The orbit method
for reductive
groups
David Vogan
Introduction
The orbit method for reductive groups
David Vogan
Department of Mathematics
Massachusetts Institute of Technology
Lie Theory and Geometry, May 19–24 2008
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Outline
The orbit method
for reductive
groups
David Vogan
Introduction: a few things I didn’t learn from Bert
Introduction
Commuting
algebras
Commuting algebras: how representation theory works
Differential operator algebras: how orbit method works
Hamiltonian G-spaces: how Bert does the orbit method
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Orbits for reductive groups: what else to steal from Bert
Meaning of it all
References (more theorems, fewer jokes)
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Abstract harmonic analysis
The orbit method
for reductive
groups
David Vogan
Say Lie group G acts on manifold M. Can ask about
I
I
I
topology of M
solutions of G-invariant differential equations
special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbert
space L2 (M). Now G acts by unitary operators.
Method step 2: DIAGONALIZE. Decompose L2 (M)
into minimal G-invariant subspaces.
Method step 3: REPRESENTATION THEORY. Study
minimal pieces: irreducible unitary repns of G.
Difficult questions: how does DIAGONALIZE work,
and what do minimal pieces look like?
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
The orbit method
for reductive
groups
Plan of talk
David Vogan
Introduction
I
Outline strategy for decomposing L2 (M): analogy
with “double centralizers” in finite-diml algebra.
I
Strategy
Philosophy of coadjoint orbits:
irreducible unitary representations
of Lie group G
m
(nearly) symplectic manifolds with
(nearly) transitive Hamiltonian action of G
I
“Strategy” and “philosophy” have a lot of wishful
thinking. Describe theorems supporting m.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
The orbit method
for reductive
groups
Plan of talk
David Vogan
Introduction
I
Outline strategy for decomposing L2 (M): analogy
with “double centralizers” in finite-diml algebra.
I
Strategy
Philosophy of coadjoint orbits:
irreducible unitary representations
of Lie group G
m
(nearly) symplectic manifolds with
(nearly) transitive Hamiltonian action of G
I
“Strategy” and “philosophy” have a lot of wishful
thinking. Describe theorems supporting m.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
The orbit method
for reductive
groups
Plan of talk
David Vogan
Introduction
I
Outline strategy for decomposing L2 (M): analogy
with “double centralizers” in finite-diml algebra.
I
Strategy
Philosophy of coadjoint orbits:
irreducible unitary representations
of Lie group G
m
(nearly) symplectic manifolds with
(nearly) transitive Hamiltonian action of G
I
“Strategy” and “philosophy” have a lot of wishful
thinking. Describe theorems supporting m.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
The orbit method
for reductive
groups
Plan of talk
David Vogan
Introduction
I
Outline strategy for decomposing L2 (M): analogy
with “double centralizers” in finite-diml algebra.
I
Strategy
Philosophy of coadjoint orbits:
irreducible unitary representations
of Lie group G
m
(nearly) symplectic manifolds with
(nearly) transitive Hamiltonian action of G
I
“Strategy” and “philosophy” have a lot of wishful
thinking. Describe theorems supporting m.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
The orbit method
for reductive
groups
Plan of talk
David Vogan
Introduction
I
Outline strategy for decomposing L2 (M): analogy
with “double centralizers” in finite-diml algebra.
I
Strategy
Philosophy of coadjoint orbits:
irreducible unitary representations
of Lie group G
m
(nearly) symplectic manifolds with
(nearly) transitive Hamiltonian action of G
I
“Strategy” and “philosophy” have a lot of wishful
thinking. Describe theorems supporting m.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Decomposing a representation
The orbit method
for reductive
groups
David Vogan
Given: interesting operators A on Hilbert space H.
Goal: decompose H in A-invt way.
Finite-dimensional case:
V /C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.
Classical structure theorem:
W1 , . . . , Wr list of all simple A-modules; then
A ' End(W1 ) × · · · × End(Wr ) V ' m1 W1 + · · · + mr Wr .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity space
Mi = HomA (Wi , V ); then mi = dim Mi , and
V ' M1 ⊗ W1 + · · · + Mr ⊗ Wr .
Slickest version: COMMUTING ALGEBRAS. . .
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Introduction
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
Commuting
algebras
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Commuting algebras and all that
The orbit method
for reductive
groups
David Vogan
Theorem
Suppose A is semisimple algebras of operators on V as
above; define Z = Cent End(V ) (A), a second semisimple
algebra of operators on V .
1. Relation between A and Z is symmetric:
A = Cent End(V ) (Z).
2. There is a natural bijection between irr modules Wi
for A and irr modules Mi for Z, given by
Mi ' HomA (Wi , V ),
Wi ' HomZ (Mi , V ).
P
3. V ' i Mi ⊗ Wi as a module for A × Z.
Example 1: finite G acts left and right on C[G].
Example 2: Sn and GL(E) act on V = T n (E).
But those are stories for other days. . .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Infinite-dimensional representations
The orbit method
for reductive
groups
David Vogan
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionary
Introduction
Commuting
algebras
Differential
operator algebras
finite-diml V
↔
C ∞ (M)
repn of G on V
End(V )
↔
↔
↔
↔
action of G on M
Diff(M)
A = im(C[G]) ⊂ End(V )
Z = CentEnd(V ) (A)
A = im(U(g)) ⊂ Diff(M)
Z = G-invt diff ops
Suggests: G-irreducible pieces of function space
correspond to simple modules for G-invt diff ops.
Which differential operators commute with G?
Answer leads to generalizations of dictionary. . .
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Differential operators and symbols
Diffn (M) = diff operators of order ≤ n.
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
σ : gr Diff(M) → Poly(T ∗ (M))
to fns on T ∗ (M) that are polynomial in fibers.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Differential operators and symbols
Diffn (M) = diff operators of order ≤ n.
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
σ : gr Diff(M) → Poly(T ∗ (M))
to fns on T ∗ (M) that are polynomial in fibers.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Differential operators and symbols
Diffn (M) = diff operators of order ≤ n.
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
σ : gr Diff(M) → Poly(T ∗ (M))
to fns on T ∗ (M) that are polynomial in fibers.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Differential operators and symbols
The orbit method
for reductive
groups
David Vogan
Diffn (M) = diff operators of order ≤ n.
Introduction
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
∗
σ : gr Diff(M) → Poly(T (M))
∗
to fns on T (M) that are polynomial in fibers.
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Differential operators and symbols
The orbit method
for reductive
groups
David Vogan
Diffn (M) = diff operators of order ≤ n.
Introduction
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
∗
σ : gr Diff(M) → Poly(T (M))
∗
to fns on T (M) that are polynomial in fibers.
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Differential operators and symbols
The orbit method
for reductive
groups
David Vogan
Diffn (M) = diff operators of order ≤ n.
Introduction
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
∗
σ : gr Diff(M) → Poly(T (M))
∗
to fns on T (M) that are polynomial in fibers.
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Differential operators and symbols
The orbit method
for reductive
groups
David Vogan
Diffn (M) = diff operators of order ≤ n.
Introduction
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
∗
σ : gr Diff(M) → Poly(T (M))
∗
to fns on T (M) that are polynomial in fibers.
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Differential operators and symbols
The orbit method
for reductive
groups
David Vogan
Diffn (M) = diff operators of order ≤ n.
Introduction
Increasing filtration, (Diffp )(Diffq ) ⊂ Diffp+q .
Theorem (Symbol calculus)
1. There is an isomorphism of graded algebras
∗
σ : gr Diff(M) → Poly(T (M))
∗
to fns on T (M) that are polynomial in fibers.
2.
σn : Diffn (M)/ Diffn−1 (M) → Polyn (T ∗ (M)).
3. Commutator of diff ops
Poisson bracket {, } on
T ∗ (M): for D ∈ Diffp (M), D 0 ∈ Diffq (M),
σp+q−1 ([D, D 0 ]) = {σp (D), σq (D 0 )}.
Diff ops comm with G ! symbols Poisson-comm with g.
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗ (M)).
ξf ∈ Vect(X ): ξf (g) = {f , g}.
Bracket with f
Lie alg hom g → Vect(X ), Y 7→ ξY .
Call X Hamiltonian G-space if the Lie alg action lifts
C ∞ (X )
%
↓
g → Vect(X )
Y
fY
% ↓
→ ξY
Map g → C ∞ (X ) same as moment map µ : X → g∗ .
Example. G acts on M ⇒ T ∗ (M) is Hamiltonian G-space: Lie alg elt
Y
vec fld ξY on M
function fY on T ∗ (M):
fY (m, λ) = λ(ξY (m))
David Vogan
Introduction
Vector flds ξf called Hamiltonian; preserve {, }. Map C ∞ (X ) → Vect(X ),
f 7→ ξf is Lie alg hom.
G action on X
The orbit method
for reductive
groups
(m ∈ M, λ ∈ Tm∗ (M)).
function f on X with {f , g} = 0 ⇔ f constant on G orbits.
?
G action transitive ⇒ only [C, G] = 0 ! irr repn of G
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Our story so far. . .
G acts on M ! T ∗ (M) Hamiltonian G-space.
G-decomp of C ∞ (M) ! (Diff M)G -modules.
σ
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(Diff M)G ! C ∞ (T ∗ (M))G ! C ∞ ((T ∗ (M))/G).
Hope C ∞ (M) irr ⇔ G has dense orbit on T ∗ (M).
Differential
operator algebras
Suggests generalization. . .
Hamiltonian
G-spaces
Hamiltonian G-cone X
graded alg Poly(X ).
σ
Seek filtered alg D, symbol calc gr D→ Poly(X )
carrying [, ] on D to {, } on Poly(X ).
Seek to lift G action on Poly(X ) to G action on D via
Lie alg hom g → D1 .
Seek simple D-module W (analogue of C ∞ (M)).
Hope W irr for G ⇔ G has dense orbit on X .
Suggests: irreducible representations of G !
homogeneous Hamiltonian G-spaces.
Coadjoint orbits for
reductive groups
Conclusion
References
Method of coadjoint orbits
The orbit method
for reductive
groups
David Vogan
Recall: Hamiltonian G-space X comes with
(G-equivariant) moment map µ : X → g∗ .
Kostant’s theorem: homogeneous Hamiltonian
G-space = covering of G-orbit on g∗ .
Includes classification of symp homog spaces for G.
(Riem homog spaces hopelessly complicated.)
Recall: commuting algebra formalism for diff operators
suggests irreducible representations ! homogeneous
Hamiltonian G-spaces.
Kirillov-Kostant philosophy of coadjt orbits suggests
b ! g∗ /G.
{irr unitary reps of G} = G
(?)
MORE PRECISELY. . . restrict right side to “admissible”
b
orbits (integrality cond). Expect to find “almost all” of G:
enough for interesting harmonic analysis.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Method of coadjoint orbits
The orbit method
for reductive
groups
David Vogan
Recall: Hamiltonian G-space X comes with
(G-equivariant) moment map µ : X → g∗ .
Kostant’s theorem: homogeneous Hamiltonian
G-space = covering of G-orbit on g∗ .
Includes classification of symp homog spaces for G.
(Riem homog spaces hopelessly complicated.)
Recall: commuting algebra formalism for diff operators
suggests irreducible representations ! homogeneous
Hamiltonian G-spaces.
Kirillov-Kostant philosophy of coadjt orbits suggests
b ! g∗ /G.
{irr unitary reps of G} = G
(?)
MORE PRECISELY. . . restrict right side to “admissible”
b
orbits (integrality cond). Expect to find “almost all” of G:
enough for interesting harmonic analysis.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Method of coadjoint orbits
The orbit method
for reductive
groups
David Vogan
Recall: Hamiltonian G-space X comes with
(G-equivariant) moment map µ : X → g∗ .
Kostant’s theorem: homogeneous Hamiltonian
G-space = covering of G-orbit on g∗ .
Includes classification of symp homog spaces for G.
(Riem homog spaces hopelessly complicated.)
Recall: commuting algebra formalism for diff operators
suggests irreducible representations ! homogeneous
Hamiltonian G-spaces.
Kirillov-Kostant philosophy of coadjt orbits suggests
b ! g∗ /G.
{irr unitary reps of G} = G
(?)
MORE PRECISELY. . . restrict right side to “admissible”
b
orbits (integrality cond). Expect to find “almost all” of G:
enough for interesting harmonic analysis.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Method of coadjoint orbits
The orbit method
for reductive
groups
David Vogan
Recall: Hamiltonian G-space X comes with
(G-equivariant) moment map µ : X → g∗ .
Kostant’s theorem: homogeneous Hamiltonian
G-space = covering of G-orbit on g∗ .
Includes classification of symp homog spaces for G.
(Riem homog spaces hopelessly complicated.)
Recall: commuting algebra formalism for diff operators
suggests irreducible representations ! homogeneous
Hamiltonian G-spaces.
Kirillov-Kostant philosophy of coadjt orbits suggests
b ! g∗ /G.
{irr unitary reps of G} = G
(?)
MORE PRECISELY. . . restrict right side to “admissible”
b
orbits (integrality cond). Expect to find “almost all” of G:
enough for interesting harmonic analysis.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Method of coadjoint orbits
The orbit method
for reductive
groups
David Vogan
Recall: Hamiltonian G-space X comes with
(G-equivariant) moment map µ : X → g∗ .
Kostant’s theorem: homogeneous Hamiltonian
G-space = covering of G-orbit on g∗ .
Includes classification of symp homog spaces for G.
(Riem homog spaces hopelessly complicated.)
Recall: commuting algebra formalism for diff operators
suggests irreducible representations ! homogeneous
Hamiltonian G-spaces.
Kirillov-Kostant philosophy of coadjt orbits suggests
b ! g∗ /G.
{irr unitary reps of G} = G
(?)
MORE PRECISELY. . . restrict right side to “admissible”
b
orbits (integrality cond). Expect to find “almost all” of G:
enough for interesting harmonic analysis.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Method of coadjoint orbits
The orbit method
for reductive
groups
David Vogan
Recall: Hamiltonian G-space X comes with
(G-equivariant) moment map µ : X → g∗ .
Kostant’s theorem: homogeneous Hamiltonian
G-space = covering of G-orbit on g∗ .
Includes classification of symp homog spaces for G.
(Riem homog spaces hopelessly complicated.)
Recall: commuting algebra formalism for diff operators
suggests irreducible representations ! homogeneous
Hamiltonian G-spaces.
Kirillov-Kostant philosophy of coadjt orbits suggests
b ! g∗ /G.
{irr unitary reps of G} = G
(?)
MORE PRECISELY. . . restrict right side to “admissible”
b
orbits (integrality cond). Expect to find “almost all” of G:
enough for interesting harmonic analysis.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Evidence for orbit method
With the caveat about restricting to admissible orbits. . .
b ! g∗ /G.
G
(?)
(?) is true for G simply conn nilpotent (Kirillov).
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
(?) is true for G type I solvable (Auslander-Kostant).
Differential
operator algebras
(?) for algebraic G reduces to reductive G (Duflo).
Hamiltonian
G-spaces
Case of reductive G is still open.
Coadjoint orbits for
reductive groups
Actually (?) is false for connected nonabelian reductive G.
Conclusion
But there are still theorems close to (?).
References
Two ways to do repn theory for reductive G:
1. start with coadjt orbit, look for repn. Hard.
2. start with repn, look for coadjt orbit. Easy.
Really need to do both things at once. Having started
to do mathematics in the Ford administration, I find
this challenging. (Gave up chewing gum at that time.)
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Structure theory for reductive Lie groups
The orbit method
for reductive
groups
David Vogan
Reductive Lie group G = closed subgp of GL(n, R)
| {z }
Introduction
s.t. G closed under transpose, and #G/G0 < ∞.
Commuting
algebras
main ex
From now on G is reductive.
Lie(G) = g ⊂ n × n matrices. Bilinear form
G-eqvt
T (X , Y ) = tr(XY ) ⇒ g '
g∗
Orbits of G on g∗ ⊂ conjugacy classes of matrices.
Orbits of GL(n, R) on g∗ = conj classes of matrices.
Family of orbits: for real numbers λ1 and λn−1 ,
O(λ1 , λn−1 ) = matrices, eigenvalue λp has mult p.
Base point in family:
O1,n−1 = nilp matrices, Jordan blocks 1, n − 1.
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
One irreducible unitary representation
V n-diml real. G = GL(V ) acts on M = PV = lines in V .
The orbit method
for reductive
groups
David Vogan
Introduction
X = T ∗ (M) = {(v , λ) ∈ (V − 0) × V ∗ | λ(v ) = 0}/ ∼ .
Relation is (v , λ) ∼ (tv , t −1 λ).
Commuting
algebras
Differential
operator algebras
Orbits of G on X : zero sec M, all else X1,n−1 .
Moment map µ : T ∗ (M) → gl(V )∗ ' End(V ),
µ(v , λ)(w) = λ(w)v .
∼
Have µ : X1,n−1 → O1,n−1 : one coadjoint orbit!
hope
X1,n−1 dense ⇒ C ∞ (T ∗ (M))G = C ⇒ C ∞ (M) irr.
This hope does disappoint us: C ∞ (M) ⊃ constants, so
rep is reducible. Also there’s no G-invt msre on M, so no
unitary Hilbert space version L2 (M).
Fix both problems: δ 1/2 = half-density bdle on PV .
Smooth half densities C ∞ (M, δ 1/2 ) are irr rep of
GL(n, R),
irr unitary rep π1,n−1 on L2 (M, δ 1/2 ).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Family of irr unitary representations
Natural generalization: replace functions on M = PV
by sections of Hermitian line bundle.
Two natural (GL(V )-eqvt) real bdles on PV :
tautological line bdle L (fiber at line L is L); and Q
((n − 1) -diml real bundle, fiber at L is V /L).
Given real parameters λ1 and λn−1 , get Hermitian
line bundle H(λ1 , λn−1 ) = Liλ1 ⊗ (∧n−1 Q)iλn−1 .
Define
2
π1,n−1 (λ1 , λn−1 ) = rep on L (M, δ
1/2
⊗ H(λ1 , λn−1 )).
These are irr unitary representations of GL(V );
naturally assoc to coadjt orbits O(λ1 , λn−1 ).
Same techniques (still for reductive G) deal with all
hyperbolic coadjt orbits (that is, orbits of matrices
diagonalizable over R.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Family of irr unitary representations
Natural generalization: replace functions on M = PV
by sections of Hermitian line bundle.
Two natural (GL(V )-eqvt) real bdles on PV :
tautological line bdle L (fiber at line L is L); and Q
((n − 1) -diml real bundle, fiber at L is V /L).
Given real parameters λ1 and λn−1 , get Hermitian
line bundle H(λ1 , λn−1 ) = Liλ1 ⊗ (∧n−1 Q)iλn−1 .
Define
2
π1,n−1 (λ1 , λn−1 ) = rep on L (M, δ
1/2
⊗ H(λ1 , λn−1 )).
These are irr unitary representations of GL(V );
naturally assoc to coadjt orbits O(λ1 , λn−1 ).
Same techniques (still for reductive G) deal with all
hyperbolic coadjt orbits (that is, orbits of matrices
diagonalizable over R.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Family of irr unitary representations
Natural generalization: replace functions on M = PV
by sections of Hermitian line bundle.
Two natural (GL(V )-eqvt) real bdles on PV :
tautological line bdle L (fiber at line L is L); and Q
((n − 1) -diml real bundle, fiber at L is V /L).
Given real parameters λ1 and λn−1 , get Hermitian
line bundle H(λ1 , λn−1 ) = Liλ1 ⊗ (∧n−1 Q)iλn−1 .
Define
2
π1,n−1 (λ1 , λn−1 ) = rep on L (M, δ
1/2
⊗ H(λ1 , λn−1 )).
These are irr unitary representations of GL(V );
naturally assoc to coadjt orbits O(λ1 , λn−1 ).
Same techniques (still for reductive G) deal with all
hyperbolic coadjt orbits (that is, orbits of matrices
diagonalizable over R.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Family of irr unitary representations
Natural generalization: replace functions on M = PV
by sections of Hermitian line bundle.
Two natural (GL(V )-eqvt) real bdles on PV :
tautological line bdle L (fiber at line L is L); and Q
((n − 1) -diml real bundle, fiber at L is V /L).
Given real parameters λ1 and λn−1 , get Hermitian
line bundle H(λ1 , λn−1 ) = Liλ1 ⊗ (∧n−1 Q)iλn−1 .
Define
2
π1,n−1 (λ1 , λn−1 ) = rep on L (M, δ
1/2
⊗ H(λ1 , λn−1 )).
These are irr unitary representations of GL(V );
naturally assoc to coadjt orbits O(λ1 , λn−1 ).
Same techniques (still for reductive G) deal with all
hyperbolic coadjt orbits (that is, orbits of matrices
diagonalizable over R.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Family of irr unitary representations
Natural generalization: replace functions on M = PV
by sections of Hermitian line bundle.
Two natural (GL(V )-eqvt) real bdles on PV :
tautological line bdle L (fiber at line L is L); and Q
((n − 1) -diml real bundle, fiber at L is V /L).
Given real parameters λ1 and λn−1 , get Hermitian
line bundle H(λ1 , λn−1 ) = Liλ1 ⊗ (∧n−1 Q)iλn−1 .
Define
2
π1,n−1 (λ1 , λn−1 ) = rep on L (M, δ
1/2
⊗ H(λ1 , λn−1 )).
These are irr unitary representations of GL(V );
naturally assoc to coadjt orbits O(λ1 , λn−1 ).
Same techniques (still for reductive G) deal with all
hyperbolic coadjt orbits (that is, orbits of matrices
diagonalizable over R.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Family of irr unitary representations
Natural generalization: replace functions on M = PV
by sections of Hermitian line bundle.
Two natural (GL(V )-eqvt) real bdles on PV :
tautological line bdle L (fiber at line L is L); and Q
((n − 1) -diml real bundle, fiber at L is V /L).
Given real parameters λ1 and λn−1 , get Hermitian
line bundle H(λ1 , λn−1 ) = Liλ1 ⊗ (∧n−1 Q)iλn−1 .
Define
2
π1,n−1 (λ1 , λn−1 ) = rep on L (M, δ
1/2
⊗ H(λ1 , λn−1 )).
These are irr unitary representations of GL(V );
naturally assoc to coadjt orbits O(λ1 , λn−1 ).
Same techniques (still for reductive G) deal with all
hyperbolic coadjt orbits (that is, orbits of matrices
diagonalizable over R.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, real sm , define coadjt orbit
The orbit method
for reductive
groups
David Vogan
Introduction
O(sm +itm , sm −itm ) = {A ∈ End(V ) | eigval sm ± itm mult m}.
Commuting
algebras
Base point in family:
Differential
operator algebras
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Parameter sm corresponds to twisting by one-diml char of GL(V ):
cumbersome and dull. So pretend it doesn’t exist.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, real sm , define coadjt orbit
The orbit method
for reductive
groups
David Vogan
Introduction
O(sm +itm , sm −itm ) = {A ∈ End(V ) | eigval sm ± itm mult m}.
Commuting
algebras
Base point in family:
Differential
operator algebras
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Parameter sm corresponds to twisting by one-diml char of GL(V ):
cumbersome and dull. So pretend it doesn’t exist.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, real sm , define coadjt orbit
The orbit method
for reductive
groups
David Vogan
Introduction
O(sm +itm , sm −itm ) = {A ∈ End(V ) | eigval sm ± itm mult m}.
Commuting
algebras
Base point in family:
Differential
operator algebras
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Parameter sm corresponds to twisting by one-diml char of GL(V ):
cumbersome and dull. So pretend it doesn’t exist.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, real sm , define coadjt orbit
The orbit method
for reductive
groups
David Vogan
Introduction
O(sm +itm , sm −itm ) = {A ∈ End(V ) | eigval sm ± itm mult m}.
Commuting
algebras
Base point in family:
Differential
operator algebras
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Parameter sm corresponds to twisting by one-diml char of GL(V ):
cumbersome and dull. So pretend it doesn’t exist.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, define coadjt orbit
O(itm , − itm ) = {A ∈ End(V ) | eigval ± itm mult m}.
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Base point in family:
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
The orbit method
for reductive
groups
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, define coadjt orbit
O(itm , − itm ) = {A ∈ End(V ) | eigval ± itm mult m}.
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Base point in family:
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
The orbit method
for reductive
groups
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, define coadjt orbit
O(itm , − itm ) = {A ∈ End(V ) | eigval ± itm mult m}.
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Base point in family:
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
The orbit method
for reductive
groups
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, define coadjt orbit
O(itm , − itm ) = {A ∈ End(V ) | eigval ± itm mult m}.
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Base point in family:
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
The orbit method
for reductive
groups
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
And now for something completely different. . .
V 2m-dimensional real vector space, G = GL(V ). Fix real
tm ≥ 0, define coadjt orbit
O(itm , − itm ) = {A ∈ End(V ) | eigval ± itm mult m}.
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Base point in family:
Om,m = {A ∈ End(V ) nilpotent, Jordan blocks m, m}.
Corresponding repns related to cplx alg variety
X = complex structures on V ,
dim X = m2 .
Have K -invariant projective subvariety
Z = orthogonal cplx structures
The orbit method
for reductive
groups
dim Z = (m2 −m)/2 = s.
Turns out (Schmid, Wolf) X is (s + 1)-complete, which
means Stein away from Z .
X has G-invt indef Kähler structure, signature ((m2 − m)/2, (m2 + m)/2);
underlying real symplectic mfld is O(itm , −itm ) (any tm > 0).
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Actually can also twist by character | det |ism of GL(V ), but we’re
pretending that doesn’t exist.
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
References
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
Best: O(itm , −itm ) ! repn
Differential
operator algebras
Conclusion
Canonical bdle is ωX = L−2m .
Hcn,n−s (X , Ltm
Introduction
Commuting
algebras
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Actually can also twist by character | det |ism of GL(V ), but we’re
pretending that doesn’t exist.
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
References
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
Best: O(itm , −itm ) ! repn
Differential
operator algebras
Conclusion
Canonical bdle is ωX = L−2m .
Hcn,n−s (X , Ltm
Introduction
Commuting
algebras
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Actually can also twist by character | det |ism of GL(V ), but we’re
pretending that doesn’t exist.
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
References
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
Best: O(itm , −itm ) ! repn
Differential
operator algebras
Conclusion
Canonical bdle is ωX = L−2m .
Hcn,n−s (X , Ltm
Introduction
Commuting
algebras
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Actually can also twist by character | det |ism of GL(V ), but we’re
pretending that doesn’t exist.
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
References
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
Best: O(itm , −itm ) ! repn
Differential
operator algebras
Conclusion
Canonical bdle is ωX = L−2m .
Hcn,n−s (X , Ltm
Introduction
Commuting
algebras
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗ ωX ).
Call this (last) representation π(tm )
(tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗ ωX ).
Call this (last) representation π(tm )
(tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ).
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗ ωX ).
Call this (last) representation π(tm )
(tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m, 2 X = space of cplx structures2on V .
n = dimC (X ) = m ,
s = dimC (maxl cpt subvar) = (m − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX =
The orbit method
for reductive
groups
L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Problem: holomorphic sections (case m > 1) exist only for p ≥ 0;
Conclusion
they extend to cplx Grassmannian of m-planes in VC , give fin diml
References
(non-unitary) rep of GL(V ).
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
n,n−s
t
Best: O(itm , −itm ) ! repn Hc
(X , L m ⊗ ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗ ωX ).
Call this (last) representation π(tm )
(tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ).
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗ ωX ).
Call this (last) representation π(tm )
(tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m, 2 X = space of cplx structures2on V .
n = dimC (X ) = m ,
The orbit method
for reductive
groups
s = dimC (maxl cpt subvar) = (m − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Canonical bdle is ωX =
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Coadjoint orbits for
reductive groups
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
Good news: infinitesimally unitary for suff neg L.
References
Bad news: cond on p complicated, Hilb space is tiny (dense)
subspace.
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
1/2
n,n−s
t
Best: O(itm , −itm ) ! repn Hc
(X , L m ⊗ ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Conclusion
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
Best: O(itm , −itm ) ! repn
Hcn,n−s (X , Ltm
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
The orbit method
for reductive
groups
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ).
Best: O(itm , −itm ) ! repn
Hcn,n−s (X , Ltm
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
≥ 0.
What we learn: interesting orbits are those with tm + m ∈ Z.
These are admissible orbits. (Duflo).
1/2
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗ ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn
Hcn,n−s (X , Ltm
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn
Hcn,n−s (X , Ltm
⊗
1/2
ωX ).
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn
Hcn,n−s (X , Ltm
⊗
1/2
ωX ).
Pre-unit for tm
This is Serre duality plus analytic results of Hon-Wai Wong.
Call this (last) representation π(tm )
(tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
≥ 0.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗
1/2
ωX ).
Pre-unit for tm
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
≥ 0.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗
1/2
ωX ).
Pre-unit for tm
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
≥ 0.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗
1/2
ωX ).
Pre-unit for tm
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
≥ 0.
The orbit method
for reductive
groups
Representations attached to O(itm , −itm )
David Vogan
Brought to you by Birgit Speh.
dim V = 2m,
n = dimC (X ) = m2 ,
X = space of cplx structures on V .
s = dimC (maxl cpt subvar) = (m2 − m)/2.
Point x ∈ X interprets V as m-diml complex vector space
Vx . Defines (tautological) holomorphic vector bundle V on
X . Top exterior power of V is a holomorphic line bundle L.
Every eqvt hol line bdle on X is Lp , some p ∈ Z.
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Canonical bdle is ωX = L−2m .
Very rough idea: O(itm , −itm ) ! repn Γ(Lp ). Fin-diml, not unitary.
Conclusion
Better: O(itm , −itm ) ! repn H 0,s (X , Lp ). Inf unit for p ≤ −m.
References
1/2
Better: O(itm , −itm ) ! repn H 0,s (X , L−tm ⊗ ωX ). Inf unit for tm
Best: O(itm , −itm ) ! repn Hcn,n−s (X , Ltm ⊗
1/2
ωX ).
Pre-unit for tm
Call this (last) representation π(tm ) (tm = 0, 1, 2, . . .).
Inclusion of compact subvariety Z gives lowest O(V )-type:
V
(tm + 1)-Cartan power of m (V ). (Shift +1 since ωZ = ωX1/2 ⊗ L−1 .)
Parallel techniques deal with elliptic coadjt orbits (that is, orbits of
semisimple matrices with purely imaginary eigenvalues.
≥ 0.
≥ 0.
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Quantizing nilpotent orbits
For GL(n, R), nilp coadjt orbits = special points in families
quantize by continuity (see below).
of semisimp orbits
For other reductive groups, not true: many nilpotent orbits
have no deformation to semsimple orbits.
Kostant-Rallis idea: nilp coadjt orbit OR has natural K -invt
cplx structure Oθ
holomorphic action of KC .
Get repn of K that quantizes Oθ ; look for a way to extend it to G. Carried
out by Brylinski and Kostant for minimal coadjt orbit in many cases.
Rossi-Vergne idea: Given semisimple orbits, quantizations
{O(λ) | λ dom reg}
{π(λ) | λ dom reg adm}.
Repns make sense (but may not be unitary) for “all”
admissible λ (not dominant or regular).
Continuity above means limiting nilpotent orbit O(0) quantized by π(0).
Rossi-Vergne idea: smaller nilpotent orbits O0 (contained
in O(0)) should be quantized by smaller constituents of
representations π(λ0 ), with λ0 admissible, not dominant.
The orbit method
for reductive
groups
David Vogan
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Meaning of it all
The orbit method
for reductive
groups
David Vogan
My first class from Bert began February 4, 1975. In the first
hour he defined symplectic forms; symplectic manifolds;
symplectic structure on cotangent bundles; Lagrangian
submanifolds; and proved coadjoint orbits were symplectic.
After that the class picked up speed.
Many years later I took another class from Bert. At some point
he needed differential operators. So he gave an introduction:
“You form the algebra generated by the derivations of C ∞ .”
That’s Bert: mathematics at Mach 2, always exciting, and the
explanations are always complete; you’ll figure them out
eventually. The first third of a century has been fantastic, and I
hope to keep listening for a very long time.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Meaning of it all
The orbit method
for reductive
groups
David Vogan
My first class from Bert began February 4, 1975. In the first
hour he defined symplectic forms; symplectic manifolds;
symplectic structure on cotangent bundles; Lagrangian
submanifolds; and proved coadjoint orbits were symplectic.
After that the class picked up speed.
Many years later I took another class from Bert. At some point
he needed differential operators. So he gave an introduction:
“You form the algebra generated by the derivations of C ∞ .”
That’s Bert: mathematics at Mach 2, always exciting, and the
explanations are always complete; you’ll figure them out
eventually. The first third of a century has been fantastic, and I
hope to keep listening for a very long time.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Meaning of it all
The orbit method
for reductive
groups
David Vogan
My first class from Bert began February 4, 1975. In the first
hour he defined symplectic forms; symplectic manifolds;
symplectic structure on cotangent bundles; Lagrangian
submanifolds; and proved coadjoint orbits were symplectic.
After that the class picked up speed.
Many years later I took another class from Bert. At some point
he needed differential operators. So he gave an introduction:
“You form the algebra generated by the derivations of C ∞ .”
That’s Bert: mathematics at Mach 2, always exciting, and the
explanations are always complete; you’ll figure them out
eventually. The first third of a century has been fantastic, and I
hope to keep listening for a very long time.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Meaning of it all
The orbit method
for reductive
groups
David Vogan
My first class from Bert began February 4, 1975. In the first
hour he defined symplectic forms; symplectic manifolds;
symplectic structure on cotangent bundles; Lagrangian
submanifolds; and proved coadjoint orbits were symplectic.
After that the class picked up speed.
Many years later I took another class from Bert. At some point
he needed differential operators. So he gave an introduction:
“You form the algebra generated by the derivations of C ∞ .”
That’s Bert: mathematics at Mach 2, always exciting, and the
explanations are always complete; you’ll figure them out
eventually. The first third of a century has been fantastic, and I
hope to keep listening for a very long time.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
Meaning of it all
The orbit method
for reductive
groups
David Vogan
My first class from Bert began February 4, 1975. In the first
hour he defined symplectic forms; symplectic manifolds;
symplectic structure on cotangent bundles; Lagrangian
submanifolds; and proved coadjoint orbits were symplectic.
After that the class picked up speed.
Many years later I took another class from Bert. At some point
he needed differential operators. So he gave an introduction:
“You form the algebra generated by the derivations of C ∞ .”
That’s Bert: mathematics at Mach 2, always exciting, and the
explanations are always complete; you’ll figure them out
eventually. The first third of a century has been fantastic, and I
hope to keep listening for a very long time.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
References
The orbit method
for reductive
groups
David Vogan
W. Graham and D. Vogan, “Geometric quantization for nilpotent
coadjoint orbits,” in Geometry and Representation Theory of
real and p-adic groups. Birkhauser, Boston-Basel-Berlin, 1998.
HAPPY BIRTHDAY BERT!
B. Kostant, “Quantization and unitary representations. I.
Prequantization,” pp. 87–208 in Lectures in modern analysis
and applications, III. Lecture Notes in Math., 170. Springer,
Berlin, 1970.
HAPPY BIRTHDAY BERT!
HAPPY
BIRTHDAY
D. Vogan, “The method
of coadjoint orbits for realBERT!
reductive
groups,” in Representation Theory of Lie Groups. IAS/Park City
Mathematics Series
8 (1999), 179–238.
HAPPY
BIRTHDAY
BERT!
D. Vogan, “Unitary representations and complex analysis,”
inRepresentation Theory and Complex Analysis (CIME 2004),
Andrea D’Agnolo, editor. Springer, 2008.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References
References
The orbit method
for reductive
groups
David Vogan
W. Graham and D. Vogan, “Geometric quantization for nilpotent
coadjoint orbits,” in Geometry and Representation Theory of
real and p-adic groups. Birkhauser, Boston-Basel-Berlin, 1998.
HAPPY BIRTHDAY BERT!
B. Kostant, “Quantization and unitary representations. I.
Prequantization,” pp. 87–208 in Lectures in modern analysis
and applications, III. Lecture Notes in Math., 170. Springer,
Berlin, 1970.
HAPPY BIRTHDAY BERT!
HAPPY
BIRTHDAY
D. Vogan, “The method
of coadjoint orbits for realBERT!
reductive
groups,” in Representation Theory of Lie Groups. IAS/Park City
Mathematics Series
8 (1999), 179–238.
HAPPY
BIRTHDAY
BERT!
D. Vogan, “Unitary representations and complex analysis,”
inRepresentation Theory and Complex Analysis (CIME 2004),
Andrea D’Agnolo, editor. Springer, 2008.
HAPPY BIRTHDAY BERT!
Introduction
Commuting
algebras
Differential
operator algebras
Hamiltonian
G-spaces
Coadjoint orbits for
reductive groups
Conclusion
References