computation of invariants for harish-chandra modules of su p q)...

Lie GroupsRepresentations

GeometryMore Symmetry

Results

Computation of Invariants for Harish-ChandraModules of SU(p, q) by Combining Algebraic and

Geometric Methods.

Matthew Housleyhousley@math.utah.edu

November 6th, 2010

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Slides and notes are available atwww.math.utah.edu/~housley → talks.

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DefinitionSymmetryReductive GroupsLie Algebras

Definition: Lie Group

I G is a differentiable manifold with a group operation.

I Smooth multiplication.

I Smooth inverse.

I Complex and real flavors.

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I Smooth inverse.

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I Smooth inverse.

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I Smooth inverse.

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Symmetry

Lie groups: smooth symmetries.

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Symmetry

Examples:

I GL(n,C) is the set of all invertible complex lineartransformations on Cn. (Complex)

I GL(n,R) is the set of all invertible real linear transformationson Rn. (Real)

I SO(n) is the set of orientation preserving isometries of the(n − 1)-sphere. (Real)

I E.g. SO(3) is the set of rotations of the 2-sphere.I Has applications to quantum mechanics.

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Symmetry

Examples:

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Symmetry

Examples:

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Symmetry

Examples:

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Symmetry

Examples:

I E.g. SO(3) is the set of rotations of the 2-sphere.

I Has applications to quantum mechanics.

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Symmetry

Examples:

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Symmetry

Examples:

I SU(p, q) is the set of invertible complex linear transformationsof Cp+q that preserve the hermitian form given by

〈v ,w〉 = v∗(Ip 00 −Iq

(real)

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Reductive Groups

I The above examples are reductive.

I Roughly speaking: no interesting normal subgroups.

I We’ll assume this from now on.

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Reductive Groups

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Reductive Groups

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Lie Algebras

Start with a Lie group G . Define g to be the tangent space to Gat id .

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Lie Algebras

I The group operation of G induces an operation [−,−] on g. gis a Lie algebra under this operation.

I Roughly speaking: g approximates G near the identity.

I [−,−] is bilinear.

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

I Can be derived from the Lie bracket of differential geometry.

I The structures of G and g are closely related.

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Lie Algebras

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

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Lie Algebras

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

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Lie Algebras

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

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Lie Algebras

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

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Lie Algebras

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

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Lie Algebras

I [x , y ] = −[y , x ].

I [x , [y , z ]] + [z , [x , y ]] + [y , [z , x ]] = 0.

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IrreducibilityLie Algebra RepresentationsInfinite Dimensional Case

Representations

I Let V be a (complex) vector space.

I Given a smooth group homomorphism from G to GL(V ), wecall V a representation of G .

I Reductive: more or less the whole group acts on V .

I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior

powers, etc.)

I Finite dimensional representations of real and complex Liegroups are well understood.

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Representations

I Let V be a (complex) vector space.I Given a smooth group homomorphism from G to GL(V ), we

call V a representation of G .

I Reductive: more or less the whole group acts on V .

powers, etc.)

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Representations

call V a representation of G .I Reductive: more or less the whole group acts on V .

powers, etc.)

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Representations

I Finite dimensional representations arise naturally.

I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior

powers, etc.)

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Representations

I Finite dimensional representations arise naturally.I Standard representation for matrix groups.

I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior

powers, etc.)

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Representations

I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.

I Subs: restriction to an invariant subspace.I Quotients.I Generating new representations. (Tensor products, exterior

powers, etc.)

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Representations

I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.

I Quotients.I Generating new representations. (Tensor products, exterior

powers, etc.)

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Representations

I Finite dimensional representations arise naturally.I Standard representation for matrix groups.I Adjoint representation: G acts on g.I Subs: restriction to an invariant subspace.I Quotients.

I Generating new representations. (Tensor products, exteriorpowers, etc.)

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Representations

powers, etc.)

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Representations

powers, etc.)

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Irreducibility

I V is an irreducible representation of G if it contains no propernonzero subrepresentations of G .

I Analogous to prime numbers, finite simple groups, etc.

I Roughly: irreducible representations are simplest actions ofthe symmetries in G .

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Irreducibility

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Irreducibility

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Lie Algebra Representations

I If V is a G representation, it becomes a g representation viadifferentiation.

I We complexify g to gC.

I Irreducible representations of gC are important in classifyingirreducible representations of G .

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Infinite Dimensional Motivations

I Start with a manifold M with a measure.

I Find a real Lie group G that acts on M and preserves themeasure.

I Irreducible representations of complex Lie groups are finitedimensional.

I Consider the action of G on L2(M).

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I Start with a manifold M with a measure.I Find a real Lie group G that acts on M and preserves the

measure.

I Irreducible representations of complex Lie groups are finitedimensional.

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measure.I Irreducible representations of complex Lie groups are finite

dimensional.

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measure.I Irreducible representations of complex Lie groups are finite

dimensional.

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Harish-Chandra Modules

I Harish-Chandra modules: algebraizations of infinitedimensional representations.

I We’ll ignore this distinction.

I We can study infinite dimensional representations usingalgebraic and algebro-geometric techniques.

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Geometric InvariantsSU(p, q)

Geometric Invariants

Two geometric invariants to consider:

I Associated variety AV(X ): a variety contained in gC.

I Associated cycle AC(X ): finer invariant that attaches aninteger (multiplicity) to each component of AV(X ).

I We’d like to calculate the multiplicities.

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SU(p, q)

I For this group, associated variety for irreducible X is anirreducible variety.

I We only need to compute one multiplicity to get AC(X ).

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SU(p, q)

I For this group, associated variety for irreducible X is anirreducible variety.

I We only need to compute one multiplicity to get AC(X ).

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The Weyl GroupCellsSymmetric Group Representations

More Symmetry: the Weyl Group

I The Weyl group W is a finite set of internal symmetries of G .

I W = NG (T )/ZG (T ).

I The Weyl group for SU(p, q) is the symmetric group Sp+q.

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I W = NG (T )/ZG (T ).

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I W = NG (T )/ZG (T ).

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I Let X be an infinite dimensional representation of SU(p, q).

I X is contained in a finite cell C = {X1,X2, . . . ,Xk} ofrepresentations that all have the same associated variety.

I Take the formal Z-span of the elements of C.

I spanZ C becomes an irreducible representation of the Weylgroup Sp+q.

I Let mXidenote the multiplicity in the associated variety of Xi .

I The representation relates the multiplicities mXifor the

various Xi in C.

I If we know mXjfor some Xj we can calculate mXi

for theother Xi in the cell C.

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various Xi in C.

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various Xi in C.

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various Xi in C.

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various Xi in C.

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various Xi in C.

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various Xi in C.

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Strategy

I For SU(p, q) and any cell C of representations, we can alwaysfind an Xi ∈ C so that mXi

can be computed by geometricmeans.

(Springer fiber.)

I Compute the Sp+q representation on spanZ C.

I Compute mXifor other Xi in C.

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Strategy

can be computed by geometricmeans. (Springer fiber.)

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Strategy

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Strategy

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Problem

It can be difficult to compute the Sp+q representation on spanZ C.

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Symmetric Group Representations

I Let n = p + q.

I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.I Let Sn act in the “obvious” way.

For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.This is the standard representation V of Sn.

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I Let n = p + q.

I Basis: {e1 − e2, e2 − e3, . . . , en−1 − en}.

I Let Sn act in the “obvious” way.

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I Let n = p + q.

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I Let n = p + q.

For example, acting by (12):e1 − e2 → e2 − e1 and e3 − e1 → e3 − e2.

This is the standard representation V of Sn.

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I Let n = p + q.

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More Representations of Sn

Irreducible representations of Sn are parametrized by Youngdiagrams with n boxes.

Construction: take subspaces of tensor powers of the standardrepresentation.

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Hook Type Representations

Hook type diagram: upside down L.

Two rows with one box on the bottom row: standardrepresentation.

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Hook type diagram: upside down L.

Two rows with one box on the bottom row: standardrepresentation.

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General hook type with m + 1 rows:∧m V , where V is the

standard representations.

Example: (12) action on (e1 − e2) ∧ (e2 − e3) for∧2 V :

(e2 − e1) ∧ (e1 − e3)

= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)

= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)

= −(e1 − e2) ∧ (e2 − e3).

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standard representations.Example: (12) action on (e1 − e2) ∧ (e2 − e3) for

∧2 V :

(e2 − e1) ∧ (e1 − e3)

= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)

= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)

= −(e1 − e2) ∧ (e2 − e3).

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∧2 V :

(e2 − e1) ∧ (e1 − e3)

= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)

= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)

= −(e1 − e2) ∧ (e2 − e3).

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∧2 V :

(e2 − e1) ∧ (e1 − e3)

= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)

= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)

= −(e1 − e2) ∧ (e2 − e3).

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∧2 V :

(e2 − e1) ∧ (e1 − e3)

= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)

= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)

= −(e1 − e2) ∧ (e2 − e3).

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∧2 V :

(e2 − e1) ∧ (e1 − e3)

= −(e1 − e2) ∧ (e1 − e2 + e2 − e3)

= −(e1 − e2) ∧ (e1 − e2)− (e1 − e2) ∧ (e2 − e3)

= −(e1 − e2) ∧ (e2 − e3).

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Strategy

I Find a hook type cell C.

I Find one Xj in the cell such that computation of mXjis easy.

I Find mXifor other Xi in C by using the Sn representation on

spanZ C.

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Strategy

spanZ C.

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Strategy

spanZ C.

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Result

We get a formula for mXiwhere Xi is any infinite dimensional

representation in a hook type cell:

mXi= Am

1∏|τk |!

∑σ∈Sτ

sgn(σ)σ·

∑σ′∈Sm

sgn(σ′)σ′ ·

( ∏i=1...m

xm−i+1τ(i)

)where

m! · (m − 1)! · · · 1.

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Result

We get a formula for mXiwhere Xi is any infinite dimensional

representation in a hook type cell:

mXi= Am

1∏|τk |!

∑σ∈Sτ

sgn(σ)σ·

∑σ′∈Sm

sgn(σ′)σ′ ·

( ∏i=1...m

xm−i+1τ(i)

)where

m! · (m − 1)! · · · 1.

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Details

Details are available at www.math.utah.edu/~housley →research.

housley@math.utah.edu

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