Monomial representation: The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Let H = exp( h ) ⊂ G be a closed connected subgroup. Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Monomial representation: The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Let H = exp( h ) ⊂ G be a closed connected subgroup. Index sets and representations G / H admits a G -invariant Borel measure dx . Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Monomial representation: The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Let H = exp( h ) ⊂ G be a closed connected subgroup. Index sets and representations G / H admits a G -invariant Borel measure dx . Let ℓ ∈ g ∗ Index sets and with � ℓ, [ h , h ] � = { 0 } . representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Monomial representation: The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Let H = exp( h ) ⊂ G be a closed connected subgroup. Index sets and representations G / H admits a G -invariant Borel measure dx . Let ℓ ∈ g ∗ Index sets and with � ℓ, [ h , h ] � = { 0 } . representations Index sets and representations χ ℓ ( h ) := e − 2 i π � ℓ, log( h ) � , h ∈ H . An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Monomial representation: The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Let H = exp( h ) ⊂ G be a closed connected subgroup. Index sets and representations G / H admits a G -invariant Borel measure dx . Let ℓ ∈ g ∗ Index sets and with � ℓ, [ h , h ] � = { 0 } . representations Index sets and representations χ ℓ ( h ) := e − 2 i π � ℓ, log( h ) � , h ∈ H . An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Definition nilpotent Lie group Index sets and representations L 2 ( G / H , χ ℓ ) Index sets and H ℓ, h = representations = { ξ : G → C , measurable , Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Definition nilpotent Lie group Index sets and representations L 2 ( G / H , χ ℓ ) Index sets and H ℓ, h = representations = { ξ : G → C , measurable , Index sets and representations ξ ( gh ) = χ ℓ ( h − 1 ) ξ ( g ) , g ∈ G , h ∈ H } Index sets and � representations | ξ ( g ) | 2 dg < ∞ . Index sets and representations G / H An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Definition nilpotent Lie group Index sets and representations L 2 ( G / H , χ ℓ ) Index sets and H ℓ, h = representations = { ξ : G → C , measurable , Index sets and representations ξ ( gh ) = χ ℓ ( h − 1 ) ξ ( g ) , g ∈ G , h ∈ H } Index sets and � representations | ξ ( g ) | 2 dg < ∞ . Index sets and representations G / H An example Let Variable groups Fourier Transform σ ℓ, h ( g ) ξ ( s ) := ξ ( g − 1 s ) , g , s ∈ G , ξ ∈ L 2 ( G / H , χ ℓ ) . Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Definition nilpotent Lie group Index sets and representations L 2 ( G / H , χ ℓ ) Index sets and H ℓ, h = representations = { ξ : G → C , measurable , Index sets and representations ξ ( gh ) = χ ℓ ( h − 1 ) ξ ( g ) , g ∈ G , h ∈ H } Index sets and � representations | ξ ( g ) | 2 dg < ∞ . Index sets and representations G / H An example Let Variable groups Fourier Transform σ ℓ, h ( g ) ξ ( s ) := ξ ( g − 1 s ) , g , s ∈ G , ξ ∈ L 2 ( G / H , χ ℓ ) . Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Proposition Index sets and representations For F ∈ L 1 ( G ) : Index sets and � representations Index sets and σ ℓ, h ( F ) ξ ( s ) = F ℓ, h ( s , t ) ξ ( t ) dt , representations G / H � Index sets and representations F ( sht − 1 ) χ ℓ ( h ) dh . where F ℓ, h ( s , t ) = An example H Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Orbit picture The dual space of a Theorem nilpotent Lie group ◮ Let ℓ ∈ g ∗ and let p be a polarization at ℓ . Then σ ℓ, p Index sets and representations is irreducible. Index sets and representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Orbit picture The dual space of a Theorem nilpotent Lie group ◮ Let ℓ ∈ g ∗ and let p be a polarization at ℓ . Then σ ℓ, p Index sets and representations is irreducible. Index sets and representations ◮ Let ℓ i ∈ g ∗ and let p i , i = 1 , 2 be a polarization at Index sets and representations ℓ i , i = 1 , 2 . Then Index sets and representations σ ℓ 1 , p 1 ≃ σ ℓ 2 , p 2 ⇔ Ad ∗ ( G ) ℓ 2 = Ad ∗ ( G ) ℓ 1 . Index sets and representations Write: An example Variable groups [ π ℓ ] := [ σ π, p ] Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Orbit picture The dual space of a Theorem nilpotent Lie group ◮ Let ℓ ∈ g ∗ and let p be a polarization at ℓ . Then σ ℓ, p Index sets and representations is irreducible. Index sets and representations ◮ Let ℓ i ∈ g ∗ and let p i , i = 1 , 2 be a polarization at Index sets and representations ℓ i , i = 1 , 2 . Then Index sets and representations σ ℓ 1 , p 1 ≃ σ ℓ 2 , p 2 ⇔ Ad ∗ ( G ) ℓ 2 = Ad ∗ ( G ) ℓ 1 . Index sets and representations Write: An example Variable groups [ π ℓ ] := [ σ π, p ] Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds G ⇒ ∃ ℓ ∈ g ∗ such that ◮ Let ( π, H π ) ∈ � Fourier inversion for sub-manifolds [ π ] = [ π ℓ ] Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
A homeomorphism The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets and Theorem representations The mapping K : g ∗ / G → � G defined by Index sets and representations Index sets and K ( Ad ∗ ( G ) ℓ ) := [ π ℓ ] representations An example is a homeomorphism Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
A partition of the orbit space The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets: Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
A partition of the orbit space The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets: Let Z = { Z 1 , · · · , Z n } be a Jordan-H¨ older Index sets and basis of g and let ℓ ∈ g ∗ . representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
A partition of the orbit space The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets: Let Z = { Z 1 , · · · , Z n } be a Jordan-H¨ older Index sets and basis of g and let ℓ ∈ g ∗ . The index set I ( ℓ ) = I Z ( ℓ ) of representations ℓ ∈ g ∗ is defined by: Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
A partition of the orbit space The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets: Let Z = { Z 1 , · · · , Z n } be a Jordan-H¨ older Index sets and basis of g and let ℓ ∈ g ∗ . The index set I ( ℓ ) = I Z ( ℓ ) of representations ℓ ∈ g ∗ is defined by: Index sets and representations I ( ℓ ) = ∅ if ℓ is a character. Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
A partition of the orbit space The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets: Let Z = { Z 1 , · · · , Z n } be a Jordan-H¨ older Index sets and basis of g and let ℓ ∈ g ∗ . The index set I ( ℓ ) = I Z ( ℓ ) of representations ℓ ∈ g ∗ is defined by: Index sets and representations I ( ℓ ) = ∅ if ℓ is a character. Otherwise, let Index sets and representations j 1 = j 1 ( ℓ ) = max { j ∈ { 1 , . . . , n } | Z j �∈ a ( ℓ ) } An example Variable groups k 1 = k 1 ( ℓ ) = max { k ∈ { 1 , . . . , n } | < l , [ Z j 1 ( ℓ ) , Z k ] > � = 0 } . Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
We let ν 1 ( ℓ ) : = � ℓ, [ Z k 1 , Z j 1 ] � The dual space of a nilpotent Lie group 1 S 1 = S 1 ( ℓ ) : = ν 1 ( ℓ )[ Z k 1 , Z j 1 ] , Index sets and representations = Z j 1 − � ℓ, Y 1 � Index sets and Y 1 = Y 1 ( ℓ ) : ν 1 ( ℓ ) S 1 representations Index sets and representations = Z k 1 − � ℓ, Z k 1 � X 1 = X 1 ( ℓ ) : ν 1 ( ℓ ) S 1 . Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
We let ν 1 ( ℓ ) : = � ℓ, [ Z k 1 , Z j 1 ] � The dual space of a nilpotent Lie group 1 S 1 = S 1 ( ℓ ) : = ν 1 ( ℓ )[ Z k 1 , Z j 1 ] , Index sets and representations = Z j 1 − � ℓ, Y 1 � Index sets and Y 1 = Y 1 ( ℓ ) : ν 1 ( ℓ ) S 1 representations Index sets and representations = Z k 1 − � ℓ, Z k 1 � X 1 = X 1 ( ℓ ) : ν 1 ( ℓ ) S 1 . Index sets and representations Index sets and Then we have that: representations An example � ℓ, X 1 � = � ℓ, Y 1 � = 0 , (0.1) Variable groups Fourier Transform � ℓ, [ X 1 , Y 1 ] � = 1 . Un-sufficient data Fourier inversion for We consider sub-manifolds Fourier inversion for g 1 ( ℓ ) := { U ∈ g | < l , [ U , Y 1 ( ℓ )] > = 0 } (0.2) sub-manifolds Fourier inversion for sub-manifolds which is an ideal of codimension one in g . Fourier inversion for sub-manifolds
older basis of ( g 1 ( ℓ ) , [ · , · ]) is given by A Jordan-H¨ { Z 1 i ( ℓ ) | i � = k 1 ( ℓ ) } defined by The dual space of a nilpotent Lie group Index sets and representations i ( ℓ ) = Z i − < l , [ Z i , Y 1 ( ℓ )] > Z 1 X 1 ( ℓ ) , i � = k 1 ( ℓ ) . (0.3) Index sets and ν 1 ( ℓ ) representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
older basis of ( g 1 ( ℓ ) , [ · , · ]) is given by A Jordan-H¨ { Z 1 i ( ℓ ) | i � = k 1 ( ℓ ) } defined by The dual space of a nilpotent Lie group Index sets and representations i ( ℓ ) = Z i − < l , [ Z i , Y 1 ( ℓ )] > Z 1 X 1 ( ℓ ) , i � = k 1 ( ℓ ) . (0.3) Index sets and ν 1 ( ℓ ) representations Index sets and representations As previously we may now compute the indices Index sets and j 2 ( ℓ ) , k 2 ( ℓ ) of l 1 := l | g 1 ( ℓ ) with respect to this new basis representations and construct the corresponding subalgebra g 2 ( ℓ ) with its Index sets and representations associated basis { Z 2 i ( ℓ ) | i � = k 1 ( ℓ ) , k 2 ( ℓ ) } . An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
older basis of ( g 1 ( ℓ ) , [ · , · ]) is given by A Jordan-H¨ { Z 1 i ( ℓ ) | i � = k 1 ( ℓ ) } defined by The dual space of a nilpotent Lie group Index sets and representations i ( ℓ ) = Z i − < l , [ Z i , Y 1 ( ℓ )] > Z 1 X 1 ( ℓ ) , i � = k 1 ( ℓ ) . (0.3) Index sets and ν 1 ( ℓ ) representations Index sets and representations As previously we may now compute the indices Index sets and j 2 ( ℓ ) , k 2 ( ℓ ) of l 1 := l | g 1 ( ℓ ) with respect to this new basis representations and construct the corresponding subalgebra g 2 ( ℓ ) with its Index sets and representations associated basis { Z 2 i ( ℓ ) | i � = k 1 ( ℓ ) , k 2 ( ℓ ) } . An example This procedure stops after a finite number r ℓ = r of Variable groups steps. Let Fourier Transform � � Un-sufficient data I Z ( ℓ ) = I ( ℓ ) = ( j 1 ( ℓ ) , k 1 ( ℓ )) , . . . , ( j r ( ℓ ) , k r ( ℓ )) Fourier inversion for sub-manifolds is called the index of ℓ in g with respect to the basis Fourier inversion for sub-manifolds Z = { Z 1 , . . . . Z n } . Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and representations It is known that the last subalgebra g r ( ℓ ) obtained by this Index sets and construction coincides with the Vergne polarization of ℓ representations in g with respect to the basis Z . Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and representations It is known that the last subalgebra g r ( ℓ ) obtained by this Index sets and construction coincides with the Vergne polarization of ℓ representations in g with respect to the basis Z . Index sets and representations The length | I | = 2 r of the index set I ( ℓ ) gives us the Index sets and dimension of the coadjoint orbit Ad ∗ ( G ) ℓ . representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Partition of g ∗ / G The dual space of a nilpotent Lie group Index sets and For an index set I ∈ N 2 j , j = 0 , · · · , dim( g / 2): representations Index sets and g ∗ I := { ℓ ∈ g ∗ , I ( ℓ ) = I , � l , X i ( ℓ ) � = 0 , � l , Y i ( ℓ ) � = 0 , i = 1 , · · · , r } . representations Index sets and representations Let Index sets and representations dim ( g / 2) � Index sets and N j , g ∗ representations I := { I ∈ I � = ∅} . An example j =0 Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Partition of g ∗ / G The dual space of a nilpotent Lie group Index sets and For an index set I ∈ N 2 j , j = 0 , · · · , dim( g / 2): representations Index sets and g ∗ I := { ℓ ∈ g ∗ , I ( ℓ ) = I , � l , X i ( ℓ ) � = 0 , � l , Y i ( ℓ ) � = 0 , i = 1 , · · · , r } . representations Index sets and representations Let Index sets and representations dim ( g / 2) � Index sets and N j , g ∗ representations I := { I ∈ I � = ∅} . An example j =0 Variable groups Then: Fourier Transform Un-sufficient data � I := ˙ g ∗ / G ≃ g ∗ I ∈I g ∗ Fourier inversion for I sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properties of the g ∗ I : The dual space of a nilpotent Lie group There exists an index I gen ∈ I such that Index sets and representations Index sets and g ∗ gen := { ℓ ∈ g ∗ , I ( ℓ ) = I gen } representations Index sets and representations is G -invariant and Zariski open in g ∗ . Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properties of the g ∗ I : The dual space of a nilpotent Lie group There exists an index I gen ∈ I such that Index sets and representations Index sets and g ∗ gen := { ℓ ∈ g ∗ , I ( ℓ ) = I gen } representations Index sets and representations is G -invariant and Zariski open in g ∗ . Index sets and representations There exists an order on I such that Index sets and ◮ I gen is maximal for this order, representations ◮ such that An example Variable groups � g ∗ g ∗ Fourier Transform ≤ I := I ′ Un-sufficient data I ′ ≤ I Fourier inversion for sub-manifolds is Zariski closed in g ∗ . Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Realization on L 2 ( R r ) The dual space of a nilpotent Lie group Proposition Index sets and representations ◮ For every I ∈ I the mappings Index sets and representations g ∗ I ∋ ℓ �→ X j ( ℓ ) , ℓ �→ Y j ( ℓ ) , ℓ �→ p Z ( ℓ ) Index sets and representations Index sets and are smooth. representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Realization on L 2 ( R r ) The dual space of a nilpotent Lie group Proposition Index sets and representations ◮ For every I ∈ I the mappings Index sets and representations g ∗ I ∋ ℓ �→ X j ( ℓ ) , ℓ �→ Y j ( ℓ ) , ℓ �→ p Z ( ℓ ) Index sets and representations Index sets and are smooth. representations ◮ The family of vectors X ( ℓ ) = { X j ( ℓ ) , j = 1 , · · · , r } Index sets and representations form a Malcev-basis of g modulo p Z ( ℓ ) , An example the vectors { Y j ( ℓ ) , j = 1 , · · · , r } form a Malcev basis Variable groups of p Z ( ℓ ) modulo g ( ℓ ) . Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Realization on L 2 ( R r ) The dual space of a nilpotent Lie group Proposition Index sets and representations ◮ For every I ∈ I the mappings Index sets and representations g ∗ I ∋ ℓ �→ X j ( ℓ ) , ℓ �→ Y j ( ℓ ) , ℓ �→ p Z ( ℓ ) Index sets and representations Index sets and are smooth. representations ◮ The family of vectors X ( ℓ ) = { X j ( ℓ ) , j = 1 , · · · , r } Index sets and representations form a Malcev-basis of g modulo p Z ( ℓ ) , An example the vectors { Y j ( ℓ ) , j = 1 , · · · , r } form a Malcev basis Variable groups of p Z ( ℓ ) modulo g ( ℓ ) . Fourier Transform ◮ We identify the Hilbert space L 2 ( G / P Z ( ℓ ) , χ ℓ ) with Un-sufficient data L 2 ( R r ℓ ) using the unitary operator: Fourier inversion for sub-manifolds Fourier inversion for U ℓ ( η ) = η ◦ E Z ℓ ∈ L 2 ( R r ℓ ) , η ∈ L 2 ( G / P Z ( ℓ ) , χ ℓ ) . sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
An example The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets and representations Let g = span { A , B , C , D , U , V } . Index sets and representations [ A , B ] = U , [ C , D ] = V , [ A , C ] = V , [ B , D ] = sU Index sets and representations ( s ∈ R ∗ ). An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Let ℓ ∈ g ∗ nilpotent Lie group Index sets and representations µ = � ℓ, U � , � ℓ, V � = ν. Index sets and representations Index sets and representations ◮ ν � = 0 ⇒ Index sets and representations span { A , B − s µ g 1 ( ℓ ) = ν C , D , U , V } , Index sets and representations j 1 ( ℓ ) = 4 , k 1 ( ℓ ) = 3 An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Let ℓ ∈ g ∗ nilpotent Lie group Index sets and representations µ = � ℓ, U � , � ℓ, V � = ν. Index sets and representations Index sets and representations ◮ ν � = 0 ⇒ Index sets and representations span { A , B − s µ g 1 ( ℓ ) = ν C , D , U , V } , Index sets and representations j 1 ( ℓ ) = 4 , k 1 ( ℓ ) = 3 An example 2 = B − s µ Variable groups Z 1 1 = A , Z 1 ν C , Fourier Transform Z 1 4 = D , Z 1 5 = U , Z 1 6 = V . Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and representations 5 − s µ [ Z 1 1 , Z 1 2 ] s ,µ,ν = Z 1 ν Z 1 Index sets and 6 , representations 5 − s µ Index sets and [ Z 1 2 , Z 1 4 ] s ,µ,ν = sZ 1 ν Z 1 6 . representations Index sets and j 2 ( ℓ ) = 2 , k 2 ( ℓ ) = 1 , if s � = 1 . representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets and If ν = 0 , µ � = 0 ⇒ g 1 ( ℓ ) = span { A , C , D , U , V } representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets and If ν = 0 , µ � = 0 ⇒ g 1 ( ℓ ) = span { A , C , D , U , V } and representations j 1 ( ℓ ) = 4 , k 1 ( ℓ ) = 2. Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Variable groups. The dual space of a nilpotent Lie group Index sets and Definition representations A variable locally compact group is a pair Index sets and representations Index sets and ( B , G ) representations Index sets and representations where B and G are locally compact topological spaces, Index sets and such that for every β ∈ B there exists a group representations multiplication · β on G , which turns ( G , · β ) into a An example topological group, such that Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Variable groups. The dual space of a nilpotent Lie group Index sets and Definition representations A variable locally compact group is a pair Index sets and representations Index sets and ( B , G ) representations Index sets and representations where B and G are locally compact topological spaces, Index sets and such that for every β ∈ B there exists a group representations multiplication · β on G , which turns ( G , · β ) into a An example topological group, such that Variable groups Fourier Transform Un-sufficient data B × ( G × G ) �→ G , ( β, ( s , t )) → s · β t Fourier inversion for sub-manifolds Fourier inversion for is continuous. sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a Definition nilpotent Lie group A variable nilpotent Lie algebra is a triple Index sets and representations Index sets and ( g , Z , B ) representations Index sets and of a real finite dimensional vector space g , of a basis representations Index sets and Z = { Z 1 , · · · , Z n } of g and a smooth manifold B , such representations that Index sets and representations ◮ for every β ∈ B there is a Lie algebra product [ , ] β An example on g , Variable groups ◮ [ Z i , Z j ] β = � n k = j +1 c i , j k ( β ) Z k , 1 ≤ i < j ≤ n Fourier Transform Un-sufficient data ◮ and such that the functions β → c i , j k ( β ) are all Fourier inversion for smooth. sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier transform The dual space of a nilpotent Lie group Definition Index sets and representations Index sets and representations l ∞ ( � G ) := { ( ϕ ( ℓ ) ∈ K ( H ℓ ) ℓ ∈ g ∗ I , � ϕ � ∞ := sup � ϕ ( ℓ ) � op < ∞} . Index sets and ℓ ∈ g ∗ representations I Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier transform The dual space of a nilpotent Lie group Definition Index sets and representations Index sets and representations l ∞ ( � G ) := { ( ϕ ( ℓ ) ∈ K ( H ℓ ) ℓ ∈ g ∗ I , � ϕ � ∞ := sup � ϕ ( ℓ ) � op < ∞} . Index sets and ℓ ∈ g ∗ representations I Index sets and representations Write for ℓ ∈ g ∗ I , ( π ℓ , H ℓ ) = ( σ ℓ, p Z ( ℓ ) , L 2 ( R r ℓ ) . Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier transform The dual space of a nilpotent Lie group Definition Index sets and representations Index sets and representations l ∞ ( � G ) := { ( ϕ ( ℓ ) ∈ K ( H ℓ ) ℓ ∈ g ∗ I , � ϕ � ∞ := sup � ϕ ( ℓ ) � op < ∞} . Index sets and ℓ ∈ g ∗ representations I Index sets and representations Write for ℓ ∈ g ∗ I , ( π ℓ , H ℓ ) = ( σ ℓ, p Z ( ℓ ) , L 2 ( R r ℓ ) . Index sets and For F ∈ L 1 ( G ) , let representations An example F ( F )( ℓ ) = � F ( ℓ ) := π ℓ ( F ) , ℓ ∈ g ∗ I . Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier transform The dual space of a nilpotent Lie group Definition Index sets and representations Index sets and representations l ∞ ( � G ) := { ( ϕ ( ℓ ) ∈ K ( H ℓ ) ℓ ∈ g ∗ I , � ϕ � ∞ := sup � ϕ ( ℓ ) � op < ∞} . Index sets and ℓ ∈ g ∗ representations I Index sets and representations Write for ℓ ∈ g ∗ I , ( π ℓ , H ℓ ) = ( σ ℓ, p Z ( ℓ ) , L 2 ( R r ℓ ) . Index sets and For F ∈ L 1 ( G ) , let representations An example F ( F )( ℓ ) = � F ( ℓ ) := π ℓ ( F ) , ℓ ∈ g ∗ I . Variable groups Fourier Transform For u ∈ U ( g ) let Un-sufficient data Fourier inversion for sub-manifolds u ( ℓ ) = d π ℓ ( u ) ∈ PD ( R r I ) , ℓ ∈ g ∗ � I Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properties of � u The dual space of a nilpotent Lie group ◮ For every u ∈ U ( g ), for ℓ ∈ g I , Index sets and representations � Index sets and α ( ℓ ) ∂ α p u representations d σ ℓ, p Z ( ℓ ) ( u ) = � u ( ℓ ) = Index sets and α ∈ R rI representations Index sets and with polynomial coefficients p u α ( ℓ ) which depend representations smoothly on ℓ ∈ g ∗ I . Index sets and representations Let An example Variable groups d µ ( u ) := ( d σ ℓ, p Z ( ℓ ) ( u )) ℓ ∈ I gen Fourier Transform ◮ For every D = � α ∈ N rI p α ∂ α there exists a smooth Un-sufficient data Fourier inversion for mapping ρ D , I : g ∗ I → U ( g ), such that sub-manifolds Fourier inversion for d σ ℓ, p Z ( ℓ ) ( ρ D , I ( ℓ )) = D , ℓ ∈ g ∗ sub-manifolds I . Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properties of � F , F ∈ S ( G ) The dual space of a nilpotent Lie group ◮ With respect to the basis X ( ℓ ) = { X 1 ( ℓ ) , · · · , X r ( ℓ ) } Index sets and the kernel functions of the operators σ ℓ, p Z ( ℓ ) ( F ) : representations � Index sets and representations F Z ( ℓ, x , x ′ ) := F ( E X ( ℓ ) ( x ) hE X ( ℓ ) ( x ′ ) − 1 ) χ ℓ ( h ) dh Index sets and P Z ( ℓ ) representations I × R r × R r are smooth and Schwartz in defined on g ∗ Index sets and representations x , x ′ . Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properties of � F , F ∈ S ( G ) The dual space of a nilpotent Lie group ◮ With respect to the basis X ( ℓ ) = { X 1 ( ℓ ) , · · · , X r ( ℓ ) } Index sets and the kernel functions of the operators σ ℓ, p Z ( ℓ ) ( F ) : representations � Index sets and representations F Z ( ℓ, x , x ′ ) := F ( E X ( ℓ ) ( x ) hE X ( ℓ ) ( x ′ ) − 1 ) χ ℓ ( h ) dh Index sets and P Z ( ℓ ) representations I × R r × R r are smooth and Schwartz in defined on g ∗ Index sets and representations x , x ′ . Index sets and ◮ Let Q ∈ C [ g ]. For every I = I gen , there exists a representations I × R r I with partial differential operator D Q ( I ) on g ∗ An example Variable groups polynomial coefficients in the variable ( x , x ′ ) ∈ R r I × R r I and smooth coefficients in ℓ ∈ g ∗ Fourier Transform I , Un-sufficient data such that for every F ∈ S ( G ): Fourier inversion for ( QF ) Z ( ℓ, x , x ′ ) = D Q ( ℓ )( F Z )( ℓ, x , x ′ ) . sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properties of � F , F ∈ S ( G ) The dual space of a nilpotent Lie group ◮ With respect to the basis X ( ℓ ) = { X 1 ( ℓ ) , · · · , X r ( ℓ ) } Index sets and the kernel functions of the operators σ ℓ, p Z ( ℓ ) ( F ) : representations � Index sets and representations F Z ( ℓ, x , x ′ ) := F ( E X ( ℓ ) ( x ) hE X ( ℓ ) ( x ′ ) − 1 ) χ ℓ ( h ) dh Index sets and P Z ( ℓ ) representations I × R r × R r are smooth and Schwartz in defined on g ∗ Index sets and representations x , x ′ . Index sets and ◮ Let Q ∈ C [ g ]. For every I = I gen , there exists a representations I × R r I with partial differential operator D Q ( I ) on g ∗ An example Variable groups polynomial coefficients in the variable ( x , x ′ ) ∈ R r I × R r I and smooth coefficients in ℓ ∈ g ∗ Fourier Transform I , Un-sufficient data such that for every F ∈ S ( G ): Fourier inversion for ( QF ) Z ( ℓ, x , x ′ ) = D Q ( ℓ )( F Z )( ℓ, x , x ′ ) . sub-manifolds Fourier inversion for sub-manifolds Let Fourier inversion for sub-manifolds δ ( Q ) := ( D Q ( ℓ )) ℓ ∈ I gen Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Properties of � F , F ∈ L 1 ( G ): Index sets and representations 1. the operator field � F is contained in l ∞ ( � G ). Index sets and representations 2. on the subsets g ∗ I , I ∈ I , the mappings Index sets and representations ℓ �→ � F ( ℓ ) ∈ K ( L 2 ( R r I )) are operator -norm continuous . Index sets and representations 3. For every sequence ( Ad ∗ ( G ) ℓ k ) k ∈ N which goes to Index sets and representations infinity in g ∗ / G , we have that An example Variable groups k →∞ � � lim F ( ℓ k ) � op = 0 . Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Questions: Index sets and representations ◮ Characterize the image of C ∗ ( G ) in l ∞ ( � G ) under the Index sets and Fourier transform,i.e. representations understand how π ℓ ( F ) varies if ℓ ∈ g ∗ I approaches Index sets and representations the boundary of g ∗ I . Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Questions: Index sets and representations ◮ Characterize the image of C ∗ ( G ) in l ∞ ( � G ) under the Index sets and Fourier transform,i.e. representations understand how π ℓ ( F ) varies if ℓ ∈ g ∗ I approaches Index sets and representations the boundary of g ∗ I . Index sets and representations ◮ Characterize the image of S ( G ) in l ∞ ( � G ) under the An example Fourier transform. Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properly converging sequences in � G The dual space of a Let I ∈ I and let O = ( π O k ) be a properly converging nilpotent Lie group sequence in � G I with limit set L ( O ) contained in � G < I , Index sets and representations then the elements ρ ∈ L ( O ) are “entangled ” by O : Index sets and representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properly converging sequences in � G The dual space of a Let I ∈ I and let O = ( π O k ) be a properly converging nilpotent Lie group sequence in � G I with limit set L ( O ) contained in � G < I , Index sets and representations then the elements ρ ∈ L ( O ) are “entangled ” by O : Index sets and For instance if for some F ∈ C ∗ ( G ) we have that representations Index sets and π O k ( F ) = 0 for an infinity of k ’s then representations Index sets and ρ ( F ) = 0 , ∀ ρ ∈ L ( O ) . representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Properly converging sequences in � G The dual space of a Let I ∈ I and let O = ( π O k ) be a properly converging nilpotent Lie group sequence in � G I with limit set L ( O ) contained in � G < I , Index sets and representations then the elements ρ ∈ L ( O ) are “entangled ” by O : Index sets and For instance if for some F ∈ C ∗ ( G ) we have that representations Index sets and π O k ( F ) = 0 for an infinity of k ’s then representations Index sets and ρ ( F ) = 0 , ∀ ρ ∈ L ( O ) . representations Index sets and representations Question: What is the relation between the sequence of An example operators Variable groups Fourier Transform ( π O k ( F ) ∈ B ( L 2 ( R r I ))) k Un-sufficient data Fourier inversion for and the operator field sub-manifolds Fourier inversion for sub-manifolds ( ρ ( F )) ρ ∈ L ( O ) ? Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
S ( � G ) The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and L 2 ( � representations { ( ϕ ( ℓ )) ℓ ∈ g ∗ Igen , ℓ → ϕ ( ℓ ) measurable , G ) = � Index sets and representations � ϕ ( ℓ ) � 2 H − S d µ ( ℓ ) < ∞} Index sets and � G representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
S ( � G ) The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and L 2 ( � representations { ( ϕ ( ℓ )) ℓ ∈ g ∗ Igen , ℓ → ϕ ( ℓ ) measurable , G ) = � Index sets and representations � ϕ ( ℓ ) � 2 H − S d µ ( ℓ ) < ∞} Index sets and � G representations Index sets and Let representations An example S ( � { ϕ ∈ L 2 ( � G ) = G ) , Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
S ( � G ) The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and L 2 ( � representations { ( ϕ ( ℓ )) ℓ ∈ g ∗ Igen , ℓ → ϕ ( ℓ ) measurable , G ) = � Index sets and representations � ϕ ( ℓ ) � 2 H − S d µ ( ℓ ) < ∞} Index sets and � G representations Index sets and Let representations An example S ( � { ϕ ∈ L 2 ( � G ) = G ) , Variable groups d µ ( u )( ϕ ) ∈ L 2 ( � G ) , u ∈ U ( g ) , Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
S ( � G ) The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and L 2 ( � representations { ( ϕ ( ℓ )) ℓ ∈ g ∗ Igen , ℓ → ϕ ( ℓ ) measurable , G ) = � Index sets and representations � ϕ ( ℓ ) � 2 H − S d µ ( ℓ ) < ∞} Index sets and � G representations Index sets and Let representations An example S ( � { ϕ ∈ L 2 ( � G ) = G ) , Variable groups d µ ( u )( ϕ ) ∈ L 2 ( � G ) , u ∈ U ( g ) , Fourier Transform Un-sufficient data δ ( Q ) ϕ ∈ L 2 ( � G ) , Q ∈ C [ g ] } . Fourier inversion for sub-manifolds Fourier inversion for Theorem sub-manifolds The Fourier transform maps S ( G ) onto S ( � Fourier inversion for G ) . sub-manifolds Fourier inversion for sub-manifolds
Inverse Fourier transform The dual space of a nilpotent Lie group Index sets and representations Index sets and Theorem representations There exists a G-invariant polynomial function P gen on g ∗ Index sets and representations such that for every F ∈ S ( G ) : Index sets and representations � Index sets and tr ( π ℓ ( g − 1 ) ◦ � F ( g ) = F ( ℓ )) | P gen ( ℓ ) | d ℓ, representations g ∗ Igen An example � Variable groups tr ( π ( g − 1 ) ◦ π ( F )) d µ ( π ) , g ∈ G . = Fourier Transform � G Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Smooth compactly supported operator fields The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and c ( � C ∞ { ( ϕ ( ℓ ) ∈ K ( R r Igen )) , ℓ ∈ g ∗ G ) = I gen ; representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Smooth compactly supported operator fields The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and c ( � C ∞ { ( ϕ ( ℓ ) ∈ K ( R r Igen )) , ℓ ∈ g ∗ G ) = I gen ; representations Index sets and support ( ϕ ) compact in g ∗ I gen , representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Smooth compactly supported operator fields The dual space of a nilpotent Lie group Definition Index sets and Let representations Index sets and c ( � C ∞ { ( ϕ ( ℓ ) ∈ K ( R r Igen )) , ℓ ∈ g ∗ G ) = I gen ; representations Index sets and support ( ϕ ) compact in g ∗ I gen , representations the function ( ℓ, x , x ′ ) → ϕ ( ℓ )( x , x ′ ) Index sets and representations is smooth in ℓ Index sets and representations and Schwartz in ( x , x ′ ) ∈ R r gen × R r gen . } An example Variable groups Fourier Transform Theorem Un-sufficient data c ( � For every ϕ ∈ C ∞ G ) there exists a unique F ∈ S ( G ) , Fourier inversion for such that sub-manifolds Fourier inversion for sub-manifolds � F = ϕ. Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Un-sufficient data The dual space of a What can we do, if we have only a smooth field nilpotent Lie group ( ϕ ( ℓ ) ∈ K ( L 2 ( R r ))) ℓ ∈ M defined on a smooth submanifold Index sets and representations of � G ? Index sets and representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Un-sufficient data The dual space of a What can we do, if we have only a smooth field nilpotent Lie group ( ϕ ( ℓ ) ∈ K ( L 2 ( R r ))) ℓ ∈ M defined on a smooth submanifold Index sets and representations of � G ? Index sets and Example: M is the one point set { π ℓ } representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Un-sufficient data The dual space of a What can we do, if we have only a smooth field nilpotent Lie group ( ϕ ( ℓ ) ∈ K ( L 2 ( R r ))) ℓ ∈ M defined on a smooth submanifold Index sets and representations of � G ? Index sets and Example: M is the one point set { π ℓ } representations Index sets and Let p be a polarization at ℓ , X = { X 1 , · · · , X r } Malcev representations basis with respect to p . Index sets and representations Theorem Index sets and (R. Howe) For every ϕ ∈ S ( R r × R r ) there exists representations An example F ∈ S ( G ) such that Variable groups F ℓ, p ( E X ( x ) , E X ( x ′ )) = ϕ ( x , x ′ ) , x , x ′ ∈ R r . Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Un-sufficient data The dual space of a What can we do, if we have only a smooth field nilpotent Lie group ( ϕ ( ℓ ) ∈ K ( L 2 ( R r ))) ℓ ∈ M defined on a smooth submanifold Index sets and representations of � G ? Index sets and Example: M is the one point set { π ℓ } representations Index sets and Let p be a polarization at ℓ , X = { X 1 , · · · , X r } Malcev representations basis with respect to p . Index sets and representations Theorem Index sets and (R. Howe) For every ϕ ∈ S ( R r × R r ) there exists representations An example F ∈ S ( G ) such that Variable groups F ℓ, p ( E X ( x ) , E X ( x ′ )) = ϕ ( x , x ′ ) , x , x ′ ∈ R r . Fourier Transform Un-sufficient data Fourier inversion for This means that sub-manifolds Fourier inversion for σ ℓ, p ( S ( G )) = B ( H ℓ, p ) ∞ . sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier inversion for sub-manifolds The dual space of a nilpotent Lie group Index sets and representations Index sets and representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier inversion for sub-manifolds The dual space of a nilpotent Lie group Index sets and representations Theorem Index sets and representations (Currey-L-Molitor-Braun) Let g ∗ I be a fixed layer of g ∗ . Index sets and Let M be a smooth sub-manifold of g ∗ I . representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier inversion for sub-manifolds The dual space of a nilpotent Lie group Index sets and representations Theorem Index sets and representations (Currey-L-Molitor-Braun) Let g ∗ I be a fixed layer of g ∗ . Index sets and Let M be a smooth sub-manifold of g ∗ I . representations There exists an open subset M 0 of M such that for any Index sets and representations smooth kernel function Φ with compact support C ⊂ M 0 , Index sets and there is a function F in the Schwartz space S ( G ) such representations that π ℓ ( F ) has Φ( ℓ ) as an operator kernel for all ℓ ∈ M 0 . An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier inversion for sub-manifolds The dual space of a nilpotent Lie group Index sets and representations Theorem Index sets and representations (Currey-L-Molitor-Braun) Let g ∗ I be a fixed layer of g ∗ . Index sets and Let M be a smooth sub-manifold of g ∗ I . representations There exists an open subset M 0 of M such that for any Index sets and representations smooth kernel function Φ with compact support C ⊂ M 0 , Index sets and there is a function F in the Schwartz space S ( G ) such representations that π ℓ ( F ) has Φ( ℓ ) as an operator kernel for all ℓ ∈ M 0 . An example Moreover, the Schwartz function F may be chosen such Variable groups that π ℓ ( F ) = 0 for all ℓ ∈ M \ M 0 and for any ℓ in g ∗ Fourier Transform < I Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
Fourier inversion for sub-manifolds The dual space of a nilpotent Lie group Index sets and representations Theorem Index sets and representations (Currey-L-Molitor-Braun) Let g ∗ I be a fixed layer of g ∗ . Index sets and Let M be a smooth sub-manifold of g ∗ I . representations There exists an open subset M 0 of M such that for any Index sets and representations smooth kernel function Φ with compact support C ⊂ M 0 , Index sets and there is a function F in the Schwartz space S ( G ) such representations that π ℓ ( F ) has Φ( ℓ ) as an operator kernel for all ℓ ∈ M 0 . An example Moreover, the Schwartz function F may be chosen such Variable groups that π ℓ ( F ) = 0 for all ℓ ∈ M \ M 0 and for any ℓ in g ∗ Fourier Transform < I Un-sufficient data and such that the map Φ �→ F is continuous with respect Fourier inversion for to the corresponding function space topologies. sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
An application The dual space of a nilpotent Lie group Index sets and representations Let A ⊂ Aut ( G ) be a Lie group of auto-morphisms of G Index sets and representations acting smoothly on G . Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
An application The dual space of a nilpotent Lie group Index sets and representations Let A ⊂ Aut ( G ) be a Lie group of auto-morphisms of G Index sets and representations acting smoothly on G . Index sets and representations For instance if G is connected Lie group containing G as Index sets and nil-radical and A = Ad ( G ). representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
An application The dual space of a nilpotent Lie group Index sets and representations Let A ⊂ Aut ( G ) be a Lie group of auto-morphisms of G Index sets and representations acting smoothly on G . Index sets and representations For instance if G is connected Lie group containing G as Index sets and nil-radical and A = Ad ( G ). representations Let J ⊂ L 1 ( G ) be a closed A -prime ideal. Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
An application The dual space of a nilpotent Lie group Index sets and representations Let A ⊂ Aut ( G ) be a Lie group of auto-morphisms of G Index sets and representations acting smoothly on G . Index sets and representations For instance if G is connected Lie group containing G as Index sets and nil-radical and A = Ad ( G ). representations Let J ⊂ L 1 ( G ) be a closed A -prime ideal. Index sets and representations For instance : ( ρ, E ) an irreducible bounded An example representation ρ of G on a Banach space E and Variable groups Fourier Transform J = ker( ρ | G ) L 1 ( G ) . Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and � G is Baire space, L 1 ( G ) has the Wiener property and J is representations A -prime Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and � G is Baire space, L 1 ( G ) has the Wiener property and J is representations A -prime ⇒ the hull h ( J ) of J in � G is the closure of an Index sets and representations A -orbit in � G : Index sets and representations h ( J ) = A · π ℓ for some ℓ ∈ g ∗ . Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Let Index sets and representations Index sets and J S := J ∩ S ( G ) . representations Index sets and representations Theorem Index sets and representations The ideal J S is a closed A-prime ideal in S ( G ) . Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Let Index sets and representations Index sets and J S := J ∩ S ( G ) . representations Index sets and representations Theorem Index sets and representations The ideal J S is a closed A-prime ideal in S ( G ) . Index sets and representations An example ker ( h ( J )) S / j ( h ( J )) S is nilpotent ⇒ J S = ker ( h ( J )) S . Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and Problem: representations Is J S dense in J ? Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and Problem: representations Is J S dense in J ? Let ϕ ∈ L ∞ ( G ), such that Index sets and representations Index sets and � ϕ, J S � = { 0 } . representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and representations Index sets and Problem: representations Is J S dense in J ? Let ϕ ∈ L ∞ ( G ), such that Index sets and representations Index sets and � ϕ, J S � = { 0 } . representations Index sets and representations Is ϕ = 0 on J ? An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
The dual space of a nilpotent Lie group Index sets and If A · π ℓ is closed (or locally closed) in � G , then A · π ℓ is a representations smooth manifold Index sets and representations Index sets and representations Index sets and representations Index sets and representations An example Variable groups Fourier Transform Un-sufficient data Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds Fourier inversion for sub-manifolds
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