Representation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College

Size: px
Start display at page:

Download "Representation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College"

Transcription

1 Representation Theory and Orbital Varieties Thomas Pietraho Bowdoin College

2 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible linear operators of a space V.

3 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible linear operators of a space V. Irreducible : V and 0 are the only closed linear subspaces of V invariant under π.

4 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible linear operators of a space V. Irreducible : V and 0 are the only closed linear subspaces of V invariant under π. Unitary : Each π(g) is a unitary operator.

5 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible linear operators of a space V. Irreducible : V and 0 are the only closed linear subspaces of V invariant under π. Unitary : Each π(g) is a unitary operator. Main Problem: Describe the set of all irreducible unitary representations of G, which we denote by Ĝunitary.

6 2 Lie Groups and the Orbit Method g - the Lie algebra of G g - its dual.

7 2 Lie Groups and the Orbit Method g - the Lie algebra of G g - its dual. Idea: For Lie groups, Ĝunitary should have something to do with the orbits of the G action on g, called coadjoint orbits.

8 2 Lie Groups and the Orbit Method g - the Lie algebra of G g - its dual. Idea: For Lie groups, Ĝunitary should have something to do with the orbits of the G action on g, called coadjoint orbits. Inspired by physics: Classical mechanical systems symplectic manifolds Quantum mechanical systems Hilbert spaces

9 3 Wishful thinking Classical Mechanics Coadjoint Orbits Quantum Mechanics Irred Unitary Reps

10 4 Nilpotent Lie Groups This works perfectly in the setting of nilpotent Lie groups: Theorem. [Kostant-Kirillov] If G is a connected and simply connected nilpotent Lie group, then there is a bijective correspondence g /G Ĝunitary between the set of coadjoint orbits of G and the set of its irreducible unitary representations.

11 5 Semisimple Lie Groups The problem is harder for semi-simple Lie groups. From now on, assume that G is semi-simple (or really reductive).

12 5 Semisimple Lie Groups The problem is harder for semi-simple Lie groups. From now on, assume that G is semi-simple (or really reductive). Upshot: Killing form is non-degenerate,

13 5 Semisimple Lie Groups The problem is harder for semi-simple Lie groups. From now on, assume that G is semi-simple (or really reductive). Upshot: Killing form is non-degenerate, Can identify g with g, and

14 5 Semisimple Lie Groups The problem is harder for semi-simple Lie groups. From now on, assume that G is semi-simple (or really reductive). Upshot: Killing form is non-degenerate, Can identify g with g, and coadjoint orbits are same as adjoint orbits.

15 6 Three Flavors of Coadjoint Orbits Suppose G GL n has discrete kernel. This leads to an inclusion An element X g is: g M n

16 6 Three Flavors of Coadjoint Orbits Suppose G GL n has discrete kernel. This leads to an inclusion An element X g is: g M n hyperbolic if matrix is diagonalizable

17 6 Three Flavors of Coadjoint Orbits Suppose G GL n has discrete kernel. This leads to an inclusion An element X g is: g M n hyperbolic if matrix is diagonalizable elliptic if diag over C, e-values ir

18 6 Three Flavors of Coadjoint Orbits Suppose G GL n has discrete kernel. This leads to an inclusion An element X g is: g M n hyperbolic if matrix is diagonalizable elliptic if diag over C, e-values ir nilpotent if matrix is nilpotent

19 7 Three Flavors of Coadjoint Orbits All coadjoint orbits are built in a simple manner from these three types. Fact. X g. Then X = X h + X e + X n, with X n nilpotent, X e elliptic, and X h hyperbolic.

20 8 Three Flavors of Quantization Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X Z with Z compact and each fiber Lagrangian.

21 8 Three Flavors of Quantization Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X Z with Z compact and each fiber Lagrangian. Attached to O X : unitary representation on space of L 2 sections of a line bundle on Z

22 8 Three Flavors of Quantization Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X Z with Z compact and each fiber Lagrangian. Attached to O X : unitary representation on space of L 2 sections of a line bundle on Z Theorem. [Cohomological Induction] If X is elliptic, there is a G-equivariant complex structure on O X making O X Kähler.

23 8 Three Flavors of Quantization Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X Z with Z compact and each fiber Lagrangian. Attached to O X : unitary representation on space of L 2 sections of a line bundle on Z Theorem. [Cohomological Induction] If X is elliptic, there is a G-equivariant complex structure on O X making O X Kähler. Attached to O X : unitary representation on Dolbeault cohomology of O X with coefficients in holomorphic line bundle.

24 9 Three Flavors of Quantization Theorem. If X is nilpotent, then O X is a cone.

25 9 Three Flavors of Quantization Theorem. If X is nilpotent, then O X is a cone. Attached to O X :??? One proposed construction by Graham and Vogan (at least for G complex).

26 10 Building Representations The orbit method philosophy can be summarized by Build Representation Build Orbit 1 Find rigid reps X n 2 Cohom induce from 1 X n + X e 3 Parab induce from 2 X n + X e + X h

27 10 Building Representations The orbit method philosophy can be summarized by Build Representation Build Orbit 1 Find rigid reps X n 2 Cohom induce from 1 X n + X e 3 Parab induce from 2 X n + X e + X h Reality: This process will not produce all irreducible unitary representations for semisimple groups. Example: Compl series in SL 2 (R). However, it s better than anything else.

28 11 A Bit about Graham-Vogan Construction O X, a nilpotent, orbit is a symplectic manifold. Study Lagrangian submanifolds.

29 11 A Bit about Graham-Vogan Construction O X, a nilpotent, orbit is a symplectic manifold. Study Lagrangian submanifolds. Fix Borel B, unipotent radical N. O X n is locally closed alg variety. Write as components (orbital varieties): O X n = i V i

30 11 A Bit about Graham-Vogan Construction O X, a nilpotent, orbit is a symplectic manifold. Study Lagrangian submanifolds. Fix Borel B, unipotent radical N. O X n is locally closed alg variety. Write as components (orbital varieties): O X n = i V i Theorem. [Ginzburg] V Lagrangian in O.

31 12 A Bit about Graham-Vogan Construction The G-V space, V (V, Q, π), lies in smooth sections of line bundle over a flag variety. Ingredients: orbital variety, V

32 12 A Bit about Graham-Vogan Construction The G-V space, V (V, Q, π), lies in smooth sections of line bundle over a flag variety. Ingredients: orbital variety, V Q, its stabilizer in G,

33 12 A Bit about Graham-Vogan Construction The G-V space, V (V, Q, π), lies in smooth sections of line bundle over a flag variety. Ingredients: orbital variety, V Q, its stabilizer in G, π, admissible orbit datum.

34 12 A Bit about Graham-Vogan Construction The G-V space, V (V, Q, π), lies in smooth sections of line bundle over a flag variety. Ingredients: orbital variety, V Q, its stabilizer in G, π, admissible orbit datum. Is it any good? Infinitesimal character and algebraic considerations (McGovern).

35 13 Nilpotent Orbits First, we would like to know what these things look like. Take G complex and classical.

36 13 Nilpotent Orbits First, we would like to know what these things look like. Take G complex and classical. Start with G = GL n (C).

37 13 Nilpotent Orbits First, we would like to know what these things look like. Take G complex and classical. Start with G = GL n (C). Fact: Conjugacy classes (adjoint orbits) in gl n C are determined by the Jordan canonical form.

38 13 Nilpotent Orbits First, we would like to know what these things look like. Take G complex and classical. Start with G = GL n (C). Fact: Conjugacy classes (adjoint orbits) in gl n C are determined by the Jordan canonical form. For nilpotent conjugacy classes, this says: nilpotent orbits in gl n C

39 13 Nilpotent Orbits First, we would like to know what these things look like. Take G complex and classical. Start with G = GL n (C). Fact: Conjugacy classes (adjoint orbits) in gl n C are determined by the Jordan canonical form. For nilpotent conjugacy classes, this says: nilpotent orbits in gl n C sizes of the Jordan blocks

40 13 Nilpotent Orbits First, we would like to know what these things look like. Take G complex and classical. Start with G = GL n (C). Fact: Conjugacy classes (adjoint orbits) in gl n C are determined by the Jordan canonical form. For nilpotent conjugacy classes, this says: nilpotent orbits in gl n C sizes of the Jordan blocks partitions of n

41 14 Nilpotent Orbits Example: There are five nilpotent orbits in gl 4 (C) corresponding to the five partitions of 4: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1], ie:

42 14 Nilpotent Orbits Example: There are five nilpotent orbits in gl 4 (C) corresponding to the five partitions of 4: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1], ie:

43 15 Nilpotent Orbits (cont.) Theorem. [Gerstenhaber] G a complex classical reductive Lie group of rank n. The set of nilpotent orbits in g is parameterized by

44 15 Nilpotent Orbits (cont.) Theorem. [Gerstenhaber] G a complex classical reductive Lie group of rank n. The set of nilpotent orbits in g is parameterized by (GL n ) partitions of n

45 15 Nilpotent Orbits (cont.) Theorem. [Gerstenhaber] G a complex classical reductive Lie group of rank n. The set of nilpotent orbits in g is parameterized by (GL n ) partitions of n (SO 2n+1 ) partitions of 2n + 1 whose even parts occur with even multiplicity

46 15 Nilpotent Orbits (cont.) Theorem. [Gerstenhaber] G a complex classical reductive Lie group of rank n. The set of nilpotent orbits in g is parameterized by (GL n ) partitions of n (SO 2n+1 ) partitions of 2n + 1 whose even parts occur with even multiplicity (Sp 2n ) partitions of 2n whose odd parts occur with even multiplicity

47 15 Nilpotent Orbits (cont.) Theorem. [Gerstenhaber] G a complex classical reductive Lie group of rank n. The set of nilpotent orbits in g is parameterized by (GL n ) partitions of n (SO 2n+1 ) partitions of 2n + 1 whose even parts occur with even multiplicity (Sp 2n ) partitions of 2n whose odd parts occur with even multiplicity (SO 2n ) partitions of 2n whose even parts occur with even multiplicity (*)

48 16 Nilpotent Orbits (cont.) Example: G = Sp(6) has eight nilpotent orbits corresponding to the Young diagrams:

49 16 Nilpotent Orbits (cont.) Example: G = Sp(6) has eight nilpotent orbits corresponding to the Young diagrams:

50 17 Orbital Varieties Theorem. [Spaltenstein] There is a bijection Irr(O X n) Irr(F X )/A X

51 17 Orbital Varieties Theorem. [Spaltenstein] There is a bijection Irr(O X n) Irr(F X )/A X F is the variety of (isotropic) flags.

52 17 Orbital Varieties Theorem. [Spaltenstein] There is a bijection Irr(O X n) Irr(F X )/A X F is the variety of (isotropic) flags. F X are those fixed by X.

53 17 Orbital Varieties Theorem. [Spaltenstein] There is a bijection Irr(O X n) Irr(F X )/A X F is the variety of (isotropic) flags. F X are those fixed by X. A X = G X /G o X.

54 17 Orbital Varieties Theorem. [Spaltenstein] There is a bijection Irr(O X n) Irr(F X )/A X F is the variety of (isotropic) flags. F X are those fixed by X. A X = G X /G o X. G X acts on F X, and so A X acts on Irr(F X ).

55 17 Orbital Varieties Theorem. [Spaltenstein] There is a bijection Irr(O X n) Irr(F X )/A X F is the variety of (isotropic) flags. F X are those fixed by X. A X = G X /G o X. G X acts on F X, and so A X acts on Irr(F X ). Fact. A X is trivial in type A and a 2-group in the other classical types.

56 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n.

57 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n. Fix a flag F F X

58 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n. Fix a flag F F X F : F 1 F 2... F n

59 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n. Fix a flag F F X F : F 1 F 2... F n Define a new (smaller) flag F by F : F 2 /F 1 F 3 /F 1... F n /F 1

60 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n. Fix a flag F F X F : F 1 F 2... F n Define a new (smaller) flag F by F : F 2 /F 1 F 3 /F 1... F n /F 1 and a nilpotent element X by X = X F.

61 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n. Fix a flag F F X F : F 1 F 2... F n Define a new (smaller) flag F by F : F 2 /F 1 F 3 /F 1... F n /F 1 and a nilpotent element X by X = X F. Then F F X.

62 18 Irreducible Components of F X First, let G = GL n (C). Nilpotent orbit O X corresponds to a partition P of n. Fix a flag F F X F : F 1 F 2... F n Define a new (smaller) flag F by F : F 2 /F 1 F 3 /F 1... F n /F 1 and a nilpotent element X by X = X F. Then F F X. Similarly, define F, X, etc.

63 19 Irreducible Components of F X Fact: If O X corresponds to the Young diagram D and O X corresponds to the Young diagram D, then D \ D is a square.

64 19 Irreducible Components of F X Fact: If O X corresponds to the Young diagram D and O X corresponds to the Young diagram D, then D \ D is a square. Example: (G = GL 5 ) One possibility is:

65 20 Labelling the squares: Label the square D \ D with an n Label the square D \ D with an n 1, etc. becomes:

66 20 Labelling the squares: Label the square D \ D with an n Label the square D \ D with an n 1, etc. becomes:

67 20 Labelling the squares: Label the square D \ D with an n Label the square D \ D with an n 1, etc. becomes: (Standard Young Tableau, write SY T ([3, 2]))

68 21 Irreducible Components of F X We ve defined a map: Φ : F X SY T (P ).

69 21 Irreducible Components of F X We ve defined a map: Φ : F X SY T (P ). Fact: Φ defines a bijection Irr(F X ) SY T (P )

70 21 Irreducible Components of F X We ve defined a map: Φ : F X SY T (P ). Fact: Φ defines a bijection Irr(F X ) SY T (P ) Corollary: Φ defines a bijection Irr(O X n) SY T (P )

71 22 Other Classical Groups Flags are isotropic flags

72 22 Other Classical Groups Flags are isotropic flags Can define F similarly

73 22 Other Classical Groups Flags are isotropic flags Can define F similarly D \ D is a domino!

74 22 Other Classical Groups Flags are isotropic flags Can define F similarly D \ D is a domino! Can define a map Φ : F X SDT (P )

75 22 Other Classical Groups Flags are isotropic flags Can define F similarly D \ D is a domino! Can define a map Φ : F X SDT (P ) Two Problems: Φ not surjective,

76 22 Other Classical Groups Flags are isotropic flags Can define F similarly D \ D is a domino! Can define a map Φ : F X SDT (P ) Two Problems: Φ not surjective, Φ does not separate the components Irr(F X ).

77 23 Example: G = Sp(6), then is not in the image of Φ.

78 Other Classical Groups (cont.) Nevertheless: Theorem. [] G a classical complex reductive Lie group of rank n. The above map Φ can be refined to a bijection Ψ : {Irr(F X ) O X nilpotent} ST (n). 24

79 24 Other Classical Groups (cont.) Nevertheless: Theorem. [] G a classical complex reductive Lie group of rank n. The above map Φ can be refined to a bijection Ψ : {Irr(F X ) O X nilpotent} ST (n). Lemma. Suppose P is the partition of the nilpotent orbit O X. For a fixed A X orbit on Irr(F X ), there is a unique tableau in the image of Ψ of shape P.

80 24 Other Classical Groups (cont.) Nevertheless: Theorem. [] G a classical complex reductive Lie group of rank n. The above map Φ can be refined to a bijection Ψ : {Irr(F X ) O X nilpotent} ST (n). Lemma. Suppose P is the partition of the nilpotent orbit O X. For a fixed A X orbit on Irr(F X ), there is a unique tableau in the image of Ψ of shape P. Corollary. Ψ defines a bijection Irr(O X n) ST (P ).

81 25 Example: Let G = Sp(6) and let X lie in the orbit corresponding to the partition [4, 2]. The group A X has order 2. There are four irreducible components of Irr(F X ), corresponding to the tableaux:

82 25 Example: Let G = Sp(6) and let X lie in the orbit corresponding to the partition [4, 2]. The group A X has order 2. There are four irreducible components of Irr(F X ), corresponding to the tableaux: Each component is fixed by the action of A X, except the first two, which are interchanged. By our corollary, there are three orbital varieties in O X, corresponding to the tableaux

83 25 Example: Let G = Sp(6) and let X lie in the orbit corresponding to the partition [4, 2]. The group A X has order 2. There are four irreducible components of Irr(F X ), corresponding to the tableaux: Each component is fixed by the action of A X, except the first two, which are interchanged. By our corollary, there are three orbital varieties in O X, corresponding to the tableaux

84 26 Back to Graham-Vogan Theorem. [] The stabilizer subgroup Q of an orbital variety V can be read off from its tableau T.

85 26 Back to Graham-Vogan Theorem. [] The stabilizer subgroup Q of an orbital variety V can be read off from its tableau T. This, along with a few other useful properties of these tableaux, allows us to calculate the infinitesimal characters of the Graham-Vogan spaces. After a few modifications of the original construction, we obtain:

86 26 Back to Graham-Vogan Theorem. [] The stabilizer subgroup Q of an orbital variety V can be read off from its tableau T. This, along with a few other useful properties of these tableaux, allows us to calculate the infinitesimal characters of the Graham-Vogan spaces. After a few modifications of the original construction, we obtain: Theorem. [] Take G as before and O X a small nilpotent orbit. The infinitesimal characters of the Graham-Vogan spaces attached to O X have precisely the infinitesimal characters attached to O X by McGovern.

COMPONENTS OF THE SPRINGER FIBER AND DOMINO TABLEAUX. Key words: Orbit Method, Orbital Varieties, Domino Tableaux

COMPONENTS OF THE SPRINGER FIBER AND DOMINO TABLEAUX. Key words: Orbit Method, Orbital Varieties, Domino Tableaux COMPONENTS OF THE SPRINGER FIBER AND DOMINO TABLEAUX THOMAS PIETRAHO Abstract. Consider a complex classical semi-simple Lie group along with the set of its nilpotent coadjoint orbits. When the group is

More information

Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups. Thomas Pietraho

Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups. Thomas Pietraho Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups by Thomas Pietraho M.S., University of Chicago, 1996 B.A., University of Chicago, 1996 Submitted to the Department of

More information

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1

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

Lectures on the Orbit Method

Lectures on the Orbit Method Lectures on the Orbit Method A. A. Kirillov Graduate Studies in Mathematics Volume 64 American Mathematical Society Providence, Rhode Island Preface Introduction xv xvii Chapter 1. Geometry of Coadjoint

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

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

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

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

A CHARACTERIZATION OF DYNKIN ELEMENTS

A CHARACTERIZATION OF DYNKIN ELEMENTS A CHARACTERIZATION OF DYNKIN ELEMENTS PAUL E. GUNNELLS AND ERIC SOMMERS ABSTRACT. We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element

More information

Multiplicity free actions of simple algebraic groups

Multiplicity free actions of simple algebraic groups Multiplicity free actions of simple algebraic groups D. Testerman (with M. Liebeck and G. Seitz) EPF Lausanne Edinburgh, April 2016 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity

More information

Workshop on B-stable ideals and nilpotent orbits

Workshop on B-stable ideals and nilpotent orbits Workshop on B-stable ideals and nilpotent orbits October 8-12, Roma, Italia Opening October 8, 10:15 Schedule of Talks Mon: 10:30-11:30 11:30-12 12-13 15-16 16-17 Kumar Coffee Break Möseneder Panyushev

More information

Nilpotent Orbits and Weyl Group Representations, I

Nilpotent Orbits and Weyl Group Representations, I Nilpotent Orbits and Weyl Group Representations, I O.S.U. Lie Groups Seminar April 12, 2017 1. Introduction Today, G will denote a complex Lie group and g the complexification of its Lie algebra. Moreover,

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

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

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

Subsystems, Nilpotent Orbits, and Weyl Group Representations

Subsystems, Nilpotent Orbits, and Weyl Group Representations Subsystems, Nilpotent Orbits, and Weyl Group Representations OSU Lie Groups Seminar November 18, 2009 1. Introduction Let g be a complex semisimple Lie algebra. (Everything I say will also be true for

More information

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N 74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

More information

THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES

THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main

More information

arxiv:math/ v1 [math.rt] 3 Feb 2006

arxiv:math/ v1 [math.rt] 3 Feb 2006 KYUNGPOOK Math J. 42(2002), 199-272 The Method of Orbits for eal Lie Groups arxiv:math/0602056v1 [math.t] 3 Feb 2006 Jae-Hyun Yang Department of Mathematics, Inha University, Incheon 402-751, Korea e-mail

More information

Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

More information

Uniformization in several complex variables.

Uniformization in several complex variables. Uniformization in several complex variables. Grenoble. April, 16, 2009. Transmitted to IMPA (Rio de Janeiro). Philippe Eyssidieux. Institut Fourier, Université Joseph Fourier (Grenoble 1). 1 The uniformization

More information

Lecture 5: Admissible Representations in the Atlas Framework

Lecture 5: Admissible Representations in the Atlas Framework Lecture 5: Admissible Representations in the Atlas Framework B. Binegar Department of Mathematics Oklahoma State University Stillwater, OK 74078, USA Nankai Summer School in Representation Theory and Harmonic

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 method of coadjoint orbits for real reductive groups. David A. Vogan, Jr.

The method of coadjoint orbits for real reductive groups. David A. Vogan, Jr. The method of coadjoint orbits for real reductive groups David A. Vogan, Jr. IAS/Park City Mathematics Series Volume 6, 1998 The method of coadjoint orbits for real reductive groups David A. Vogan, Jr.

More information

UNITARY REPRESENTATIONS OF REDUCTIVE LIE GROUPS David A. Vogan, Jr. Abstract. One of the fundamental problems of abstract harmonic analysis is the det

UNITARY REPRESENTATIONS OF REDUCTIVE LIE GROUPS David A. Vogan, Jr. Abstract. One of the fundamental problems of abstract harmonic analysis is the det UNITARY REPRESENTATIONS OF REDUCTIVE LIE GROUPS David A. Vogan, Jr. Abstract. One of the fundamental problems of abstract harmonic analysis is the determination of the irreducible unitary representations

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

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O). 9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson

More information

Lecture 4: LS Cells, Twisted Induction, and Duality

Lecture 4: LS Cells, Twisted Induction, and Duality Lecture 4: LS Cells, Twisted Induction, and Duality B. Binegar Department of Mathematics Oklahoma State University Stillwater, OK 74078, USA Nankai Summer School in Representation Theory and Harmonic Analysis

More information

The Contragredient. Spherical Unitary Dual for Complex Classical Groups

The Contragredient. Spherical Unitary Dual for Complex Classical Groups The Contragredient Joint with D. Vogan Spherical Unitary Dual for Complex Classical Groups Joint with D. Barbasch The Contragredient Problem: Compute the involution φ φ of the space of L-homomorphisms

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

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

TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS. Alfred G. Noël Department of Mathematics Northeastern University Boston, MA

TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS. Alfred G. Noël Department of Mathematics Northeastern University Boston, MA TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS Alfred G. Noël Department of Mathematics Northeastern University Boston, MA In this paper we give a brief survey of the basic results on B-N pairs

More information

Reductive group actions and some problems concerning their quotients

Reductive group actions and some problems concerning their quotients Reductive group actions and some problems concerning their quotients Brandeis University January 2014 Linear Algebraic Groups A complex linear algebraic group G is an affine variety such that the mappings

More information

DERIVED HAMILTONIAN REDUCTION

DERIVED HAMILTONIAN REDUCTION DERIVED HAMILTONIAN REDUCTION PAVEL SAFRONOV 1. Classical definitions 1.1. Motivation. In classical mechanics the main object of study is a symplectic manifold X together with a Hamiltonian function H

More information

An Invitation to Geometric Quantization

An Invitation to Geometric Quantization An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to

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

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

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

More information

Gauge Theory and Mirror Symmetry

Gauge Theory and Mirror Symmetry Gauge Theory and Mirror Symmetry Constantin Teleman UC Berkeley ICM 2014, Seoul C. Teleman (Berkeley) Gauge theory, Mirror symmetry ICM Seoul, 2014 1 / 14 Character space for SO(3) and Toda foliation Support

More information

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS Contents 1. Regular elements in semisimple Lie algebras 1 2. The flag variety and the Bruhat decomposition 3 3. The Grothendieck-Springer resolution 6 4. The

More information

September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds

September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds International Journal of Mathematics c World Scientific Publishing Company Hermitian Einstein connections on principal bundles over flat affine manifolds Indranil Biswas School of Mathematics Tata Institute

More information

Symplectic varieties and Poisson deformations

Symplectic varieties and Poisson deformations Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the

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

Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups. Dihua Jiang University of Minnesota

Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups. Dihua Jiang University of Minnesota Fourier Coefficients and Automorphic Discrete Spectrum of Classical Groups Dihua Jiang University of Minnesota KIAS, Seoul November 16, 2015 Square-Integrable Automorphic Forms G a reductive algebraic

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

Decay to zero of matrix coefficients at Adjoint infinity by Scot Adams

Decay to zero of matrix coefficients at Adjoint infinity by Scot Adams Decay to zero of matrix coefficients at Adjoint infinity by Scot Adams I. Introduction The main theorems below are Theorem 9 and Theorem 11. As far as I know, Theorem 9 represents a slight improvement

More information

Maximal subgroups of exceptional groups: representing groups in groups

Maximal subgroups of exceptional groups: representing groups in groups Maximal subgroups of exceptional groups: representing groups in groups David A. Craven University of Birmingham Representations of Finite and Algebraic Groups, Les Houches. 9th February, 2015. Joint with

More information

Introduction to the Baum-Connes conjecture

Introduction to the Baum-Connes conjecture Introduction to the Baum-Connes conjecture Nigel Higson, John Roe PSU NCGOA07 Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 1 / 15 History of the BC conjecture Lecture

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

12 Geometric quantization

12 Geometric quantization 12 Geometric quantization 12.1 Remarks on quantization and representation theory Definition 12.1 Let M be a symplectic manifold. A prequantum line bundle with connection on M is a line bundle L M equipped

More information

NILPOTENT ORBITS: GEOMETRY AND COMBINATORICS

NILPOTENT ORBITS: GEOMETRY AND COMBINATORICS NILPOTENT ORBITS: GEOMETRY AND COMBINATORICS YUZHOU GU MENTOR: KONSTANTIN TOLMACHOV PROJECT SUGGESTED BY: ROMAN BEZRUKAVNIKOV Abstract. We review the geometry of nilpotent orbits, and then restrict to

More information

Rigid Schubert classes in compact Hermitian symmetric spaces

Rigid Schubert classes in compact Hermitian symmetric spaces Rigid Schubert classes in compact Hermitian symmetric spaces Colleen Robles joint work with Dennis The Texas A&M University April 10, 2011 Part A: Question of Borel & Haefliger 1. Compact Hermitian symmetric

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

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS PETER E. TRAPA Abstract. For classical real Lie groups, we compute the annihilators and associated varieti

RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS PETER E. TRAPA Abstract. For classical real Lie groups, we compute the annihilators and associated varieti RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS PETER E. TRAPA Abstract. For classical real Lie groups, we compute the annihilators and associated varieties of the derived functor modules cohomologically induced

More information

PREQUANTIZATION OF SYMPLECTIC SUPERMANIFOLDS

PREQUANTIZATION OF SYMPLECTIC SUPERMANIFOLDS Ninth International Conference on Geometry, Integrability and Quantization June 8 13, 2007, Varna, Bulgaria Ivaïlo M. Mladenov, Editor SOFTEX, Sofia 2008, pp 301 307 Geometry, Integrability and Quantization

More information

Patrick Iglesias-Zemmour

Patrick Iglesias-Zemmour Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

More information

Reductive subgroup schemes of a parahoric group scheme

Reductive subgroup schemes of a parahoric group scheme Reductive subgroup schemes of a parahoric group scheme George McNinch Department of Mathematics Tufts University Medford Massachusetts USA June 2018 Contents 1 Levi factors 2 Parahoric group schemes 3

More information

On Cuspidal Spectrum of Classical Groups

On Cuspidal Spectrum of Classical Groups On Cuspidal Spectrum of Classical Groups Dihua Jiang University of Minnesota Simons Symposia on Geometric Aspects of the Trace Formula April 10-16, 2016 Square-Integrable Automorphic Forms G a reductive

More information

Techniques of computations of Dolbeault cohomology of solvmanifolds

Techniques of computations of Dolbeault cohomology of solvmanifolds .. Techniques of computations of Dolbeault cohomology of solvmanifolds Hisashi Kasuya Graduate School of Mathematical Sciences, The University of Tokyo. Hisashi Kasuya (Graduate School of Mathematical

More information

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

More information

Birational geometry and deformations of nilpotent orbits

Birational geometry and deformations of nilpotent orbits arxiv:math/0611129v1 [math.ag] 6 Nov 2006 Birational geometry and deformations of nilpotent orbits Yoshinori Namikawa In order to explain what we want to do in this paper, let us begin with an explicit

More information

Polar orbitopes. Leonardo Biliotti, Alessandro Ghigi and Peter Heinzner. Università di Parma - Università di Milano Bicocca - Ruhr Universität Bochum

Polar orbitopes. Leonardo Biliotti, Alessandro Ghigi and Peter Heinzner. Università di Parma - Università di Milano Bicocca - Ruhr Universität Bochum Polar orbitopes Leonardo Biliotti, Alessandro Ghigi and Peter Heinzner Università di Parma - Università di Milano Bicocca - Ruhr Universität Bochum Workshop su varietà reali e complesse: geometria, topologia

More information

MUMFORD-TATE GROUPS AND ABELIAN VARIETIES. 1. Introduction These are notes for a lecture in Elham Izadi s 2006 VIGRE seminar on the Hodge Conjecture.

MUMFORD-TATE GROUPS AND ABELIAN VARIETIES. 1. Introduction These are notes for a lecture in Elham Izadi s 2006 VIGRE seminar on the Hodge Conjecture. MUMFORD-TATE GROUPS AND ABELIAN VARIETIES PETE L. CLARK 1. Introduction These are notes for a lecture in Elham Izadi s 2006 VIGRE seminar on the Hodge Conjecture. Let us recall what we have done so far:

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

30 Surfaces and nondegenerate symmetric bilinear forms

30 Surfaces and nondegenerate symmetric bilinear forms 80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces

More information

LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

LECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment. LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.

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

Hodge Structures. October 8, A few examples of symmetric spaces

Hodge Structures. October 8, A few examples of symmetric spaces Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H

More information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

On the geometric Langlands duality

On the geometric Langlands duality On the geometric Langlands duality Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Schwerpunkttagung Bad Honnef April 2010 Outline This lecture will give an overview on the following topics:

More information

Half the sum of positive roots, the Coxeter element, and a theorem of Kostant

Half the sum of positive roots, the Coxeter element, and a theorem of Kostant DOI 10.1515/forum-2014-0052 Forum Math. 2014; aop Research Article Dipendra Prasad Half the sum of positive roots, the Coxeter element, and a theorem of Kostant Abstract: Interchanging the character and

More information

Spherical varieties and arc spaces

Spherical varieties and arc spaces Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January 2017 1 Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected

More information

ON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig

ON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig ON SOME PARTITIONS OF A FLAG MANIFOLD G. Lusztig Introduction Let G be a connected reductive group over an algebraically closed field k of characteristic p 0. Let W be the Weyl group of G. Let W be the

More information

Geometric Structure and the Local Langlands Conjecture

Geometric Structure and the Local Langlands Conjecture Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure

More information

Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky

Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey

More information

Ascona Conference Algebraic Groups and Invariant Theory. Springer fibers admitting singular components. Lucas Fresse, joint work with Anna Melnikov

Ascona Conference Algebraic Groups and Invariant Theory. Springer fibers admitting singular components. Lucas Fresse, joint work with Anna Melnikov Ascona Conference Algebraic Groups and Invariant Theory Springer fibers admitting singular components Lucas Fresse, joint work with Anna Melnikov Definition of Springer fibers Let V = C n, let u End(V

More information

Topics in Representation Theory: Cultural Background

Topics in Representation Theory: Cultural Background Topics in Representation Theory: Cultural Background This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that

More information

The Spinor Representation

The Spinor Representation The Spinor Representation Math G4344, Spring 2012 As we have seen, the groups Spin(n) have a representation on R n given by identifying v R n as an element of the Clifford algebra C(n) and having g Spin(n)

More information

A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface

A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface Documenta Math. 111 A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface Indranil Biswas Received: August 8, 2013 Communicated by Edward Frenkel

More information

SPHERICAL UNITARY DUAL FOR COMPLEX CLASSICAL GROUPS

SPHERICAL UNITARY DUAL FOR COMPLEX CLASSICAL GROUPS October 3, 008 SPHERICAL UNITARY DUAL FOR COMPLEX CLASSICAL GROUPS DAN BARBASCH 1. Introduction The full unitary dual for the complex classical groups viewed as real Lie groups is computed in [B1]. This

More information

HERMITIAN SYMMETRIC DOMAINS. 1. Introduction

HERMITIAN SYMMETRIC DOMAINS. 1. Introduction HERMITIAN SYMMETRI DOMAINS BRANDON LEVIN 1. Introduction Warning: these are rough notes based on the two lectures in the Shimura varieties seminar. After the unitary Hermitian symmetric spaces example,

More information

Multiplicity One Theorem in the Orbit Method

Multiplicity One Theorem in the Orbit Method Amer. Math. Soc. Transl. (2) Vol. 00, XXXX Multiplicity One Theorem in the Orbit Method Toshiyuki Kobayashi and Salma Nasrin In memory of Professor F. Karpelevič Abstract. Let G H be Lie groups, g h their

More information

A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS

A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions

More information

Moment map flows and the Hecke correspondence for quivers

Moment map flows and the Hecke correspondence for quivers and the Hecke correspondence for quivers International Workshop on Geometry and Representation Theory Hong Kong University, 6 November 2013 The Grassmannian and flag varieties Correspondences in Morse

More information

Local systems on nilpotent orbits and weighted Dynkin diagrams

Local systems on nilpotent orbits and weighted Dynkin diagrams University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 2002 Local systems on nilpotent orbits and weighted

More information

ON NEARLY SEMIFREE CIRCLE ACTIONS

ON NEARLY SEMIFREE CIRCLE ACTIONS ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)

More information

Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

More information

About polynomiality of the Poisson semicentre for parabolic subalgebras

About polynomiality of the Poisson semicentre for parabolic subalgebras About polynomiality of the Poisson semicentre for parabolic subalgebras University of Saint-Etienne, ICJ, LYON - France The Canicular Days - Haifa - July 2017 - Celebrating A. Joseph s 75th birthday Aim.

More information

Unipotent Representations and the Dual Pairs Correspondence

Unipotent Representations and the Dual Pairs Correspondence Unipotent Representations and the Dual Pairs Correspondence Dan Barbasch Yale June 015 August 7, 015 1 / 35 Introduction I first met Roger Howe at a conference in Luminy in 1978. At the time I knew some

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

LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES

LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES BENJAMIN HOFFMAN 1. Outline Lie algebroids are the infinitesimal counterpart of Lie groupoids, which generalize how we can talk about symmetries

More information

Notes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson

Notes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson Notes on Partial Resolutions of Nilpotent Varieties by Borho and Macpherson Chris Elliott January 14th, 2014 1 Setup Let G be a complex reductive Lie group with Lie algebra g. The paper [BM83] relates

More information

ALGEBRAIC GROUPS: PART IV

ALGEBRAIC GROUPS: PART IV ALGEBRAIC GROUPS: PART IV EYAL Z. GOREN, MCGILL UNIVERSITY Contents 11. Quotients 60 11.1. Some general comments 60 11.2. The quotient of a linear group by a subgroup 61 12. Parabolic subgroups, Borel

More information

Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =

Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg = Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for

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

arxiv: v1 [math.rt] 28 Jan 2009

arxiv: v1 [math.rt] 28 Jan 2009 Geometry of the Borel de Siebenthal Discrete Series Bent Ørsted & Joseph A Wolf arxiv:0904505v [mathrt] 28 Jan 2009 27 January 2009 Abstract Let G 0 be a connected, simply connected real simple Lie group

More information

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is

More information

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-

More information

Hodge Theory of Maps

Hodge Theory of Maps Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

More information