Unitary Representations of Nilpotent Super Lie groups Hadi - PowerPoint PPT Presentation

Unitary Representations of Nilpotent Super Lie groups Hadi Salmasian February 6, 2010

Basic Definitions and Notation Let G be a Lie group and H be a Hilbert space. A unitary representation π of G in H is a map π : G → U( H ) where U( H ) is the group of linear isometries of H , such that : π ( g 1 g 2 ) = π ( g 1 ) π ( g 2 ) π is strongly continuous, i.e., the map g �→ π ( g ) v is continuous for every v ∈ H

Example : the Schr¨ odinger model Suppose ( W , Ω ) is a finite dimensional symplectic vector space, i.e., Ω is nondegenerate, Ω ( v , w ) = − Ω ( w , v ).

Example : the Schr¨ odinger model Suppose ( W , Ω ) is a finite dimensional symplectic vector space, i.e., Ω is nondegenerate, Ω ( v , w ) = − Ω ( w , v ). Set G = H n where H n = { ( v , s ) | v ∈ W and s ∈ R } and the group law is defined by ( v 1 , s 1 ) • ( v 2 , s 2 ) = ( v 1 + v 2 , s 1 + s 2 + 1 2 Ω ( v 1 , v 2 )) .

Example : the Schr¨ odinger model Suppose ( W , Ω ) is a finite dimensional symplectic vector space, i.e., Ω is nondegenerate, Ω ( v , w ) = − Ω ( w , v ). Set G = H n where H n = { ( v , s ) | v ∈ W and s ∈ R } and the group law is defined by ( v 1 , s 1 ) • ( v 2 , s 2 ) = ( v 1 + v 2 , s 1 + s 2 + 1 2 Ω ( v 1 , v 2 )) . We know that dim Z ( H n ) = 1 and H n / Z ( H n ) is commutative (i.e., H n is two-step nilpotent).

Example : the Schr¨ odinger model (cont.) H n = { ( v , s ) | v ∈ W and s ∈ R }

Example : the Schr¨ odinger model (cont.) H n = { ( v , s ) | v ∈ W and s ∈ R } Consider a polarization of ( W , Ω ), i.e., a direct sum decomposition W = X ⊕ Y such that Ω ( X , X ) = Ω ( Y , Y ) = 0 .

Example : the Schr¨ odinger model (cont.) H n = { ( v , s ) | v ∈ W and s ∈ R } Consider a polarization of ( W , Ω ), i.e., a direct sum decomposition W = X ⊕ Y such that Ω ( X , X ) = Ω ( Y , Y ) = 0 . � Set H : = L 2 ( Y ) : = { f : Y → C | Y | f | 2 d µ < ∞ } .

Example : the Schr¨ odinger model (cont.) H n = { ( v , s ) | v ∈ W and s ∈ R } Consider a polarization of ( W , Ω ), i.e., a direct sum decomposition W = X ⊕ Y such that Ω ( X , X ) = Ω ( Y , Y ) = 0 . � Set H : = L 2 ( Y ) : = { f : Y → C | Y | f | 2 d µ < ∞ } . Fix a nonzero a ∈ R and define a representation π a of H n on H via √ � � e a Ω ( y , v ) − 1 f ( y ) π a ( v , 0) f ( y ) = if v ∈ X , � � if v ∈ Y , π a (0 , v ) f ( y ) = f ( y + v ) √ � � e at − 1 f ( y ) π a (0 , s ) f ( y ) = otherwise.

Example : the Schr¨ odinger model (cont.) √ � � e a Ω ( y , v ) − 1 f ( y ) π a ( v , 0) f ( y ) = if v ∈ X , � � π a (0 , v ) f ( y ) = f ( y + v ) if v ∈ Y , √ � � e at − 1 f ( y ) π a (0 , s ) f ( y ) = otherwise.

Example : the Schr¨ odinger model (cont.) √ � � e a Ω ( y , v ) − 1 f ( y ) π a ( v , 0) f ( y ) = if v ∈ X , � � π a (0 , v ) f ( y ) = f ( y + v ) if v ∈ Y , √ � � e at − 1 f ( y ) π a (0 , s ) f ( y ) = otherwise. Facts: For every a ∈ R , π a is an irreducible unitary representation of H n . ( i.e., H does not have nontrivial H n -invariant closed subspaces. )

Example : the Schr¨ odinger model (cont.) √ � � e a Ω ( y , v ) − 1 f ( y ) π a ( v , 0) f ( y ) = if v ∈ X , � � π a (0 , v ) f ( y ) = f ( y + v ) if v ∈ Y , √ � � e at − 1 f ( y ) π a (0 , s ) f ( y ) = otherwise. Facts: For every a ∈ R , π a is an irreducible unitary representation of H n . ( i.e., H does not have nontrivial H n -invariant closed subspaces. ) If a � b , the representations π a and π b are not (unitarily) equivalent.

Example : the Schr¨ odinger model (cont.) √ � � e a Ω ( y , v ) − 1 f ( y ) π a ( v , 0) f ( y ) = if v ∈ X , � � π a (0 , v ) f ( y ) = f ( y + v ) if v ∈ Y , √ � � e at − 1 f ( y ) π a (0 , s ) f ( y ) = otherwise. Facts: For every a ∈ R , π a is an irreducible unitary representation of H n . ( i.e., H does not have nontrivial H n -invariant closed subspaces. ) If a � b , the representations π a and π b are not (unitarily) equivalent. (Stone-von Neumann, 1930’s) Up to unitary equivalence, the irreducible unitary representations of H n are : one-dimensional representations (which factor through 1 H n / Z ( H n )), The representations π a , a ∈ R × . 2

A bit of history... Gelfand (1940’s) : Unitary representations Quantization of ← − − − − − − − → of G G − spaces

A bit of history... Gelfand (1940’s) : Unitary representations Quantization of ← − − − − − − − → of G G − spaces Kirillov (1950’s) If G is a nilpotent simply connected Lie group, then there exists a bijective correspondence Irreducible unitary G − orbits � in g ∗ representations of G

A bit of history... Gelfand (1940’s) : Unitary representations Quantization of ← − − − − − − − → of G G − spaces Kirillov (1950’s) If G is a nilpotent simply connected Lie group, then there exists a bijective correspondence Irreducible unitary G − orbits � in g ∗ representations of G There is also a dictionary : Algebraic operation Geometric operation p ( O ) where p : g ∗ → h ∗ Res G H π p − 1 ( O ) where p : g ∗ → h ∗ Ind G H π π 1 ⊗ π 2 O 1 + O 2 ... ... Note that the algebraic operations should be understood in the context of direct � integrals, i.e. : Res G H π = H n ( σ ) σ d µ ( σ ), etc. ˆ

Kirillov’s orbit method Suppose G is nilpotent and simply connected. Set g = Lie( G ).

Kirillov’s orbit method Suppose G is nilpotent and simply connected. Set g = Lie( G ). Recipe to construct π from O :

Kirillov’s orbit method Suppose G is nilpotent and simply connected. Set g = Lie( G ). Recipe to construct π from O : Fix λ ∈ O . Consider the skew-symmetric form 1 Ω λ : g × g → R defined by Ω λ ( X , Y ) = λ ([ X , Y ]).

Kirillov’s orbit method Suppose G is nilpotent and simply connected. Set g = Lie( G ). Recipe to construct π from O : Fix λ ∈ O . Consider the skew-symmetric form 1 Ω λ : g × g → R defined by Ω λ ( X , Y ) = λ ([ X , Y ]). Proposition. There exists a subalgebra m ⊂ g such that m is 2 a maximal isotropic subspace of Ω λ .

Kirillov’s orbit method Suppose G is nilpotent and simply connected. Set g = Lie( G ). Recipe to construct π from O : Fix λ ∈ O . Consider the skew-symmetric form 1 Ω λ : g × g → R defined by Ω λ ( X , Y ) = λ ([ X , Y ]). Proposition. There exists a subalgebra m ⊂ g such that m is 2 a maximal isotropic subspace of Ω λ . Set M = exp( m ) and define χ λ : M → C × by 3 √ χ λ (exp( X )) = e λ ( X ) − 1 for every X ∈ m .

Kirillov’s orbit method Suppose G is nilpotent and simply connected. Set g = Lie( G ). Recipe to construct π from O : Fix λ ∈ O . Consider the skew-symmetric form 1 Ω λ : g × g → R defined by Ω λ ( X , Y ) = λ ([ X , Y ]). Proposition. There exists a subalgebra m ⊂ g such that m is 2 a maximal isotropic subspace of Ω λ . Set M = exp( m ) and define χ λ : M → C × by 3 √ χ λ (exp( X )) = e λ ( X ) − 1 for every X ∈ m . Set π = Ind G M χ λ . 4

Example : Schr¨ odinger model revisited Recall that : H n = { ( v , s ) | v ∈ W and s ∈ R } Set h n = Lie( H n ) and fix Z ∈ Z ( h n ).

Example : Schr¨ odinger model revisited Recall that : H n = { ( v , s ) | v ∈ W and s ∈ R } Set h n = Lie( H n ) and fix Z ∈ Z ( h n ). H n -orbits in h ∗ n are : one-dimensional { λ } where λ ( Z ) = 0 � representations of H n . { λ ∈ h ∗ n | λ ( Z ) = a } the representation π a . �

Lie superalgebras : introduction g = g 0 ⊕ g 1 and [ · , · ] : g × g → g where ( − 1) | x |·| z | [ X , [ Y , Z ]] + ( − 1) | y |·| x | [ Y , [ Z , X ]] + ( − 1) | z |·| y | [ Z , [ X , Y ]] = 0

Lie superalgebras : introduction g = g 0 ⊕ g 1 and [ · , · ] : g × g → g where ( − 1) | x |·| z | [ X , [ Y , Z ]] + ( − 1) | y |·| x | [ Y , [ Z , X ]] + ( − 1) | z |·| y | [ Z , [ X , Y ]] = 0 Examples gl ( m | n ) : V = V 0 ⊕ V 1 and g = End( V ) = End 0 ( V ) ⊕ End 1 ( V ) with [ X , Y ] = XY − ( − 1) | x |·| y | YX

Lie superalgebras : introduction g = g 0 ⊕ g 1 and [ · , · ] : g × g → g where ( − 1) | x |·| z | [ X , [ Y , Z ]] + ( − 1) | y |·| x | [ Y , [ Z , X ]] + ( − 1) | z |·| y | [ Z , [ X , Y ]] = 0 Examples gl ( m | n ) : V = V 0 ⊕ V 1 and g = End( V ) = End 0 ( V ) ⊕ End 1 ( V ) with [ X , Y ] = XY − ( − 1) | x |·| y | YX sl ( m | n ) , osp ( m | 2 n ) , p ( n ) , q ( n ) , ...

Lie superalgebras : introduction g = g 0 ⊕ g 1 and [ · , · ] : g × g → g where ( − 1) | x |·| z | [ X , [ Y , Z ]] + ( − 1) | y |·| x | [ Y , [ Z , X ]] + ( − 1) | z |·| y | [ Z , [ X , Y ]] = 0 Examples gl ( m | n ) : V = V 0 ⊕ V 1 and g = End( V ) = End 0 ( V ) ⊕ End 1 ( V ) with [ X , Y ] = XY − ( − 1) | x |·| y | YX sl ( m | n ) , osp ( m | 2 n ) , p ( n ) , q ( n ) , ... Heisenberg-Cli ff ord Lie superalgebra.

Heisenberg-Cli ff ord Lie superalgebra Let ( W , Ω ) be a supersymplectic space, i.e., W = W 0 ⊕ W 1 . Ω : W × W → R satisfies Ω ( W 0 , W 1 ) = Ω ( W 1 , W 0 ) = 0 Ω | W 1 × W 1 is a nondegenerate symmetric form. Ω | W 0 × W 0 is a symplectic form.