Representation Theory and Orbital

Varieties

Thomas Pietraho

Bowdoin College

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 .

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.

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.

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.

2

Lie Groups and the Orbit Method

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

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.

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

3

Wishful thinking

Classical Mechanics←→ Coadjoint Orbits

Quantum Mechanics←→ Irred Unitary Reps

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.

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).

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,

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

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.

6

Three Flavors of Coadjoint Orbits

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

inclusion

g∗ ⊂Mn

An element X ∈ g∗ is:

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

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

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

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.

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.

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

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.

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.

9

Three Flavors of Quantization

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

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).

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

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.

11

A Bit about Graham-Vogan Construction

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

Lagrangian submanifolds.

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

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.

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

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,

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.

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).

13

Nilpotent Orbits

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

Take G complex and classical.

13

Nilpotent Orbits

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

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

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.

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 ≈

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 ≈

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

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:

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:

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

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

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

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

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 (*)

16

Nilpotent Orbits (cont.)

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

corresponding to the Young diagrams:

16

Nilpotent Orbits (cont.)

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

corresponding to the Young diagrams:

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

17

Orbital Varieties

Theorem. [Spaltenstein] There is a bijection

Irr(OX ∩ n)→ Irr(FX)/AX

• F is the variety of (isotropic) flags.

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.

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.

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).

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.

18

Irreducible Components of FX

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

to a partition P of n.

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

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

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

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 ′.

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′.

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.

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.

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:

→ → → →

20

Labelling the squares:

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

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

→ → → →

becomes:

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

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]))

21

Irreducible Components of FX

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

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 )

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 )

22

Other Classical Groups

• Flags are isotropic flags

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

22

Other Classical Groups

• Flags are isotropic flags

• Can define F ′ similarly

• D \D′ is a domino!

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 )

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,

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).

23

Example: G = Sp(6), then

123

is not in the image of Φ.

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).

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 .

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 ).

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

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

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

26

Back to Graham-Vogan

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

be “read off” from its tableau T .

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:

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.