Representation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College

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 Ĝunitary.

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

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.

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

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 Ĝunitary 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 GL n has discrete kernel. This leads to an inclusion An element X g is: g M n

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

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

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

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.

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.

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

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.

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.

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

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).

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

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.

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

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

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.

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 = GL n (C).

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.

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

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

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

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:

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:

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

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

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

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

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 (*)

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(O X n) Irr(F X )/A X

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.

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.

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.

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 ).

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.

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

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

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

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

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.

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.

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.

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.

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:

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 4 2 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 4 2 5 (Standard Young Tableau, write SY T ([3, 2]))

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

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 )

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 )

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 : F X SDT (P )

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,

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 ).

23 Example: G = Sp(6), then 1 2 3 is not in the image of Φ.

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

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.

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 ).

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: 1 2 3 1 2 3 1 2 3 1 3 2

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: 1 2 3 1 2 3 1 2 3 1 3 2 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

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: 1 2 3 1 2 3 1 2 3 1 3 2 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 1 2 3 1 2 3 1 3 2

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 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.