedding The purpose of this note is to link the Cartan and the root decompositions. In addition, it explains how we can view a Hermitian symmetric domain as G C /P where P is a

More information

ON DEGENERATION OF THE SPECTRAL SEQUENCE FOR THE COMPOSITION OF ZUCKERMAN FUNCTORS

ON DEGENERATION OF THE SPECTRAL SEQUENCE FOR THE COMPOSITION OF ZUCKERMAN FUNCTORS ON DEGENERATION OF THE SPECTRAL SEQUENCE FOR THE COMPOSITION OF ZUCKERMAN FUNCTORS Dragan Miličić and Pavle Pandžić Department of Mathematics, University of Utah Department of Mathematics, Massachusetts

More information

The Structure of Compact Groups

The Structure of Compact Groups Karl H. Hofmann Sidney A. Morris The Structure of Compact Groups A Primer for the Student A Handbook for the Expert wde G Walter de Gruyter Berlin New York 1998 Chapter 1. Basic Topics and Examples 1 Definitions

More information

Branching rules of unitary representations: Examples and applications to automorphic forms.

Branching rules of unitary representations: Examples and applications to automorphic forms. Branching rules of unitary representations: Examples and applications to automorphic forms. Basic Notions: Jerusalem, March 2010 Birgit Speh Cornell University 1 Let G be a group and V a vector space.

More information

THE CENTRALIZER OF K IN U(g) C(p) FOR THE GROUP SO e (4,1) Ana Prlić University of Zagreb, Croatia

THE CENTRALIZER OF K IN U(g) C(p) FOR THE GROUP SO e (4,1) Ana Prlić University of Zagreb, Croatia GLASNIK MATEMATIČKI Vol. 5(7)(017), 75 88 THE CENTRALIZER OF K IN U(g) C(p) FOR THE GROUP SO e (4,1) Ana Prlić University of Zagreb, Croatia Abstract. Let G be the Lie group SO e(4,1), with maximal compact

More information

Weyl Group Representations and Unitarity of Spherical Representations.

Weyl Group Representations and Unitarity of Spherical Representations. Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν

More information

Fixed Point Theorem and Character Formula

Fixed Point Theorem and Character Formula Fixed Point Theorem and Character Formula Hang Wang University of Adelaide Index Theory and Singular Structures Institut de Mathématiques de Toulouse 29 May, 2017 Outline Aim: Study representation theory

More information

BRST and Dirac Cohomology

BRST and Dirac Cohomology BRST and Dirac Cohomology Peter Woit Columbia University Dartmouth Math Dept., October 23, 2008 Peter Woit (Columbia University) BRST and Dirac Cohomology October 2008 1 / 23 Outline 1 Introduction 2 Representation

More information

A partition of the set of enhanced Langlands parameters of a reductive p-adic group

A partition of the set of enhanced Langlands parameters of a reductive p-adic group A partition of the set of enhanced Langlands parameters of a reductive p-adic group joint work with Ahmed Moussaoui and Maarten Solleveld Anne-Marie Aubert Institut de Mathématiques de Jussieu - Paris

More information

DIRAC OPERATORS AND LIE ALGEBRA COHOMOLOGY

DIRAC OPERATORS AND LIE ALGEBRA COHOMOLOGY DIRAC OPERATORS AND LIE ALGEBRA COHOMOLOGY JING-SONG HUANG, PAVLE PANDŽIĆ, AND DAVID RENARD Abstract. Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups.

More information

Primitive Ideals and Unitarity

Primitive Ideals and Unitarity Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)

More information

Lie algebra cohomology

Lie algebra cohomology Lie algebra cohomology November 16, 2018 1 History Citing [1]: In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the

More information

Variations on a Casselman-Osborne theme

Variations on a Casselman-Osborne theme Variations on a Casselman-Osborne theme Dragan Miličić Introduction This paper is inspired by two classical results in homological algebra o modules over an enveloping algebra lemmas o Casselman-Osborne

More information

k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula 20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

More information

Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs UNIVERSITY OF TOKYO

Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs UNIVERSITY OF TOKYO UTMS 2011 8 April 22, 2011 Classification of discretely decomposable A q (λ) with respect to reductive symmetric pairs by Toshiyuki Kobayashi and Yoshiki Oshima T UNIVERSITY OF TOKYO GRADUATE SCHOOL OF

More information

ON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT

ON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT ON THE MAXIMAL PRIMITIVE IDEAL CORRESPONDING TO THE MODEL NILPOTENT ORBIT HUNG YEAN LOKE AND GORDAN SAVIN Abstract. Let g = k s be a Cartan decomposition of a simple complex Lie algebra corresponding to

More information

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

Extended groups and representation theory

Extended groups and representation theory Extended groups and representation theory Jeffrey Adams David Vogan University of Maryland Massachusetts Institute of Technology CUNY Representation Theory Seminar April 19, 2013 Outline Classification

More information

IVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) =

IVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) = LECTURE 7: CATEGORY O AND REPRESENTATIONS OF ALGEBRAIC GROUPS IVAN LOSEV Introduction We continue our study of the representation theory of a finite dimensional semisimple Lie algebra g by introducing

More information

C*-Algebras and Group Representations

C*-Algebras and Group Representations C*-Algebras and Department of Mathematics Pennsylvania State University EMS Joint Mathematical Weekend University of Copenhagen, February 29, 2008 Outline Summary Mackey pointed out an analogy between

More information

Gelfand Pairs and Invariant Distributions

Gelfand Pairs and Invariant Distributions Gelfand Pairs and Invariant Distributions A. Aizenbud Massachusetts Institute of Technology http://math.mit.edu/~aizenr Examples Example (Fourier Series) Examples Example (Fourier Series) L 2 (S 1 ) =

More information

Math 210C. The representation ring

Math 210C. The representation ring Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

More information

Spin norm: combinatorics and representations

Spin norm: combinatorics and representations Spin norm: combinatorics and representations Chao-Ping Dong Institute of Mathematics Hunan University September 11, 2018 Chao-Ping Dong (HNU) Spin norm September 11, 2018 1 / 38 Overview This talk aims

More information

REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES. Notation. 1. GL n

REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES. Notation. 1. GL n REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES ZHIWEI YUN Fix a prime number p and a power q of p. k = F q ; k d = F q d. ν n means ν is a partition of n. Notation Conjugacy classes 1. GL n 1.1.

More information

Joint work with Milen Yakimov. Kent Vashaw. November 19, Louisiana State University Prime Spectra of 2-Categories.

Joint work with Milen Yakimov. Kent Vashaw. November 19, Louisiana State University Prime Spectra of 2-Categories. Joint work with Milen Yakimov Louisiana State University kvasha1@lsu.edu November 19, 2016 Overview 1 2 3 A 2-category is a category enriched over the category small categories. So a 2-category T has:

More information

BRANCHING LAWS FOR SOME UNITARY REPRESENTATIONS OF SL(4,R)

BRANCHING LAWS FOR SOME UNITARY REPRESENTATIONS OF SL(4,R) BRANCHING LAWS FOR SOME UNITARY REPRESENTATIONS OF SL(4,R) BENT ØRSTED AND BIRGIT SPEH Abstract. In this paper we consider the restriction of a unitary irreducible representation of type A q (λ) of GL(4,

More information

9. The Lie group Lie algebra correspondence

9. The Lie group Lie algebra correspondence 9. The Lie group Lie algebra correspondence 9.1. The functor Lie. The fundamental theorems of Lie concern the correspondence G Lie(G). The work of Lie was essentially local and led to the following fundamental

More information

Non separated points in the dual spaces of semi simple Lie groups

Non separated points in the dual spaces of semi simple Lie groups 1 Non separated points in the dual spaces of semi simple Lie groups Let G be a connected semi simple Lie group with finite center, g its Lie algebra, g c = g R C, G the universal enveloping algebra of

More information

An introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109

An introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups

More information

Intersection of stable and unstable manifolds for invariant Morse functions

Intersection of stable and unstable manifolds for invariant Morse functions Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and

More information

THE HARISH-CHANDRA ISOMORPHISM FOR sl(2, C)

THE HARISH-CHANDRA ISOMORPHISM FOR sl(2, C) THE HARISH-CHANDRA ISOMORPHISM FOR sl(2, C) SEAN MCAFEE 1. summary Given a reductive Lie algebra g and choice of Cartan subalgebra h, the Harish- Chandra map gives an isomorphism between the center z of

More information

Kazhdan-Lusztig polynomials for

Kazhdan-Lusztig polynomials for Kazhdan-Lusztig polynomials for Department of Mathematics Massachusetts Institute of Technology Trends in Representation Theory January 4, 2012, Boston Outline What are KL polynomials for? How to compute

More information

CHAPTER 6. Representations of compact groups

CHAPTER 6. Representations of compact groups CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive

More information

BINYONG SUN AND CHEN-BO ZHU

BINYONG SUN AND CHEN-BO ZHU A GENERAL FORM OF GELFAND-KAZHDAN CRITERION BINYONG SUN AND CHEN-BO ZHU Abstract. We formalize the notion of matrix coefficients for distributional vectors in a representation of a real reductive group,

More information

Bruhat Tits buildings and representations of reductive p-adic groups

Bruhat Tits buildings and representations of reductive p-adic groups Bruhat Tits buildings and representations of reductive p-adic groups Maarten Solleveld Radboud Universiteit Nijmegen joint work with Ralf Meyer 26 November 2013 Starting point Let G be a reductive p-adic

More information

Fundamental Lemma and Hitchin Fibration

Fundamental Lemma and Hitchin Fibration Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental

More information

ON THE STANDARD MODULES CONJECTURE. V. Heiermann and G. Muić

ON THE STANDARD MODULES CONJECTURE. V. Heiermann and G. Muić ON THE STANDARD MODULES CONJECTURE V. Heiermann and G. Muić Abstract. Let G be a quasi-split p-adic group. Under the assumption that the local coefficients C defined with respect to -generic tempered representations

More information

DIRAC OPERATORS IN REPRESENTATION THEORY A SELECTED SURVEY D. RENARD

DIRAC OPERATORS IN REPRESENTATION THEORY A SELECTED SURVEY D. RENARD DIRAC OPERATORS IN REPRESENTATION THEORY A SELECTED SURVEY D. RENARD 1. Introduction Dirac operators were introduced into representation theory of real reductive groups by Parthasarathy [24] with the aim

More information

Iwasawa algebras and duality

Iwasawa algebras and duality Iwasawa algebras and duality Romyar Sharifi University of Arizona March 6, 2013 Idea of the main result Goal of Talk (joint with Meng Fai Lim) Provide an analogue of Poitou-Tate duality which 1 takes place

More information

Tilting exotic sheaves, parity sheaves on affine Grassmannians, and the Mirković Vilonen conjecture

Tilting exotic sheaves, parity sheaves on affine Grassmannians, and the Mirković Vilonen conjecture Tilting exotic sheaves, parity sheaves on affine Grassmannians, and the Mirković Vilonen conjecture (joint work with C. Mautner) Simon Riche CNRS Université Blaise Pascal (Clermont-Ferrand 2) Feb. 17th,

More information

arxiv:dg-ga/ v2 15 Dec 1995

arxiv:dg-ga/ v2 15 Dec 1995 arxiv:dg-ga/9512001v2 15 Dec 1995 On the index of Dirac operators on arithmetic quotients Anton Deitmar Math. Inst. d. Univ., INF 288, 69126 Heidelberg, Germany ABSTRACT: Using the Arthur-Selberg trace

More information

Category O and its basic properties

Category O and its basic properties Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α

More information

Part III Symmetries, Fields and Particles

Part III Symmetries, Fields and Particles Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often

More information

NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD. by Daniel Bump

NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD. by Daniel Bump NOTES ON REPRESENTATIONS OF GL(r) OVER A FINITE FIELD by Daniel Bump 1 Induced representations of finite groups Let G be a finite group, and H a subgroup Let V be a finite-dimensional H-module The induced

More information

Geometric proof of the local Jacquet-Langlands correspondence for GL(n) for prime n

Geometric proof of the local Jacquet-Langlands correspondence for GL(n) for prime n Geometric proof of the local Jacquet-Langlands correspondence for GL(n) for prime n Yoichi Mieda Abstract. In this paper, we give a purely geometric proof of the local Jacquet-Langlands correspondence

More information

DIRAC COHOMOLOGY FOR GRADED AFFINE HECKE ALGEBRAS

DIRAC COHOMOLOGY FOR GRADED AFFINE HECKE ALGEBRAS DIRAC COHOMOLOGY FOR GRADED AFFINE HECKE ALGEBRAS DAN BARBASCH, DAN CIUBOTARU, AND PETER E. TRAPA Abstract. We define an analogue of the Casimir element for a graded affine Hecke algebra H, and then introduce

More information

Jean-Pierre Serre. Galois Cohomology. translated from the French by Patrick Ion. Springer

Jean-Pierre Serre. Galois Cohomology. translated from the French by Patrick Ion. Springer Jean-Pierre Serre Galois Cohomology translated from the French by Patrick Ion Springer Table of Contents Foreword Chapter I. Cohomology of profinite groups 1. Profinite groups 3 1.1 Definition 3 1.2 Subgroups

More information

Langlands parameters and finite-dimensional representations

Langlands parameters and finite-dimensional representations Langlands parameters and finite-dimensional representations Department of Mathematics Massachusetts Institute of Technology March 21, 2016 Outline What Langlands can do for you Representations of compact

More information

Parameters for Representations of Real Groups Atlas Workshop, July 2004 updated for Workshop, July 2005

Parameters for Representations of Real Groups Atlas Workshop, July 2004 updated for Workshop, July 2005 Parameters for Representations of Real Groups Atlas Workshop, July 2004 updated for Workshop, July 2005 Jeffrey Adams July 21, 2005 The basic references are [7] and [6]. The parameters given in these notes

More information

The local Langlands correspondence for inner forms of SL n. Plymen, Roger. MIMS EPrint:

The local Langlands correspondence for inner forms of SL n. Plymen, Roger. MIMS EPrint: The local Langlands correspondence for inner forms of SL n Plymen, Roger 2013 MIMS EPrint: 2013.43 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports

More information

Lie groupoids, cyclic homology and index theory

Lie groupoids, cyclic homology and index theory Lie groupoids, cyclic homology and index theory (Based on joint work with M. Pflaum and X. Tang) H. Posthuma University of Amsterdam Kyoto, December 18, 2013 H. Posthuma (University of Amsterdam) Lie groupoids

More information

Weyl orbits of π-systems in Kac-Moody algebras

Weyl orbits of π-systems in Kac-Moody algebras Weyl orbits of π-systems in Kac-Moody algebras Krishanu Roy The Institute of Mathematical Sciences, Chennai, India (Joint with Lisa Carbone, K N Raghavan, Biswajit Ransingh and Sankaran Viswanath) June

More information

A relative version of Kostant s theorem

A relative version of Kostant s theorem A relative version of Kostant s theorem 1 University of Vienna Faculty of Mathematics Srni, January 2015 1 supported by project P27072 N25 of the Austrian Science Fund (FWF) This talk reports on joint

More information

arxiv: v1 [math.rt] 14 Nov 2007

arxiv: v1 [math.rt] 14 Nov 2007 arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof

More information

A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds

A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn

More information

Math 249B. Applications of Borel s theorem on Borel subgroups

Math 249B. Applications of Borel s theorem on Borel subgroups Math 249B. Applications of Borel s theorem on Borel subgroups 1. Motivation In class we proved the important theorem of Borel that if G is a connected linear algebraic group over an algebraically closed

More information

LECTURE 11: SOERGEL BIMODULES

LECTURE 11: SOERGEL BIMODULES LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing

More information

Discrete Series and Characters of the Component Group

Discrete Series and Characters of the Component Group Discrete Series and Characters of the Component Group Jeffrey Adams April 9, 2007 Suppose φ : W R L G is an L-homomorphism. There is a close relationship between the L-packet associated to φ and characters

More information

A GEOMETRIC JACQUET FUNCTOR

A GEOMETRIC JACQUET FUNCTOR A GEOMETRIC JACQUET FUNCTOR M. EMERTON, D. NADLER, AND K. VILONEN 1. Introduction In the paper [BB1], Beilinson and Bernstein used the method of localisation to give a new proof and generalisation of Casselman

More information

Topics in Representation Theory: Roots and Complex Structures

Topics in Representation Theory: Roots and Complex Structures Topics in Representation Theory: Roots and Complex Structures 1 More About Roots To recap our story so far: we began by identifying an important abelian subgroup of G, the maximal torus T. By restriction

More information

LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O

LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O LECTURE 3: TENSORING WITH FINITE DIMENSIONAL MODULES IN CATEGORY O CHRISTOPHER RYBA Abstract. These are notes for a seminar talk given at the MIT-Northeastern Category O and Soergel Bimodule seminar (Autumn

More information

Quaternionic Complexes

Quaternionic Complexes Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

More information

Representations of semisimple Lie algebras

Representations of semisimple Lie algebras Chapter 14 Representations of semisimple Lie algebras In this chapter we study a special type of representations of semisimple Lie algberas: the so called highest weight representations. In particular

More information

On the Self-dual Representations of a p-adic Group

On the Self-dual Representations of a p-adic Group IMRN International Mathematics Research Notices 1999, No. 8 On the Self-dual Representations of a p-adic Group Dipendra Prasad In an earlier paper [P1], we studied self-dual complex representations of

More information

ALGEBRAIC AND ANALYTIC DIRAC INDUCTION FOR GRADED AFFINE HECKE ALGEBRAS

ALGEBRAIC AND ANALYTIC DIRAC INDUCTION FOR GRADED AFFINE HECKE ALGEBRAS ALGEBRAIC AND ANALYTIC DIRAC INDUCTION FOR GRADED AFFINE HECKE ALGEBRAS DAN CIUBOTARU, ERIC M. OPDAM, AND PETER E. TRAPA Abstract. We define the algebraic Dirac induction map Ind D for graded affine Hecke

More information

Whittaker models and Fourier coeffi cients of automorphic forms

Whittaker models and Fourier coeffi cients of automorphic forms Whittaker models and Fourier coeffi cients of automorphic forms Nolan R. Wallach May 2013 N. Wallach () Whittaker models 5/13 1 / 20 G be a real reductive group with compact center and let K be a maximal

More information

The Lusztig-Vogan Bijection in the Case of the Trivial Representation

The Lusztig-Vogan Bijection in the Case of the Trivial Representation The Lusztig-Vogan Bijection in the Case of the Trivial Representation Alan Peng under the direction of Guangyi Yue Department of Mathematics Massachusetts Institute of Technology Research Science Institute

More information

14 From modular forms to automorphic representations

14 From modular forms to automorphic representations 14 From modular forms to automorphic representations We fix an even integer k and N > 0 as before. Let f M k (N) be a modular form. We would like to product a function on GL 2 (A Q ) out of it. Recall

More information

Twisted Harish-Chandra sheaves and Whittaker modules: The nondegenerate case

Twisted Harish-Chandra sheaves and Whittaker modules: The nondegenerate case Twisted Harish-Chandra sheaves and Whittaker modules: The nondegenerate case Dragan Miličić and Wolfgang Soergel Introduction Let g be a complex semisimple Lie algebra, U (g) its enveloping algebra and

More information

Defining equations for some nilpotent varieties

Defining equations for some nilpotent varieties 1 Defining equations for some nilpotent varieties Eric Sommers (UMass Amherst) Ben Johnson (Oklahoma State) The Mathematical Legacy of Bertram Kostant MIT June 1, 2018 Kostant s interest in the Buckyball

More information

Draft: February 26, 2010 ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-adic REDUCTIVE GROUPS I. DEFINITION AND FIRST PROPERTIES

Draft: February 26, 2010 ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-adic REDUCTIVE GROUPS I. DEFINITION AND FIRST PROPERTIES Draft: February 26, 2010 ORDINARY PARTS OF ADISSIBLE REPRESENTATIONS OF p-adic REDUCTIVE ROUPS I. DEFINITION AND FIRST PROPERTIES ATTHEW EERTON Contents 1. Introduction 1 2. Representations of p-adic analytic

More information

Cuspidality and Hecke algebras for Langlands parameters

Cuspidality and Hecke algebras for Langlands parameters Cuspidality and Hecke algebras for Langlands parameters Maarten Solleveld Universiteit Nijmegen joint with Anne-Marie Aubert and Ahmed Moussaoui 12 April 2016 Maarten Solleveld Universiteit Nijmegen Cuspidality

More information

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that

More information

Math 225B: Differential Geometry, Final

Math 225B: Differential Geometry, Final Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

REPRESENTATION THEORY. WEEK 4

REPRESENTATION THEORY. WEEK 4 REPRESENTATION THEORY. WEEK 4 VERA SERANOVA 1. uced modules Let B A be rings and M be a B-module. Then one can construct induced module A B M = A B M as the quotient of a free abelian group with generators

More information

MAT 445/ INTRODUCTION TO REPRESENTATION THEORY

MAT 445/ INTRODUCTION TO REPRESENTATION THEORY MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations

More information

Notes on D 4 May 7, 2009

Notes on D 4 May 7, 2009 Notes on D 4 May 7, 2009 Consider the simple Lie algebra g of type D 4 over an algebraically closed field K of characteristic p > h = 6 (the Coxeter number). In particular, p is a good prime. We have dim

More information

Introduction to Representations Theory of Lie Groups

Introduction to Representations Theory of Lie Groups Introduction to Representations Theory of Lie roups Raul omez October 14, 2009 Introduction The purpose of this notes is twofold. The first goal is to give a quick answer to the question What is representation

More information

DIRAC COHOMOLOGY, UNITARY REPRESENTATIONS AND A PROOF OF A CONJECTURE OF VOGAN

DIRAC COHOMOLOGY, UNITARY REPRESENTATIONS AND A PROOF OF A CONJECTURE OF VOGAN JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 15, Number 1, Pages 185 202 S 0894-0347(01)00383-6 Article electronically published on September 6, 2001 DIRAC COHOMOLOGY, UNITARY REPRESENTATIONS AND

More information

ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH. 1. Introduction

ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH. 1. Introduction ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH Abstract. We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains

More information

Invariant Distributions and Gelfand Pairs

Invariant Distributions and Gelfand Pairs Invariant Distributions and Gelfand Pairs A. Aizenbud and D. Gourevitch http : //www.wisdom.weizmann.ac.il/ aizenr/ Gelfand Pairs and distributional criterion Definition A pair of groups (G H) is called

More information

370 INDEX AND NOTATION

370 INDEX AND NOTATION Index and Notation action of a Lie algebra on a commutative! algebra 1.4.9 action of a Lie algebra on a chiral algebra 3.3.3 action of a Lie algebroid on a chiral algebra 4.5.4, twisted 4.5.6 action of

More information

Characters in Categorical Representation Theory

Characters in Categorical Representation Theory Characters in Categorical Representation Theory David Ben-Zvi University of Texas at Austin Symplectic Algebraic eometry and Representation Theory, CIRM, Luminy. July 2012 Overview Describe ongoing joint

More information

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

More information

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue

More information

Algebraic methods in the theory of generalized Harish-Chandra modules

Algebraic methods in the theory of generalized Harish-Chandra modules Algebraic methods in the theory of generalized Harish-Chandra modules Ivan Penkov and Gregg Zuckerman Abstract This paper is a review of results on generalized Harish-Chandra modules in the framework of

More information

Weyl s Character Formula for Representations of Semisimple Lie Algebras

Weyl s Character Formula for Representations of Semisimple Lie Algebras Weyl s Character Formula for Representations of Semisimple Lie Algebras Ben Reason University of Toronto December 22, 2005 1 Introduction Weyl s character formula is a useful tool in understanding the

More information

A Langlands classification for unitary representations

A Langlands classification for unitary representations Advanced Studies in Pure Mathematics 26, 1998 Analysis on Homogeneous Spaces and Representations of Lie Groups pp. 1 26 A Langlands classification for unitary representations David A. Vogan, Jr. Abstract.

More information

A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS

A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds

More information

`-modular Representations of Finite Reductive Groups

`-modular Representations of Finite Reductive Groups `-modular Representations of Finite Reductive Groups Bhama Srinivasan University of Illinois at Chicago AIM, June 2007 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations AIM,

More information

Mathematical Research Letters 3, (1996) THE DIMENSION OF THE FIXED POINT SET ON AFFINE FLAG MANIFOLDS. Roman Bezrukavnikov

Mathematical Research Letters 3, (1996) THE DIMENSION OF THE FIXED POINT SET ON AFFINE FLAG MANIFOLDS. Roman Bezrukavnikov Mathematical Research Letters 3, 185 189 (1996) THE DIMENSION OF THE FIXED POINT SET ON AFFINE FLAG MANIFOLDS Roman Bezrukavnikov Let G be a semisimple simply-connected algebraic group over C, g its Lie

More information

Associated varieties for real reductive groups

Associated varieties for real reductive groups Pure and Applied Mathematics Quarterly Volume 0, Number 0, 1, 2019 Associated varieties for real reductive groups Jeffrey Adams and David A. Vogan, Jr. In fond memory of our teacher and friend, Bert Kostant.

More information

Frobenius Green functors

Frobenius Green functors UC at Santa Cruz Algebra & Number Theory Seminar 30th April 2014 Topological Motivation: Morava K-theory and finite groups For each prime p and each natural number n there is a 2-periodic multiplicative

More information

Lecture 4 Super Lie groups

Lecture 4 Super Lie groups Lecture 4 Super Lie groups In this lecture we want to take a closer look to supermanifolds with a group structure: Lie supergroups or super Lie groups. As in the ordinary setting, a super Lie group is

More information