Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of...
Transcript of Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of...
![Page 1: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/1.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Geometry and representations ofreductive groups
David Vogan
Department of MathematicsMassachusetts Institute of Technology
Ritt Lectures, Columbia, December 13–14 2007
![Page 2: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/2.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Outline
Introduction
Commuting algebras
Differential operator algebras
Hamiltonian G-spaces
References
![Page 3: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/3.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 4: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/4.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 5: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/5.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 6: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/6.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 7: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/7.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 8: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/8.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 9: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/9.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 10: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/10.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Abstract harmonic analysis
Say Lie group G acts on manifold M. Can ask aboutI topology of MI solutions of G-invariant differential equationsI special functions on M (automorphic forms, etc.)
Method step 1: LINEARIZE. Replace M by Hilbertspace 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. Studyminimal pieces: irreducible unitary repns of G.Difficult questions: how does DIAGONALIZE work,and what kind of minimal pieces do you get?
![Page 11: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/11.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Plan of talks
I Outline strategy for decomposing L2(M), by analogywith “double centralizers” in finite-dimensionalalgebra.
I Strategy Kirillov-Kostant philosophy:irreducible unitary representationsof Lie group G
m(nearly) symplectic manifolds with (nearly)transitive Hamiltonian action of G
I “Strategy” and “philosophy” have a lot of wishfulthinking. Describe theorems supporting m.
![Page 12: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/12.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Plan of talks
I Outline strategy for decomposing L2(M), by analogywith “double centralizers” in finite-dimensionalalgebra.
I Strategy Kirillov-Kostant philosophy:irreducible unitary representationsof Lie group G
m(nearly) symplectic manifolds with (nearly)transitive Hamiltonian action of G
I “Strategy” and “philosophy” have a lot of wishfulthinking. Describe theorems supporting m.
![Page 13: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/13.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Plan of talks
I Outline strategy for decomposing L2(M), by analogywith “double centralizers” in finite-dimensionalalgebra.
I Strategy Kirillov-Kostant philosophy:irreducible unitary representationsof Lie group G
m(nearly) symplectic manifolds with (nearly)transitive Hamiltonian action of G
I “Strategy” and “philosophy” have a lot of wishfulthinking. Describe theorems supporting m.
![Page 14: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/14.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Plan of talks
I Outline strategy for decomposing L2(M), by analogywith “double centralizers” in finite-dimensionalalgebra.
I Strategy Kirillov-Kostant philosophy:irreducible unitary representationsof Lie group G
m(nearly) symplectic manifolds with (nearly)transitive Hamiltonian action of G
I “Strategy” and “philosophy” have a lot of wishfulthinking. Describe theorems supporting m.
![Page 15: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/15.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Plan of talks
I Outline strategy for decomposing L2(M), by analogywith “double centralizers” in finite-dimensionalalgebra.
I Strategy Kirillov-Kostant philosophy:irreducible unitary representationsof Lie group G
m(nearly) symplectic manifolds with (nearly)transitive Hamiltonian action of G
I “Strategy” and “philosophy” have a lot of wishfulthinking. Describe theorems supporting m.
![Page 16: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/16.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 17: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/17.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 18: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/18.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 19: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/19.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 20: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/20.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 21: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/21.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 22: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/22.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Decomposing a representation
Given ops on Hilbert space H, want to decompose Hin operator-invt way. Fin-diml theory:V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Classical structure theorem:
W1, . . . , Wr all simple A-modules; then
A ' End(W1)× · · · × End(Wr ).
V ' m1W1 + · · ·+ mr Wr .
Positive integer mi is multiplicity of Wi in V .
Slicker version: define multiplicity spaceMi = HomA(Wi , V ); then mi = dim Mi , and
V ' M1 ⊗W1 + · · ·+ Mr ⊗Wr .
Slickest version: COMMUTING ALGEBRAS. . .
![Page 23: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/23.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 24: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/24.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 25: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/25.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 26: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/26.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 27: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/27.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 28: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/28.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 29: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/29.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Commuting algebras and all that
V/C fin-diml, A ⊂ End(V ) cplx semisimple alg of ops.Define
Z = Cent End(V )(A),
a new semisimple algebra of operators on V .
TheoremSay A and Z are complex semisimple algebras ofoperators on V as above.
1. 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 ).
3. V '∑
i Mi ⊗Wi as a module for A×Z.
![Page 30: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/30.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Example of commuting algebras
G finite group, V = L2(G).
A = alg gen by left translations in G ⊂ End(V ).
A is the group algebra of G.
Z = alg gen by right translations in G ⊂ End(V ).
Z is also the group algebra of G.Set of simple A-modules is
{Wi} = all irr reps of G.
Set of simple Z-modules is
{Mi} = all irr reps of G, Mi = W ∗i .
Decomposition of L2(G) is Peter-Weyl theorem:
L2(G) =∑
Wi irr of G
Wi ⊗W ∗i .
![Page 31: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/31.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Example of commuting algebras
G finite group, V = L2(G).
A = alg gen by left translations in G ⊂ End(V ).
A is the group algebra of G.
Z = alg gen by right translations in G ⊂ End(V ).
Z is also the group algebra of G.Set of simple A-modules is
{Wi} = all irr reps of G.
Set of simple Z-modules is
{Mi} = all irr reps of G, Mi = W ∗i .
Decomposition of L2(G) is Peter-Weyl theorem:
L2(G) =∑
Wi irr of G
Wi ⊗W ∗i .
![Page 32: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/32.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Example of commuting algebras
G finite group, V = L2(G).
A = alg gen by left translations in G ⊂ End(V ).
A is the group algebra of G.
Z = alg gen by right translations in G ⊂ End(V ).
Z is also the group algebra of G.Set of simple A-modules is
{Wi} = all irr reps of G.
Set of simple Z-modules is
{Mi} = all irr reps of G, Mi = W ∗i .
Decomposition of L2(G) is Peter-Weyl theorem:
L2(G) =∑
Wi irr of G
Wi ⊗W ∗i .
![Page 33: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/33.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Example of commuting algebras
G finite group, V = L2(G).
A = alg gen by left translations in G ⊂ End(V ).
A is the group algebra of G.
Z = alg gen by right translations in G ⊂ End(V ).
Z is also the group algebra of G.Set of simple A-modules is
{Wi} = all irr reps of G.
Set of simple Z-modules is
{Mi} = all irr reps of G, Mi = W ∗i .
Decomposition of L2(G) is Peter-Weyl theorem:
L2(G) =∑
Wi irr of G
Wi ⊗W ∗i .
![Page 34: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/34.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Example of commuting algebras
G finite group, V = L2(G).
A = alg gen by left translations in G ⊂ End(V ).
A is the group algebra of G.
Z = alg gen by right translations in G ⊂ End(V ).
Z is also the group algebra of G.Set of simple A-modules is
{Wi} = all irr reps of G.
Set of simple Z-modules is
{Mi} = all irr reps of G, Mi = W ∗i .
Decomposition of L2(G) is Peter-Weyl theorem:
L2(G) =∑
Wi irr of G
Wi ⊗W ∗i .
![Page 35: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/35.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Another example of commuting algebras
GL(V ) acts on nth tensor power T n(V ): defineA = ends of T n(V ) gen by GL(V ).
Quotient of group alg of GL(V );
simple A-mods {Wi} = irr reps of GL(V ) on T n(V ).
Symmetric group Sn also acts on T n(V ): defineZ = ends of T n(V ) gen by symm group Sn.
Quotient of group alg of Sn;simple Z-mods {Mi} = irr reps of Sn on T n(V ).
Theorem (Schur-Weyl duality)Algebras A and Z acting on T n(V ) as mutual centralizers:
T n(V ) =∑
Mi ⊗Wi .
Summands! partitions of n into at most dim V parts.
![Page 36: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/36.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Another example of commuting algebras
GL(V ) acts on nth tensor power T n(V ): defineA = ends of T n(V ) gen by GL(V ).
Quotient of group alg of GL(V );
simple A-mods {Wi} = irr reps of GL(V ) on T n(V ).
Symmetric group Sn also acts on T n(V ): defineZ = ends of T n(V ) gen by symm group Sn.
Quotient of group alg of Sn;simple Z-mods {Mi} = irr reps of Sn on T n(V ).
Theorem (Schur-Weyl duality)Algebras A and Z acting on T n(V ) as mutual centralizers:
T n(V ) =∑
Mi ⊗Wi .
Summands! partitions of n into at most dim V parts.
![Page 37: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/37.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Another example of commuting algebras
GL(V ) acts on nth tensor power T n(V ): defineA = ends of T n(V ) gen by GL(V ).
Quotient of group alg of GL(V );
simple A-mods {Wi} = irr reps of GL(V ) on T n(V ).
Symmetric group Sn also acts on T n(V ): defineZ = ends of T n(V ) gen by symm group Sn.
Quotient of group alg of Sn;simple Z-mods {Mi} = irr reps of Sn on T n(V ).
Theorem (Schur-Weyl duality)Algebras A and Z acting on T n(V ) as mutual centralizers:
T n(V ) =∑
Mi ⊗Wi .
Summands! partitions of n into at most dim V parts.
![Page 38: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/38.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Another example of commuting algebras
GL(V ) acts on nth tensor power T n(V ): defineA = ends of T n(V ) gen by GL(V ).
Quotient of group alg of GL(V );
simple A-mods {Wi} = irr reps of GL(V ) on T n(V ).
Symmetric group Sn also acts on T n(V ): defineZ = ends of T n(V ) gen by symm group Sn.
Quotient of group alg of Sn;simple Z-mods {Mi} = irr reps of Sn on T n(V ).
Theorem (Schur-Weyl duality)Algebras A and Z acting on T n(V ) as mutual centralizers:
T n(V ) =∑
Mi ⊗Wi .
Summands! partitions of n into at most dim V parts.
![Page 39: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/39.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Another example of commuting algebras
GL(V ) acts on nth tensor power T n(V ): defineA = ends of T n(V ) gen by GL(V ).
Quotient of group alg of GL(V );
simple A-mods {Wi} = irr reps of GL(V ) on T n(V ).
Symmetric group Sn also acts on T n(V ): defineZ = ends of T n(V ) gen by symm group Sn.
Quotient of group alg of Sn;simple Z-mods {Mi} = irr reps of Sn on T n(V ).
Theorem (Schur-Weyl duality)Algebras A and Z acting on T n(V ) as mutual centralizers:
T n(V ) =∑
Mi ⊗Wi .
Summands! partitions of n into at most dim V parts.
![Page 40: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/40.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Another example of commuting algebras
GL(V ) acts on nth tensor power T n(V ): define
A = ends of T n(V ) gen by GL(V ).
Quotient of group alg of GL(V );
simple A-mods {Wi} = irr reps of GL(V ) on T n(V ).
Symmetric group Sn also acts on T n(V ): define
Z = ends of T n(V ) gen by symm group Sn.
Quotient of group alg of Sn;
simple Z-mods {Mi} = irr reps of Sn on T n(V ).
Theorem (Schur-Weyl duality)Algebras A and Z acting on T n(V ) as mutual centralizers:
T n(V ) =∑
Mi ⊗Wi .
Summands! partitions of n into at most dim V parts.
![Page 41: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/41.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 42: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/42.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 43: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/43.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 44: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/44.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 45: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/45.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 46: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/46.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 47: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/47.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 48: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/48.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 49: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/49.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Infinite-dimensional representations
Need framework to study ops on inf-diml V .
Finite-diml ↔ infinite-diml dictionaryfinite-diml V ↔ C∞(M)
repn of G on V ↔ action of G on MEnd(V ) ↔ Diff(M)
A = im(C[G]) ⊂ End(V ) ↔ A = im(U(g)) ⊂ Diff(M)
Z = CentEnd(V )(A) ↔ Z = diff ops comm with G
Which differential operators commute with G?
Answer generalizations of dictionary. . .
![Page 50: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/50.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 51: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/51.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 52: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/52.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 53: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/53.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 54: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/54.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 55: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/55.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 56: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/56.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 57: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/57.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
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.2.
σn : Diffn(M)/ Diffn−1(M) → Polyn(T ∗(M)).
3. Commutator of diff ops Poisson bracket {, } onT ∗(M): for D ∈ Diffp(M), D′ ∈ Diffq(M),
σp+q−1([D, D′]) = {σp(D), σq(D′)}.
Diff ops comm with G! symbols Poisson-comm with g.
![Page 58: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/58.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 59: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/59.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 60: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/60.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 61: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/61.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 62: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/62.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 63: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/63.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 64: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/64.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson structure and Lie group actions
X mfld w. Poisson {, } on fns (e.g. T ∗(M)).Bracket with f ξf ∈ Vect(X ): ξf (g) = {f , g}.Vector fields ξf called Hamiltonian; flows preserve{, }. Map f 7→ ξf is Lie alg homomomorphism.Lie group action on X Lie alg homom Y 7→ ξYfrom Lie(G) to Vect(X ).Call X Hamiltonian G-space if given Lie alg homomY 7→ fY from Lie(G) to C∞(X ) with ξY = ξfY .G acts on M T ∗(M) is Hamiltonian G-space: Liealg elt Y vec fld ξM
Y on M function fY on T ∗(M):
fY (m, λ) = λ(ξMY (m)) (m ∈ M, λ ∈ T ∗
m(M)).
f on X with {f , g} = 0! f constant on G orbits.
![Page 65: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/65.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 66: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/66.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 67: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/67.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 68: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/68.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 69: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/69.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 70: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/70.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 71: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/71.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 72: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/72.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 73: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/73.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 74: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/74.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Our story so far. . .
G acts on M ! T ∗(M) Hamiltonian G-space.G-decomp of C∞(M)! (Diff M)G-modules.
(Diff M)G σ! C∞(T ∗(M))G ! C∞((T ∗(M))//G).
Hope C∞(M) irr ⇔ G has dense orbit on T ∗(M).
Suggests generalization. . .
Hamiltonian G-cone X graded alg Poly(X ).Seek filtered alg D, symbol calc grD σ→Poly(X )carrying [, ] on D to {, } on Poly(X ).Seek to lift G action on Poly(X ) to G action on D viaLie alg hom g → D1.Seek simple D-module W (analogue of C∞(M)).Hope W irr for G ⇔ G has dense orbit on X .
![Page 75: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/75.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 76: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/76.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 77: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/77.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 78: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/78.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 79: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/79.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 80: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/80.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 81: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/81.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 82: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/82.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 83: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/83.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Poisson manifolds
Poisson mfld X : brkt {, } on fns with Lie alg axioms(skew, bilin, Jacobi) AND {f , gh} = {f , g}h + g{f , h}.Bracket with f ξf ∈ Vect X , Hamiltonian vector fld;Jacobi [ξf , ξg] = ξ{f ,g}.Vals of Hamiltonian vec flds integrable distn S foliation of X by embedded submflds.Symp form on Sx : ωx(ξf (x), ξg(x)) = {f , g}(x).
I Embedded submflds are symplectic.I Hamiltonian flows preserve embedded submflds.I Ham G-space! Lie alg map g → C∞(X )!
(Poisson) moment map map µ : X → g∗
I X Ham G-space G-orbits ⊂ embedded submflds.I Kostant: Homog Ham G-space = cover of orbit on g∗.
![Page 84: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/84.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 85: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/85.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 86: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/86.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 87: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/87.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 88: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/88.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 89: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/89.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 90: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/90.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 91: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/91.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 92: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/92.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
Method of coadjoint orbits
Kostant’s thm worth stating twice: homogeneousHamiltonian G-space = covering of G-orbit on g∗.Includes classification of symp homog spaces for G.(Riem homog spaces hopelessly complicated.)
Kirillov-Kostant philosophy of coadjt orbits suggests{irr unitary reps of G} = G! g∗/G. (?)
Bij (?) true for G simply conn nilp (Kirillov).
Other G: restr rt side to “admissible” orbits (integralitycond). Expect “almost all” of G: enough for interestingharmonic analysis.
Duflo: (?) for algebraic G reduces to reductive G.Two ways to do repn theory:
1. start with coadjt orbit, look for repn. Hard.2. start with repn, look for coadjt orbit. Easy.
![Page 93: Geometry and representations of reductive groupsdav/rittC.pdf · Geometry and representations of reductive groups David Vogan Introduction Commuting algebras Differential operator](https://reader034.fdocuments.net/reader034/viewer/2022042322/5f0c508f7e708231d434c98e/html5/thumbnails/93.jpg)
Geometry andrepresentations ofreductive groups
David Vogan
Introduction
Commutingalgebras
Differentialoperator algebras
HamiltonianG-spaces
References
References
W. Graham and D. Vogan, “Geometric quantization fornilpotent coadjoint orbits,” in Geometry andRepresentation Theory of real and p-adic groups.Birkhauser, Boston-Basel-Berlin, 1998 .
D. Vogan, “The method of coadjoint orbits for realreductive groups,” in Representation Theory of LieGroups. IAS/Park City Mathematics Series 8 (1999),179–238.