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Computation of Invariants for Harish-Chandra
Modules of SU(p, q) by Combining Algebraic and
Geometric Methods.
Matthew Housley
housley@math.utah.edu
November 6th, 2010
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Slides and notes are available at
www.math.utah.edu/~housley → talks.
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Definition
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Definition: Lie Group
I
G is a differentiable manifold with a group operation.
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Definition
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Definition: Lie Group
I
G is a differentiable manifold with a group operation.
I
Smooth multiplication.
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Definition
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Definition: Lie Group
I
G is a differentiable manifold with a group operation.
I
Smooth multiplication.
I
Smooth inverse.
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Definition
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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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Definition
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Symmetry
Lie groups: smooth symmetries.
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Definition
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Symmetry
Examples:
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Definition
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Examples:
I
GL(n, C) is the set of all invertible complex linear
transformations on Cn . (Complex)
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Examples:
I
GL(n, C) is the set of all invertible complex linear
transformations on Cn . (Complex)
I
GL(n, R) is the set of all invertible real linear transformations
on Rn . (Real)
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Definition
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Examples:
I
GL(n, C) is the set of all invertible complex linear
transformations on Cn . (Complex)
I
GL(n, R) is the set of all invertible real linear transformations
on Rn . (Real)
I
SO(n) is the set of orientation preserving isometries of the
(n − 1)-sphere. (Real)
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Definition
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Examples:
I
GL(n, C) is the set of all invertible complex linear
transformations on Cn . (Complex)
I
GL(n, R) is the set of all invertible real linear transformations
on Rn . (Real)
I
SO(n) is the set of orientation preserving isometries of the
(n − 1)-sphere. (Real)
E.g. SO(3) is the set of rotations of the 2-sphere.
I
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Definition
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Symmetry
Examples:
I
GL(n, C) is the set of all invertible complex linear
transformations on Cn . (Complex)
I
GL(n, R) is the set of all invertible real linear transformations
on Rn . (Real)
I
SO(n) is the set of orientation preserving isometries of the
(n − 1)-sphere. (Real)
E.g. SO(3) is the set of rotations of the 2-sphere.
I
I
Has applications to quantum mechanics.
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Definition
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Examples:
I
SU(p, q) is the set of invertible complex linear transformations
of Cp+q that preserve the hermitian form given by
0
∗ Ip
hv , w i = v
w.
0 −Iq
(real)
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Definition
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Reductive Groups
I
The above examples are reductive.
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Reductive Groups
I
The above examples are reductive.
I
Roughly speaking: no interesting normal subgroups.
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Reductive Groups
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The above examples are reductive.
I
Roughly speaking: no interesting normal subgroups.
I
We’ll assume this from now on.
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Definition
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Lie Algebras
Start with a Lie group G . Define g to be the tangent space to G
at id.
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Definition
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Lie Algebras
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The group operation of G induces an operation [−, −] on g. g
is a Lie algebra under this operation.
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Definition
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Lie Algebras
I
The group operation of G induces an operation [−, −] on g. g
is a Lie algebra under this operation.
I
Roughly speaking: g approximates G near the identity.
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Definition
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Lie Algebras
I
The group operation of G induces an operation [−, −] on g. g
is a Lie algebra under this operation.
I
Roughly speaking: g approximates G near the identity.
I
[−, −] is bilinear.
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Definition
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Lie Algebras
I
The group operation of G induces an operation [−, −] on g. g
is a Lie algebra under this operation.
I
Roughly speaking: g approximates G near the identity.
I
[−, −] is bilinear.
I
[x, y ] = −[y , x].
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Definition
Symmetry
Reductive Groups
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Lie Algebras
I
The group operation of G induces an operation [−, −] on g. g
is 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.
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Definition
Symmetry
Reductive Groups
Lie Algebras
Lie Algebras
I
The group operation of G induces an operation [−, −] on g. g
is 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.
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Definition
Symmetry
Reductive Groups
Lie Algebras
Lie Algebras
I
The group operation of G induces an operation [−, −] on g. g
is 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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Irreducibility
Lie Algebra Representations
Infinite Dimensional Case
Representations
I
Let V be a (complex) vector space.
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Irreducibility
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Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
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I
I
Let V be a (complex) vector space.
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 .
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Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Reductive: more or less the whole group acts on V .
Finite dimensional representations arise naturally.
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Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Reductive: more or less the whole group acts on V .
Finite dimensional representations arise naturally.
I
Standard representation for matrix groups.
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Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Reductive: more or less the whole group acts on V .
Finite dimensional representations arise naturally.
I
I
Standard representation for matrix groups.
Adjoint representation: G acts on g.
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Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Reductive: more or less the whole group acts on V .
Finite dimensional representations arise naturally.
I
I
I
Standard representation for matrix groups.
Adjoint representation: G acts on g.
Subs: restriction to an invariant subspace.
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Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Reductive: more or less the whole group acts on V .
Finite dimensional representations arise naturally.
I
I
I
I
Standard representation for matrix groups.
Adjoint representation: G acts on g.
Subs: restriction to an invariant subspace.
Quotients.
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Irreducibility
Lie Algebra Representations
Infinite Dimensional Case
Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Reductive: more or less the whole group acts on V .
Finite dimensional representations arise naturally.
I
I
I
I
I
Standard representation for matrix groups.
Adjoint representation: G acts on g.
Subs: restriction to an invariant subspace.
Quotients.
Generating new representations. (Tensor products, exterior
powers, etc.)
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Irreducibility
Lie Algebra Representations
Infinite Dimensional Case
Representations
I
I
Let V be a (complex) vector space.
Given a smooth group homomorphism from G to GL(V ), we
call V a representation of G .
I
I
Finite dimensional representations arise naturally.
I
I
I
I
I
I
Reductive: more or less the whole group acts on V .
Standard representation for matrix groups.
Adjoint representation: G acts on g.
Subs: restriction to an invariant subspace.
Quotients.
Generating new representations. (Tensor products, exterior
powers, etc.)
Finite dimensional representations of real and complex Lie
groups are well understood.
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Irreducibility
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Irreducibility
I
V is an irreducible representation of G if it contains no proper
nonzero subrepresentations of G .
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Irreducibility
I
V is an irreducible representation of G if it contains no proper
nonzero subrepresentations of G .
I
Analogous to prime numbers, finite simple groups, etc.
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Irreducibility
I
V is an irreducible representation of G if it contains no proper
nonzero subrepresentations of G .
I
Analogous to prime numbers, finite simple groups, etc.
I
Roughly: irreducible representations are simplest actions of
the symmetries in G .
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Lie Algebra Representations
I
If V is a G representation, it becomes a g representation via
differentiation.
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Lie Algebra Representations
I
If V is a G representation, it becomes a g representation via
differentiation.
I
We complexify g to gC .
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Lie Algebra Representations
I
If V is a G representation, it becomes a g representation via
differentiation.
I
We complexify g to gC .
I
Irreducible representations of gC are important in classifying
irreducible representations of G .
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Infinite Dimensional Motivations
I
Start with a manifold M with a measure.
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Infinite Dimensional Motivations
I
I
Start with a manifold M with a measure.
Find a real Lie group G that acts on M and preserves the
measure.
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Infinite Dimensional Case
Infinite Dimensional Motivations
I
I
Start with a manifold M with a measure.
Find a real Lie group G that acts on M and preserves the
measure.
I
Irreducible representations of complex Lie groups are finite
dimensional.
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Infinite Dimensional Motivations
I
I
Start with a manifold M with a measure.
Find a real Lie group G that acts on M and preserves the
measure.
I
I
Irreducible representations of complex Lie groups are finite
dimensional.
Consider the action of G on L2 (M).
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Harish-Chandra Modules
I
Harish-Chandra modules: algebraizations of infinite
dimensional representations.
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Harish-Chandra Modules
I
Harish-Chandra modules: algebraizations of infinite
dimensional representations.
I
We’ll ignore this distinction.
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Irreducibility
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Infinite Dimensional Case
Harish-Chandra Modules
I
Harish-Chandra modules: algebraizations of infinite
dimensional representations.
I
We’ll ignore this distinction.
I
We can study infinite dimensional representations using
algebraic and algebro-geometric techniques.
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Geometric Invariants
SU(p, q)
Geometric Invariants
Two geometric invariants to consider:
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Geometric Invariants
SU(p, q)
Geometric Invariants
Two geometric invariants to consider:
I
Associated variety AV(X ): a variety contained in gC .
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Geometric Invariants
SU(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 an
integer (multiplicity) to each component of AV(X ).
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Geometric Invariants
SU(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 an
integer (multiplicity) to each component of AV(X ).
I
We’d like to calculate the multiplicities.
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Geometric Invariants
SU(p, q)
SU(p, q)
I
For this group, associated variety for irreducible X is an
irreducible variety.
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Geometric Invariants
SU(p, q)
SU(p, q)
I
For this group, associated variety for irreducible X is an
irreducible variety.
I
We only need to compute one multiplicity to get AC(X ).
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The Weyl Group
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More Symmetry: the Weyl Group
I
The Weyl group W is a finite set of internal symmetries of G .
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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 ).
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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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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
I
X is contained in a finite cell C = {X1 , X2 , . . . , Xk } of
representations that all have the same associated variety.
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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
I
X is contained in a finite cell C = {X1 , X2 , . . . , Xk } of
representations that all have the same associated variety.
I
Take the formal Z-span of the elements of C.
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The Weyl Group
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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
I
X is contained in a finite cell C = {X1 , X2 , . . . , Xk } of
representations 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 Weyl
group Sp+q .
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The Weyl Group
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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
I
X is contained in a finite cell C = {X1 , X2 , . . . , Xk } of
representations 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 Weyl
group Sp+q .
I
Let mXi denote the multiplicity in the associated variety of Xi .
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The Weyl Group
Cells
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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
I
X is contained in a finite cell C = {X1 , X2 , . . . , Xk } of
representations 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 Weyl
group Sp+q .
I
Let mXi denote the multiplicity in the associated variety of Xi .
I
The representation relates the multiplicities mXi for the
various Xi in C.
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The Weyl Group
Cells
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Cells
I
Let X be an infinite dimensional representation of SU(p, q).
I
X is contained in a finite cell C = {X1 , X2 , . . . , Xk } of
representations 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 Weyl
group Sp+q .
I
Let mXi denote the multiplicity in the associated variety of Xi .
I
The representation relates the multiplicities mXi for the
various Xi in C.
I
If we know mXj for some Xj we can calculate mXi for the
other Xi in the cell C.
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Strategy
I
For SU(p, q) and any cell C of representations, we can always
find an Xi ∈ C so that mXi can be computed by geometric
means.
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The Weyl Group
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Strategy
I
For SU(p, q) and any cell C of representations, we can always
find an Xi ∈ C so that mXi can be computed by geometric
means. (Springer fiber.)
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The Weyl Group
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Strategy
I
For SU(p, q) and any cell C of representations, we can always
find an Xi ∈ C so that mXi can be computed by geometric
means. (Springer fiber.)
I
Compute the Sp+q representation on spanZ C.
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The Weyl Group
Cells
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Strategy
I
For SU(p, q) and any cell C of representations, we can always
find an Xi ∈ C so that mXi can be computed by geometric
means. (Springer fiber.)
I
Compute the Sp+q representation on spanZ C.
I
Compute mXi for other Xi in C.
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The Weyl Group
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Problem
It can be difficult to compute the Sp+q representation on spanZ C.
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The Weyl Group
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Symmetric Group Representations
I
Let n = p + q.
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Symmetric Group Representations
I
Let n = p + q.
I
Basis: {e1 − e2 , e2 − e3 , . . . , en−1 − en }.
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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.
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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 .
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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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The Weyl Group
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Symmetric Group Representations
More Representations of Sn
Irreducible representations of Sn are parametrized by Young
diagrams with n boxes.
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The Weyl Group
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More Representations of Sn
Irreducible representations of Sn are parametrized by Young
diagrams with n boxes.
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The Weyl Group
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More Representations of Sn
Irreducible representations of Sn are parametrized by Young
diagrams with n boxes.
Construction: take subspaces of tensor powers of the standard
representation.
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The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
Hook type diagram: upside down L.
78 / 91
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The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
Hook type diagram: upside down L.
Two rows with one box on the bottom row: standard
representation.
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Lie Groups
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Geometry
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The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
General hook type with m + 1 rows:
standard representations.
Vm
V , where V is the
80 / 91
Lie Groups
Representations
Geometry
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Results
The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
V
General hook type with m + 1 rows: m V , where V is the
standard representations.
V
Example: (12) action on (e1 − e2 ) ∧ (e2 − e3 ) for 2 V :
81 / 91
Lie Groups
Representations
Geometry
More Symmetry
Results
The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
V
General hook type with m + 1 rows: m V , where V is the
standard representations.
V
Example: (12) action on (e1 − e2 ) ∧ (e2 − e3 ) for 2 V :
(e2 − e1 ) ∧ (e1 − e3 )
82 / 91
Lie Groups
Representations
Geometry
More Symmetry
Results
The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
V
General hook type with m + 1 rows: m V , where V is the
standard representations.
V
Example: (12) action on (e1 − e2 ) ∧ (e2 − e3 ) for 2 V :
(e2 − e1 ) ∧ (e1 − e3 )
= −(e1 − e2 ) ∧ (e1 − e2 + e2 − e3 )
83 / 91
Lie Groups
Representations
Geometry
More Symmetry
Results
The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
V
General hook type with m + 1 rows: m V , where V is the
standard representations.
V
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 )
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Lie Groups
Representations
Geometry
More Symmetry
Results
The Weyl Group
Cells
Symmetric Group Representations
Hook Type Representations
V
General hook type with m + 1 rows: m V , where V is the
standard representations.
V
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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Lie Groups
Representations
Geometry
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Results
Strategy
I
Find a hook type cell C.
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Lie Groups
Representations
Geometry
More Symmetry
Results
Strategy
I
Find a hook type cell C.
I
Find one Xj in the cell such that computation of mXj is easy.
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Lie Groups
Representations
Geometry
More Symmetry
Results
Strategy
I
Find a hook type cell C.
I
Find one Xj in the cell such that computation of mXj is easy.
I
Find mXi for other Xi in C by using the Sn representation on
spanZ C.
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Lie Groups
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Results
Result
We get a formula for mXi where Xi is any infinite dimensional
representation in a hook type cell:
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Lie Groups
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Results
Result
We get a formula for mXi where Xi is any infinite dimensional
representation in a hook type cell:

!
X
Y
X
1

sgn(σ 0 )σ 0 ·
xτm−i+1
mXi = Am Q
sgn(σ)σ·
(i)
|τk |!
0
σ∈Sτ
where
Am =
σ ∈Sm
i=1...m
1
.
m! · (m − 1)! · · · 1
90 / 91
Lie Groups
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Details
Details are available at www.math.utah.edu/~housley →
research.
housley@math.utah.edu
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