Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem....

86
Representation Theory and Orbital Varieties Thomas Pietraho Bowdoin College

Transcript of Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem....

Page 1: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

Representation Theory and Orbital

Varieties

Thomas Pietraho

Bowdoin College

Page 2: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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 .

Page 3: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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.

Page 4: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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.

Page 5: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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 Gunitary.

Page 6: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

2

Lie Groups and the Orbit Method

g - the Lie algebra of Gg∗ - its dual.

Page 7: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

2

Lie Groups and the Orbit Method

g - the Lie algebra of Gg∗ - its dual.

Idea: For Lie groups, Gunitary should have something to

do with the orbits of the G action on g∗, called coadjointorbits.

Page 8: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

2

Lie Groups and the Orbit Method

g - the Lie algebra of Gg∗ - its dual.

Idea: For Lie groups, Gunitary should have something to

do with the orbits of the G action on g∗, called coadjointorbits.

• Inspired by physics:

Classical mechanical systems ←→ symplectic manifolds

Quantum mechanical systems ←→ Hilbert spaces

Page 9: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

3

Wishful thinking

Classical Mechanics←→ Coadjoint Orbits

Quantum Mechanics←→ Irred Unitary Reps

Page 10: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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 −→ Gunitary

between the set of coadjoint orbits of G and the set of its

irreducible unitary representations.

Page 11: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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).

Page 12: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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,

Page 13: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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

Page 14: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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.

Page 15: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

6

Three Flavors of Coadjoint Orbits

Suppose G −→ GLn has discrete kernel. This leads to an

inclusion

g∗ ⊂Mn

An element X ∈ g∗ is:

Page 16: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

6

Three Flavors of Coadjoint Orbits

Suppose G −→ GLn has discrete kernel. This leads to an

inclusion

g∗ ⊂Mn

An element X ∈ g∗ is:

• hyperbolic if matrix is diagonalizable

Page 17: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

6

Three Flavors of Coadjoint Orbits

Suppose G −→ GLn has discrete kernel. This leads to an

inclusion

g∗ ⊂Mn

An element X ∈ g∗ is:

• hyperbolic if matrix is diagonalizable

• elliptic if diag over C, e-values ∈ iR

Page 18: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

6

Three Flavors of Coadjoint Orbits

Suppose G −→ GLn has discrete kernel. This leads to an

inclusion

g∗ ⊂Mn

An element X ∈ g∗ is:

• hyperbolic if matrix is diagonalizable

• elliptic if diag over C, e-values ∈ iR

• nilpotent if matrix is nilpotent

Page 19: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

7

Three Flavors of Coadjoint Orbits

All coadjoint orbits are built in a simple manner from

these three types.

Fact. X ∈ g. Then X = Xh + Xe + Xn, with Xn

nilpotent, Xe elliptic, and Xh hyperbolic.

Page 20: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

8

Three Flavors of Quantization

Theorem. [Parabolic Induction] If X is hyperbolic,

there is a G-equivariant fibration OX → Z with Z com-

pact and each fiber Lagrangian.

Page 21: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

8

Three Flavors of Quantization

Theorem. [Parabolic Induction] If X is hyperbolic,

there is a G-equivariant fibration OX → Z with Z com-

pact and each fiber Lagrangian.Attached to OX: unitary representation on space of L2 sections of a

line bundle on Z

Page 22: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

8

Three Flavors of Quantization

Theorem. [Parabolic Induction] If X is hyperbolic,

there is a G-equivariant fibration OX → Z with Z com-

pact and each fiber Lagrangian.Attached to OX: unitary representation on space of L2 sections of a

line bundle on Z

Theorem. [Cohomological Induction] If X is elliptic,

there is a G-equivariant complex structure on OX making

OX Kahler.

Page 23: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

8

Three Flavors of Quantization

Theorem. [Parabolic Induction] If X is hyperbolic,

there is a G-equivariant fibration OX → Z with Z com-

pact and each fiber Lagrangian.Attached to OX: unitary representation on space of L2 sections of a

line bundle on Z

Theorem. [Cohomological Induction] If X is elliptic,

there is a G-equivariant complex structure on OX making

OX Kahler.

Attached to OX: unitary representation on Dolbeault cohomology of

OX with coefficients in holomorphic line bundle.

Page 24: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

9

Three Flavors of Quantization

Theorem. If X is nilpotent, then OX is a cone.

Page 25: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

9

Three Flavors of Quantization

Theorem. If X is nilpotent, then OX is a cone.

Attached to OX: ???

One proposed construction by Graham and Vogan (at

least for G complex).

Page 26: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

10

Building Representations

The orbit method philosophy can be summarized by

Build Representation Build Orbit

1 Find rigid reps Xn

2 Cohom induce from 1 Xn + Xe

3 Parab induce from 2 Xn + Xe + Xh

Page 27: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

10

Building Representations

The orbit method philosophy can be summarized by

Build Representation Build Orbit

1 Find rigid reps Xn

2 Cohom induce from 1 Xn + Xe

3 Parab induce from 2 Xn + Xe + Xh

Reality: This process will not produce all irreducible

unitary representations for semisimple groups. Example:

Compl series in SL2(R). However, it’s better than

anything else.

Page 28: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

11

A Bit about Graham-Vogan Construction

OX, a nilpotent, orbit is a symplectic manifold. Study

Lagrangian submanifolds.

Page 29: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

11

A Bit about Graham-Vogan Construction

OX, a nilpotent, orbit is a symplectic manifold. Study

Lagrangian submanifolds.

Fix Borel B, unipotent radical N . OX ∩ n is locally closed

alg variety. Write as components (orbital varieties):

OX ∩ n =⋃i

Vi

Page 30: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

11

A Bit about Graham-Vogan Construction

OX, a nilpotent, orbit is a symplectic manifold. Study

Lagrangian submanifolds.

Fix Borel B, unipotent radical N . OX ∩ n is locally closed

alg variety. Write as components (orbital varieties):

OX ∩ n =⋃i

Vi

Theorem. [Ginzburg] V Lagrangian in O.

Page 31: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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

Page 32: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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,

Page 33: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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.

Page 34: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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).

Page 35: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

13

Nilpotent Orbits

First, we would like to know what these things look like.

Take G complex and classical.

Page 36: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

13

Nilpotent Orbits

First, we would like to know what these things look like.

Take G complex and classical. Start with G = GLn(C).

Page 37: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

13

Nilpotent Orbits

First, we would like to know what these things look like.

Take G complex and classical. Start with G = GLn(C).

Fact: Conjugacy classes (adjoint orbits) in glnC are

determined by the Jordan canonical form.

Page 38: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

13

Nilpotent Orbits

First, we would like to know what these things look like.

Take G complex and classical. Start with G = GLn(C).

Fact: Conjugacy classes (adjoint orbits) in glnC are

determined by the Jordan canonical form.

For nilpotent conjugacy classes, this says:

nilpotent orbits in glnC ≈

Page 39: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

13

Nilpotent Orbits

First, we would like to know what these things look like.

Take G complex and classical. Start with G = GLn(C).

Fact: Conjugacy classes (adjoint orbits) in glnC are

determined by the Jordan canonical form.

For nilpotent conjugacy classes, this says:

nilpotent orbits in glnC ≈sizes of the Jordan blocks ≈

Page 40: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

13

Nilpotent Orbits

First, we would like to know what these things look like.

Take G complex and classical. Start with G = GLn(C).

Fact: Conjugacy classes (adjoint orbits) in glnC are

determined by the Jordan canonical form.

For nilpotent conjugacy classes, this says:

nilpotent orbits in glnC ≈sizes of the Jordan blocks ≈

partitions of n

Page 41: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

14

Nilpotent Orbits

Example: There are five nilpotent orbits in gl4(C)corresponding to the five partitions of 4:

[4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1], ie:

Page 42: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

14

Nilpotent Orbits

Example: There are five nilpotent orbits in gl4(C)corresponding to the five partitions of 4:

[4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1], ie:

Page 43: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

15

Nilpotent Orbits (cont.)

Theorem. [Gerstenhaber] G a complex classical reduc-

tive Lie group of rank n. The set of nilpotent orbits in g

is parameterized by

Page 44: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

15

Nilpotent Orbits (cont.)

Theorem. [Gerstenhaber] G a complex classical reduc-

tive Lie group of rank n. The set of nilpotent orbits in g

is parameterized by

· (GLn) partitions of n

Page 45: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

15

Nilpotent Orbits (cont.)

Theorem. [Gerstenhaber] G a complex classical reduc-

tive Lie group of rank n. The set of nilpotent orbits in g

is parameterized by

· (GLn) partitions of n

· (SO2n+1) partitions of 2n + 1 whose even parts occur with even

multiplicity

Page 46: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

15

Nilpotent Orbits (cont.)

Theorem. [Gerstenhaber] G a complex classical reduc-

tive Lie group of rank n. The set of nilpotent orbits in g

is parameterized by

· (GLn) partitions of n

· (SO2n+1) partitions of 2n + 1 whose even parts occur with even

multiplicity

· (Sp2n) partitions of 2n whose odd parts occur with even multiplicity

Page 47: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

15

Nilpotent Orbits (cont.)

Theorem. [Gerstenhaber] G a complex classical reduc-

tive Lie group of rank n. The set of nilpotent orbits in g

is parameterized by

· (GLn) partitions of n

· (SO2n+1) partitions of 2n + 1 whose even parts occur with even

multiplicity

· (Sp2n) partitions of 2n whose odd parts occur with even multiplicity

· (SO2n) partitions of 2n whose even parts occur with even multipli-

city (*)

Page 48: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

16

Nilpotent Orbits (cont.)

Example: G = Sp(6) has eight nilpotent orbits

corresponding to the Young diagrams:

Page 49: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

16

Nilpotent Orbits (cont.)

Example: G = Sp(6) has eight nilpotent orbits

corresponding to the Young diagrams:

Page 50: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

Page 51: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

• F is the variety of (isotropic) flags.

Page 52: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

• F is the variety of (isotropic) flags.

• FX are those fixed by X.

Page 53: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

• F is the variety of (isotropic) flags.

• FX are those fixed by X.

• AX = GX/GoX.

Page 54: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

• F is the variety of (isotropic) flags.

• FX are those fixed by X.

• AX = GX/GoX. GX acts on FX, and so AX acts on

Irr(FX).

Page 55: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

• F is the variety of (isotropic) flags.

• FX are those fixed by X.

• AX = GX/GoX. GX acts on FX, and so AX acts on

Irr(FX).

Fact. AX is trivial in type A and a 2-group in the other

classical types.

Page 56: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n.

Page 57: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n. Fix a flag F ∈ FX

Page 58: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n. Fix a flag F ∈ FX

F : F1 ⊂ F2 ⊂ . . . Fn

Page 59: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n. Fix a flag F ∈ FX

F : F1 ⊂ F2 ⊂ . . . Fn

Define a new (smaller) flag F ′ by

F ′ : F2/F1 ⊂ F3/F1 ⊂ . . . Fn/F1

Page 60: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n. Fix a flag F ∈ FX

F : F1 ⊂ F2 ⊂ . . . Fn

Define a new (smaller) flag F ′ by

F ′ : F2/F1 ⊂ F3/F1 ⊂ . . . Fn/F1

and a nilpotent element X ′ by X ′ = X|F ′.

Page 61: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n. Fix a flag F ∈ FX

F : F1 ⊂ F2 ⊂ . . . Fn

Define a new (smaller) flag F ′ by

F ′ : F2/F1 ⊂ F3/F1 ⊂ . . . Fn/F1

and a nilpotent element X ′ by X ′ = X|F ′. Then

F ′ ∈ FX′.

Page 62: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

18

Irreducible Components of FX

First, let G = GLn(C). Nilpotent orbit OX corresponds

to a partition P of n. Fix a flag F ∈ FX

F : F1 ⊂ F2 ⊂ . . . Fn

Define a new (smaller) flag F ′ by

F ′ : F2/F1 ⊂ F3/F1 ⊂ . . . Fn/F1

and a nilpotent element X ′ by X ′ = X|F ′. Then

F ′ ∈ FX′. Similarly, define F ′′, X ′′, etc.

Page 63: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

19

Irreducible Components of FX

Fact: If OX corresponds to the Young diagram D and

OX′ corresponds to the Young diagram D′, then D \D′

is a square.

Page 64: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

19

Irreducible Components of FX

Fact: If OX corresponds to the Young diagram D and

OX′ corresponds to the Young diagram D′, then D \D′

is a square.

Example: (G = GL5) One possibility is:

→ → → →

Page 65: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

20

Labelling the squares:

• Label the square D \D′ with an ′′n′′

• Label the square D′ \D′′ with an ′′n− 1′′, etc.

→ → → →

becomes:

Page 66: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

20

Labelling the squares:

• Label the square D \D′ with an ′′n′′

• Label the square D′ \D′′ with an ′′n− 1′′, etc.

→ → → →

becomes:

1 3 42 5

Page 67: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

20

Labelling the squares:

• Label the square D \D′ with an ′′n′′

• Label the square D′ \D′′ with an ′′n− 1′′, etc.

→ → → →

becomes:

1 3 42 5

(Standard Young Tableau, write SY T ([3, 2]))

Page 68: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

21

Irreducible Components of FX

We’ve defined a map: Φ : FX → SY T (P ).

Page 69: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

21

Irreducible Components of FX

We’ve defined a map: Φ : FX → SY T (P ).

Fact: Φ defines a bijection

Irr(FX)→ SY T (P )

Page 70: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

21

Irreducible Components of FX

We’ve defined a map: Φ : FX → SY T (P ).

Fact: Φ defines a bijection

Irr(FX)→ SY T (P )

Corollary: Φ defines a bijection

Irr(OX ∩ n) −→ SY T (P )

Page 71: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

22

Other Classical Groups

• Flags are isotropic flags

Page 72: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

Page 73: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

• D \D′ is a domino!

Page 74: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

• D \D′ is a domino!

• Can define a map Φ : FX → SDT (P )

Page 75: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

• D \D′ is a domino!

• Can define a map Φ : FX → SDT (P )

Two Problems:

• Φ not surjective,

Page 76: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

• D \D′ is a domino!

• Can define a map Φ : FX → SDT (P )

Two Problems:

• Φ not surjective,

• Φ does not separate the components Irr(FX).

Page 77: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

23

Example: G = Sp(6), then

123

is not in the image of Φ.

Page 78: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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(FX) | OX nilpotent} −→ ST (n).

Page 79: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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(FX) | OX nilpotent} −→ ST (n).

Lemma. Suppose P is the partition of the nilpotent orbit OX. For

a fixed AX orbit on Irr(FX), there is a unique tableau in the image

of Ψ of shape P .

Page 80: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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(FX) | OX nilpotent} −→ ST (n).

Lemma. Suppose P is the partition of the nilpotent orbit OX. For

a fixed AX orbit on Irr(FX), there is a unique tableau in the image

of Ψ of shape P .

Corollary. Ψ defines a bijection

Irr(OX ∩ n) −→ ST (P ).

Page 81: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

25

Example: Let G = Sp(6) and let X lie in the orbit corresponding to

the partition [4, 2]. The group AX has order 2. There are four

irreducible components of Irr(FX), corresponding to the tableaux:

1 23

12 3 1 2

31 32

Page 82: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

25

Example: Let G = Sp(6) and let X lie in the orbit corresponding to

the partition [4, 2]. The group AX has order 2. There are four

irreducible components of Irr(FX), corresponding to the tableaux:

1 23

12 3 1 2

31 32

Each component is fixed by the action of AX, except the first two,

which are interchanged. By our corollary, there are three orbital

varieties in OX, corresponding to the tableaux

Page 83: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

25

Example: Let G = Sp(6) and let X lie in the orbit corresponding to

the partition [4, 2]. The group AX has order 2. There are four

irreducible components of Irr(FX), corresponding to the tableaux:

1 23

12 3 1 2

31 32

Each component is fixed by the action of AX, except the first two,

which are interchanged. By our corollary, there are three orbital

varieties in OX, corresponding to the tableaux

1 23 1 2

31 32

Page 84: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

26

Back to Graham-Vogan

Theorem. [] The stabilizer subgroup Q of an orbital variety V can

be “read off” from its tableau T .

Page 85: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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:

Page 86: Representation Theory and Orbital Varietiestpietrah/PAPERS/tufts.pdf · 2003-11-21 · Theorem. [Parabolic Induction] If X is hyperbolic, there is a G-equivariant fibration O X →Z

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 OX a “small” nilpotent orbit.

The infinitesimal characters of the Graham-Vogan spaces attached

to OX have precisely the infinitesimal characters attached to OX by

McGovern.