Spectrum of invariant differential operators for the - PowerPoint PPT Presentation

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem multiplicity-free spaces and the Capelli basis � S ( W ) ∼ Assume W is G -multiplicity-free: = G m λ V λ where m λ ≤ 1 . λ ∈ � � � S ( W ) ∼ V λ ⇒ P ( W ) ∼ V ∗ = = λ . λ ∈I W λ ∈I W � � ∼ ∼ ∼ V ∗ Hom C ( V ∗ µ , V ∗ PD ( W ) = P ( W ) ⊗ S ( W ) λ ⊗ V µ λ ) = = λ,µ ∈I W λ,µ ∈I W � if λ = µ, C Hom G ( V ∗ µ , V ∗ D λ ↔ 1 ∈ Hom G ( V ∗ λ , V ∗ λ ) := λ ) { 0 } if λ � = µ. 15 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem multiplicity-free spaces and the Capelli basis � S ( W ) ∼ Assume W is G -multiplicity-free: = G m λ V λ where m λ ≤ 1 . λ ∈ � � � S ( W ) ∼ V λ ⇒ P ( W ) ∼ V ∗ = = λ . λ ∈I W λ ∈I W � G ∼ � � G ∼ � � � ∼ V ∗ Hom G ( V ∗ µ , V ∗ PD ( W ) P ( W ) ⊗ S ( W ) λ ⊗ V µ λ ) = = = λ,µ ∈I W λ,µ ∈I W � if λ = µ, C Hom G ( V ∗ µ , V ∗ D λ ↔ 1 ∈ Hom G ( V ∗ λ , V ∗ λ ) := λ ) { 0 } if λ � = µ. 16 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem multiplicity-free spaces and the Capelli basis � S ( W ) ∼ Assume W is G -multiplicity-free: = G m λ V λ where m λ ≤ 1 . λ ∈ � � � S ( W ) ∼ V λ ⇒ P ( W ) ∼ V ∗ = = λ . λ ∈I W λ ∈I W � G ∼ � � G ∼ � PD ( W ) G ∼ � � V ∗ Hom G ( V ∗ µ , V ∗ P ( W ) ⊗ S ( W ) λ ⊗ V µ λ ) = = = λ,µ ∈I W λ,µ ∈I W � if λ = µ, C Hom G ( V ∗ µ , V ∗ D λ ↔ 1 ∈ Hom G ( V ∗ λ , V ∗ λ ) := λ ) { 0 } if λ � = µ. 17 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem multiplicity-free spaces and the Capelli basis � S ( W ) ∼ Assume W is G -multiplicity-free: = G m λ V λ where m λ ≤ 1 . λ ∈ � � � S ( W ) ∼ V λ ⇒ P ( W ) ∼ V ∗ = = λ . λ ∈I W λ ∈I W � G ∼ � � G ∼ � PD ( W ) G ∼ � � V ∗ Hom G ( V ∗ µ , V ∗ P ( W ) ⊗ S ( W ) λ ⊗ V µ λ ) = = = λ,µ ∈I W λ,µ ∈I W � if λ = µ, C Hom G ( V ∗ µ , V ∗ D λ ↔ 1 ∈ Hom G ( V ∗ λ , V ∗ λ ) := λ ) { 0 } if λ � = µ. • The basis { D λ } λ ∈I W is called the Capelli basis for PD ( W ) G . 18 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem multiplicity-free spaces and the Capelli basis � S ( W ) ∼ Assume W is G -multiplicity-free: = G m λ V λ where m λ ≤ 1 . λ ∈ � � � S ( W ) ∼ V λ ⇒ P ( W ) ∼ V ∗ = = λ . λ ∈I W λ ∈I W � G ∼ � � G ∼ � PD ( W ) G ∼ � � V ∗ Hom G ( V ∗ µ , V ∗ P ( W ) ⊗ S ( W ) λ ⊗ V µ λ ) = = = λ,µ ∈I W λ,µ ∈I W � if λ = µ, C Hom G ( V ∗ µ , V ∗ D λ ↔ 1 ∈ Hom G ( V ∗ λ , V ∗ λ ) := λ ) { 0 } if λ � = µ. • The basis { D λ } λ ∈I W is called the Capelli basis for PD ( W ) G . λ, µ ∈ I W ⇒ D λ : V ∗ µ → V ∗ µ acts by c λ ( µ ) ∈ C . Problem (Kostant) . Give an explicit description of c λ ( µ ). 19 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 20 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 21 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 22 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 23 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 24 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 25 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Example V = C n , W = S 2 ( V ) , G = GL( V ) ∼ = GL n ( C ) , K = O( V ) ∼ = O n ( C ). � P ( W ) ∼ V ∗ = where λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. λ λ λ = h.w. of V ∗ λ = ( − 2 λ n ) ε 1 + · · · + ( − 2 λ 1 ) ε n . Every V ∗ λ ⊂ P ( W ) contains a K -invariant vector 0 � = z λ ∈ V ∗ λ . w ◦ ∈ W a K -invariant vector � ι : G/K ֒ → W , g �→ g · w ◦ . � � ⊂ gl n ∼ a := diag( a 1 , . . . , a n ) | a 1 , . . . , a n ∈ C = gl ( V ). z λ � → gl ( V ) d ι a ֒ − → W − − → C J λ := z λ a ∈ P ( a ) � � . 26 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. (a) There exists a polynomial J ⋆ λ ∈ P ( a ∗ ) S n such that deg( J ⋆ λ ) = | λ | = λ 1 + · · · + λ n and c λ ( µ ) = J ⋆ λ ( µ + ρ ) where ρ = n − 1 2 ε 1 + · · · + 1 − n 2 ε n and µ = h.w. of V ∗ µ . (b) J ⋆ λ is determined up to scalar by the following conditions: λ ∈ P ( a ∗ ) S n , J ⋆ deg( J ⋆ λ ) ≤ | λ | , J ⋆ λ ( λ + ρ ) � = 0; J ⋆ λ ( µ + ρ ) = 0 for all other µ ↔ µ s.t. | µ | ≤ | λ | . (c) Up to a scalar, J ⋆ λ ∈ P ( a ∗ ) ∼ = P ( a ) can be written as J ⋆ λ = J λ + lower degree terms Other examples: Hermitian symmetric pairs – GL( V ) × GL( V ) / GL( V ) leads to factorial Schur functions (Biedenharn, Louck, Okounkov, Olshanskii). 27 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. (a) There exists a polynomial J ⋆ λ ∈ P ( a ∗ ) S n such that deg( J ⋆ λ ) = | λ | = λ 1 + · · · + λ n and c λ ( µ ) = J ⋆ λ ( µ + ρ ) where ρ = n − 1 2 ε 1 + · · · + 1 − n 2 ε n and µ = h.w. of V ∗ µ . (b) J ⋆ λ is determined up to scalar by the following conditions: λ ∈ P ( a ∗ ) S n , J ⋆ deg( J ⋆ λ ) ≤ | λ | , J ⋆ λ ( λ + ρ ) � = 0; J ⋆ λ ( µ + ρ ) = 0 for all other µ ↔ µ s.t. | µ | ≤ | λ | . (c) Up to a scalar, J ⋆ λ ∈ P ( a ∗ ) ∼ = P ( a ) can be written as J ⋆ λ = J λ + lower degree terms Other examples: Hermitian symmetric pairs – GL( V ) × GL( V ) / GL( V ) leads to factorial Schur functions (Biedenharn, Louck, Okounkov, Olshanskii). 28 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. (a) There exists a polynomial J ⋆ λ ∈ P ( a ∗ ) S n such that deg( J ⋆ λ ) = | λ | = λ 1 + · · · + λ n and c λ ( µ ) = J ⋆ λ ( µ + ρ ) where ρ = n − 1 2 ε 1 + · · · + 1 − n 2 ε n and µ = h.w. of V ∗ µ . (b) J ⋆ λ is determined up to scalar by the following conditions: λ ∈ P ( a ∗ ) S n , J ⋆ deg( J ⋆ λ ) ≤ | λ | , J ⋆ λ ( λ + ρ ) � = 0; J ⋆ λ ( µ + ρ ) = 0 for all other µ ↔ µ s.t. | µ | ≤ | λ | . (c) Up to a scalar, J ⋆ λ ∈ P ( a ∗ ) ∼ = P ( a ) can be written as J ⋆ λ = J λ + lower degree terms Other examples: Hermitian symmetric pairs – GL( V ) × GL( V ) / GL( V ) leads to factorial Schur functions (Biedenharn, Louck, Okounkov, Olshanskii). 29 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. (a) There exists a polynomial J ⋆ λ ∈ P ( a ∗ ) S n such that deg( J ⋆ λ ) = | λ | = λ 1 + · · · + λ n and c λ ( µ ) = J ⋆ λ ( µ + ρ ) where ρ = n − 1 2 ε 1 + · · · + 1 − n 2 ε n and µ = h.w. of V ∗ µ . (b) J ⋆ λ is determined up to scalar by the following conditions: λ ∈ P ( a ∗ ) S n , J ⋆ deg( J ⋆ λ ) ≤ | λ | , J ⋆ λ ( λ + ρ ) � = 0; J ⋆ λ ( µ + ρ ) = 0 for all other µ ↔ µ s.t. | µ | ≤ | λ | . (c) Up to a scalar, J ⋆ λ ∈ P ( a ∗ ) ∼ = P ( a ) can be written as J ⋆ λ = J λ + lower degree terms Other examples: Hermitian symmetric pairs – GL( V ) × GL( V ) / GL( V ) leads to factorial Schur functions (Biedenharn, Louck, Okounkov, Olshanskii). 30 / 104

The spectrum and interpolation polynomials multiplicity-free spaces and the Capelli basis Lie superalgebras The spectrum c λ ( µ ) Main Theorem The spectrum c λ ( µ ) Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = ( λ 1 , . . . , λ n ), λ 1 ≥ · · · ≥ λ n ≥ 0. (a) There exists a polynomial J ⋆ λ ∈ P ( a ∗ ) S n such that deg( J ⋆ λ ) = | λ | = λ 1 + · · · + λ n and c λ ( µ ) = J ⋆ λ ( µ + ρ ) where ρ = n − 1 2 ε 1 + · · · + 1 − n 2 ε n and µ = h.w. of V ∗ µ . (b) J ⋆ λ is determined up to scalar by the following conditions: λ ∈ P ( a ∗ ) S n , J ⋆ deg( J ⋆ λ ) ≤ | λ | , J ⋆ λ ( λ + ρ ) � = 0; J ⋆ λ ( µ + ρ ) = 0 for all other µ ↔ µ s.t. | µ | ≤ | λ | . (c) Up to a scalar, J ⋆ λ ∈ P ( a ∗ ) ∼ = P ( a ) can be written as J ⋆ λ = J λ + lower degree terms Other examples: Hermitian symmetric pairs – GL( V ) × GL( V ) / GL( V ) leads to factorial Schur functions (Biedenharn, Louck, Okounkov, Olshanskii). 31 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Deformed CMS operators The deformation L m,n,θ (Sergeev–Veselov, 2005) m ∂ 2 n ∂ 2 � � � 2 θ ( θ − 1) L m,n,θ = − + θ + ∂x 2 ∂y 2 sin 2 ( x i − x j ) i =1 i i =1 i 1 ≤ i<j ≤ m 2( θ − 1 + 1) m n � � � 2( θ − 1) − − sin 2 ( y i − y j ) sin 2 ( x i − y j ) i =1 j =1 1 ≤ i<j ≤ n θ -supersymmetric functions Let Λ m,n,θ be the subalgebra of all f ∈ C [ x 1 , . . . , x m , y 1 , . . . , y n ] Sm × Sn such that � � ∂ ∂ + θ f = 0 on the hyperplane x i − y j = 0 . ∂x i ∂y j ϕ : Λ → Λ m,n,θ , � i �→ � m � n Λ n := C [ x 1 , . . . , x n ] Sn , i x r i =1 x r i − 1 j =1 y r Λ = lim − Λ n , j ← θ Λ Λ L ϕ ϕ L m,n,θ � Λ m,n,θ Λ m,n,θ λ = ( λ 1 , λ 2 , . . . ) ⇒ sJ λ := ϕ ( J λ ) super Jack polynomials 32 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Deformed CMS operators The deformation L m,n,θ (Sergeev–Veselov, 2005) m ∂ 2 n ∂ 2 � � � 2 θ ( θ − 1) L m,n,θ = − + θ + ∂x 2 ∂y 2 sin 2 ( x i − x j ) i =1 i i =1 i 1 ≤ i<j ≤ m 2( θ − 1 + 1) m n � � � 2( θ − 1) − − sin 2 ( y i − y j ) sin 2 ( x i − y j ) i =1 j =1 1 ≤ i<j ≤ n θ -supersymmetric functions Let Λ m,n,θ be the subalgebra of all f ∈ C [ x 1 , . . . , x m , y 1 , . . . , y n ] Sm × Sn such that � � ∂ ∂ + θ f = 0 on the hyperplane x i − y j = 0 . ∂x i ∂y j ϕ : Λ → Λ m,n,θ , � i �→ � m � n Λ n := C [ x 1 , . . . , x n ] Sn , i x r i =1 x r i − 1 j =1 y r Λ = lim − Λ n , j ← θ Λ Λ L ϕ ϕ L m,n,θ � Λ m,n,θ Λ m,n,θ λ = ( λ 1 , λ 2 , . . . ) ⇒ sJ λ := ϕ ( J λ ) super Jack polynomials 33 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Deformed CMS operators The deformation L m,n,θ (Sergeev–Veselov, 2005) m ∂ 2 n ∂ 2 � � � 2 θ ( θ − 1) L m,n,θ = − + θ + ∂x 2 ∂y 2 sin 2 ( x i − x j ) i =1 i i =1 i 1 ≤ i<j ≤ m 2( θ − 1 + 1) m n � � � 2( θ − 1) − − sin 2 ( y i − y j ) sin 2 ( x i − y j ) i =1 j =1 1 ≤ i<j ≤ n θ -supersymmetric functions Let Λ m,n,θ be the subalgebra of all f ∈ C [ x 1 , . . . , x m , y 1 , . . . , y n ] Sm × Sn such that � � ∂ ∂ + θ f = 0 on the hyperplane x i − y j = 0 . ∂x i ∂y j ϕ : Λ → Λ m,n,θ , � i �→ � m � n Λ n := C [ x 1 , . . . , x n ] Sn , i x r i =1 x r i − 1 j =1 y r Λ = lim − Λ n , j ← θ Λ Λ L ϕ ϕ L m,n,θ � Λ m,n,θ Λ m,n,θ λ = ( λ 1 , λ 2 , . . . ) ⇒ sJ λ := ϕ ( J λ ) super Jack polynomials 34 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Deformed CMS operators The deformation L m,n,θ (Sergeev–Veselov, 2005) m ∂ 2 n ∂ 2 � � � 2 θ ( θ − 1) L m,n,θ = − + θ + ∂x 2 ∂y 2 sin 2 ( x i − x j ) i =1 i i =1 i 1 ≤ i<j ≤ m 2( θ − 1 + 1) m n � � � 2( θ − 1) − − sin 2 ( y i − y j ) sin 2 ( x i − y j ) i =1 j =1 1 ≤ i<j ≤ n θ -supersymmetric functions Let Λ m,n,θ be the subalgebra of all f ∈ C [ x 1 , . . . , x m , y 1 , . . . , y n ] Sm × Sn such that � � ∂ ∂ + θ f = 0 on the hyperplane x i − y j = 0 . ∂x i ∂y j ϕ : Λ → Λ m,n,θ , � i �→ � m � n Λ n := C [ x 1 , . . . , x n ] Sn , i x r i =1 x r i − 1 j =1 y r Λ = lim − Λ n , j ← θ Λ Λ L ϕ ϕ L m,n,θ � Λ m,n,θ Λ m,n,θ λ = ( λ 1 , λ 2 , . . . ) ⇒ sJ λ := ϕ ( J λ ) super Jack polynomials 35 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Deformed CMS operators The deformation L m,n,θ (Sergeev–Veselov, 2005) m ∂ 2 n ∂ 2 � � � 2 θ ( θ − 1) L m,n,θ = − + θ + ∂x 2 ∂y 2 sin 2 ( x i − x j ) i =1 i i =1 i 1 ≤ i<j ≤ m 2( θ − 1 + 1) m n � � � 2( θ − 1) − − sin 2 ( y i − y j ) sin 2 ( x i − y j ) i =1 j =1 1 ≤ i<j ≤ n θ -supersymmetric functions Let Λ m,n,θ be the subalgebra of all f ∈ C [ x 1 , . . . , x m , y 1 , . . . , y n ] Sm × Sn such that � � ∂ ∂ + θ f = 0 on the hyperplane x i − y j = 0 . ∂x i ∂y j ϕ : Λ → Λ m,n,θ , � i �→ � m � n Λ n := C [ x 1 , . . . , x n ] Sn , i x r i =1 x r i − 1 j =1 y r Λ = lim − Λ n , j ← θ Λ Λ L ϕ ϕ L m,n,θ � Λ m,n,θ Λ m,n,θ λ = ( λ 1 , λ 2 , . . . ) ⇒ sJ λ := ϕ ( J λ ) super Jack polynomials 36 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Deformed CMS operators The deformation L m,n,θ (Sergeev–Veselov, 2005) m ∂ 2 n ∂ 2 � � � 2 θ ( θ − 1) L m,n,θ = − + θ + ∂x 2 ∂y 2 sin 2 ( x i − x j ) i =1 i i =1 i 1 ≤ i<j ≤ m 2( θ − 1 + 1) m n � � � 2( θ − 1) − − sin 2 ( y i − y j ) sin 2 ( x i − y j ) i =1 j =1 1 ≤ i<j ≤ n θ -supersymmetric functions Let Λ m,n,θ be the subalgebra of all f ∈ C [ x 1 , . . . , x m , y 1 , . . . , y n ] Sm × Sn such that � � ∂ ∂ + θ f = 0 on the hyperplane x i − y j = 0 . ∂x i ∂y j ϕ : Λ → Λ m,n,θ , � i �→ � m � n Λ n := C [ x 1 , . . . , x n ] Sn , i x r i =1 x r i − 1 j =1 y r Λ = lim − Λ n , j ← θ Λ Λ L ϕ ϕ L m,n,θ � Λ m,n,θ Λ m,n,θ λ = ( λ 1 , λ 2 , . . . ) ⇒ sJ λ := ϕ ( J λ ) super Jack polynomials 37 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials Quantum integrals Λ n,θ := C [ x 1 , . . . , x i + θ (1 − i ) , . . . , x n + θ (1 − n )] S n Λ θ := lim − Λ n,θ ← � L f L m,n,θ � L f L , m,n,θ for every f ∈ Λ θ θ L f θ L f m,n,θ sJ λ = f ( λ ) sJ λ . Λ Λ ϕ ϕ � Λ m,n,θ Λ m,n,θ L f m,n,θ λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ ϕ ( J λ ) = 0. 38 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials Quantum integrals Λ n,θ := C [ x 1 , . . . , x i + θ (1 − i ) , . . . , x n + θ (1 − n )] S n Λ θ := lim − Λ n,θ ← � L f L m,n,θ � L f L , m,n,θ for every f ∈ Λ θ θ L f θ L f m,n,θ sJ λ = f ( λ ) sJ λ . Λ Λ ϕ ϕ � Λ m,n,θ Λ m,n,θ L f m,n,θ λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ ϕ ( J λ ) = 0. 39 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials Quantum integrals Λ n,θ := C [ x 1 , . . . , x i + θ (1 − i ) , . . . , x n + θ (1 − n )] S n Λ θ := lim − Λ n,θ ← � L f L m,n,θ � L f L , m,n,θ for every f ∈ Λ θ θ L f θ L f m,n,θ sJ λ = f ( λ ) sJ λ . Λ Λ ϕ ϕ � Λ m,n,θ Λ m,n,θ L f m,n,θ λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ ϕ ( J λ ) = 0. 40 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials Quantum integrals Λ n,θ := C [ x 1 , . . . , x i + θ (1 − i ) , . . . , x n + θ (1 − n )] S n Λ θ := lim − Λ n,θ ← � L f L m,n,θ � L f L , m,n,θ for every f ∈ Λ θ θ L f θ L f m,n,θ sJ λ = f ( λ ) sJ λ . Λ Λ ϕ ϕ � Λ m,n,θ Λ m,n,θ L f m,n,θ λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ ϕ ( J λ ) = 0. 41 / 104

� � � The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials Quantum integrals Λ n,θ := C [ x 1 , . . . , x i + θ (1 − i ) , . . . , x n + θ (1 − n )] S n Λ θ := lim − Λ n,θ ← � L f L m,n,θ � L f L , m,n,θ for every f ∈ Λ θ θ L f θ L f m,n,θ sJ λ = f ( λ ) sJ λ . Λ Λ ϕ ϕ � Λ m,n,θ Λ m,n,θ L f m,n,θ λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ ϕ ( J λ ) = 0. 42 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials The algebra Λ ♮ m,n,θ and the polynomials sJ ⋆ λ Let Λ ♮ m,n,θ ⊂ C [ x 1 , . . . , x m , y 1 , . . . , y n ] S m × S n be defined as follows: � � � � 1 1 1 1 f ∈ Λ ♮ iff f x i + , y j − = f x i − , y j + on the hyperplane x i + θy j = 0 . m,n,θ 2 2 2 2 Set ϕ ♮ : Λ θ → Λ ♮ ϕ ♮ ( f )( p , q ) := f ( F − 1 ( p , q )) m,n,θ , where the map F : { ( λ 1 , . . . , λ m , µ 1 , . . . , µ n ) } → C m + n is given by “Frobenius coordinates”:  p i = λ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = µ ′ j − θ − 1 ( j − 1 2 ) + 1 2 ( θ − 1 n + m )  1 ≤ j ≤ n. sJ ⋆ λ := ϕ ♮ ( J ⋆ λ ) shifted super Jack polynomials λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ sJ ⋆ λ = 0 . 43 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials The algebra Λ ♮ m,n,θ and the polynomials sJ ⋆ λ Let Λ ♮ m,n,θ ⊂ C [ x 1 , . . . , x m , y 1 , . . . , y n ] S m × S n be defined as follows: � � � � 1 1 1 1 f ∈ Λ ♮ iff f x i + , y j − = f x i − , y j + on the hyperplane x i + θy j = 0 . m,n,θ 2 2 2 2 Set ϕ ♮ : Λ θ → Λ ♮ ϕ ♮ ( f )( p , q ) := f ( F − 1 ( p , q )) m,n,θ , where the map F : { ( λ 1 , . . . , λ m , µ 1 , . . . , µ n ) } → C m + n is given by “Frobenius coordinates”:  p i = λ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = µ ′ j − θ − 1 ( j − 1 2 ) + 1 2 ( θ − 1 n + m )  1 ≤ j ≤ n. sJ ⋆ λ := ϕ ♮ ( J ⋆ λ ) shifted super Jack polynomials λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ sJ ⋆ λ = 0 . 44 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials The algebra Λ ♮ m,n,θ and the polynomials sJ ⋆ λ Let Λ ♮ m,n,θ ⊂ C [ x 1 , . . . , x m , y 1 , . . . , y n ] S m × S n be defined as follows: � � � � 1 1 1 1 f ∈ Λ ♮ iff f x i + , y j − = f x i − , y j + on the hyperplane x i + θy j = 0 . m,n,θ 2 2 2 2 Set ϕ ♮ : Λ θ → Λ ♮ ϕ ♮ ( f )( p , q ) := f ( F − 1 ( p , q )) m,n,θ , where the map F : { ( λ 1 , . . . , λ m , µ 1 , . . . , µ n ) } → C m + n is given by “Frobenius coordinates”:  p i = λ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = µ ′ j − θ − 1 ( j − 1 2 ) + 1 2 ( θ − 1 n + m )  1 ≤ j ≤ n. sJ ⋆ λ := ϕ ♮ ( J ⋆ λ ) shifted super Jack polynomials λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ sJ ⋆ λ = 0 . 45 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials The algebra Λ ♮ m,n,θ and the polynomials sJ ⋆ λ Let Λ ♮ m,n,θ ⊂ C [ x 1 , . . . , x m , y 1 , . . . , y n ] S m × S n be defined as follows: � � � � 1 1 1 1 f ∈ Λ ♮ iff f x i + , y j − = f x i − , y j + on the hyperplane x i + θy j = 0 . m,n,θ 2 2 2 2 Set ϕ ♮ : Λ θ → Λ ♮ ϕ ♮ ( f )( p , q ) := f ( F − 1 ( p , q )) m,n,θ , where the map F : { ( λ 1 , . . . , λ m , µ 1 , . . . , µ n ) } → C m + n is given by “Frobenius coordinates”:  p i = λ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = µ ′ j − θ − 1 ( j − 1 2 ) + 1 2 ( θ − 1 n + m )  1 ≤ j ≤ n. sJ ⋆ λ := ϕ ♮ ( J ⋆ λ ) shifted super Jack polynomials λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ sJ ⋆ λ = 0 . 46 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Shifted super Jack polynomials The algebra Λ ♮ m,n,θ and the polynomials sJ ⋆ λ Let Λ ♮ m,n,θ ⊂ C [ x 1 , . . . , x m , y 1 , . . . , y n ] S m × S n be defined as follows: � � � � 1 1 1 1 f ∈ Λ ♮ iff f x i + , y j − = f x i − , y j + on the hyperplane x i + θy j = 0 . m,n,θ 2 2 2 2 Set ϕ ♮ : Λ θ → Λ ♮ ϕ ♮ ( f )( p , q ) := f ( F − 1 ( p , q )) m,n,θ , where the map F : { ( λ 1 , . . . , λ m , µ 1 , . . . , µ n ) } → C m + n is given by “Frobenius coordinates”:  p i = λ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = µ ′ j − θ − 1 ( j − 1 2 ) + 1 2 ( θ − 1 n + m )  1 ≤ j ≤ n. sJ ⋆ λ := ϕ ♮ ( J ⋆ λ ) shifted super Jack polynomials λ = ( λ 1 , λ 2 , . . . ) such that λ m +1 > n ⇒ sJ ⋆ λ = 0 . 47 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Generalities on Lie superalgebras SVec : symmetric monoidal category SVec of Z / 2-graded vector spaces. Z / 2 = { 0 , 1 } V ∈ obj SVec � V = V 0 ⊕ V 1 . � � Mor SVec ( V, W ) = T ∈ Hom C ( V, W ) : TV 0 ⊂ W 0 and TV 1 ⊂ W 1 V ⊗ W → W ⊗ V , v ⊗ w �→ ( − 1) | v |·| w | w ⊗ v. S ( V ) ∼ = S ( V 0 ) ⊗ Λ( V 1 ) , P ( V ) = S ( V ∗ ) Lie superalgebra: g = g 0 ⊕ g 1 such that ( − 1) | x |·| z | [ x, [ y, z ]] + ( − 1) | y |·| x | [ y, [ z, x ]] + ( − 1) | z |·| y | [ z, [ x, y ]] = 0 . V = V 0 ⊕ V 1 � End C ( V ) = End( V ) 0 ⊕ End( V ) 1 is a Lie superalgebra: [ S, T ] = ST − ( − 1) | S |·| T | TS. 48 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Generalities on Lie superalgebras SVec : symmetric monoidal category SVec of Z / 2-graded vector spaces. Z / 2 = { 0 , 1 } V ∈ obj SVec � V = V 0 ⊕ V 1 . � � Mor SVec ( V, W ) = T ∈ Hom C ( V, W ) : TV 0 ⊂ W 0 and TV 1 ⊂ W 1 V ⊗ W → W ⊗ V , v ⊗ w �→ ( − 1) | v |·| w | w ⊗ v. S ( V ) ∼ = S ( V 0 ) ⊗ Λ( V 1 ) , P ( V ) = S ( V ∗ ) Lie superalgebra: g = g 0 ⊕ g 1 such that ( − 1) | x |·| z | [ x, [ y, z ]] + ( − 1) | y |·| x | [ y, [ z, x ]] + ( − 1) | z |·| y | [ z, [ x, y ]] = 0 . V = V 0 ⊕ V 1 � End C ( V ) = End( V ) 0 ⊕ End( V ) 1 is a Lie superalgebra: [ S, T ] = ST − ( − 1) | S |·| T | TS. 49 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Generalities on Lie superalgebras SVec : symmetric monoidal category SVec of Z / 2-graded vector spaces. Z / 2 = { 0 , 1 } V ∈ obj SVec � V = V 0 ⊕ V 1 . � � Mor SVec ( V, W ) = T ∈ Hom C ( V, W ) : TV 0 ⊂ W 0 and TV 1 ⊂ W 1 V ⊗ W → W ⊗ V , v ⊗ w �→ ( − 1) | v |·| w | w ⊗ v. S ( V ) ∼ = S ( V 0 ) ⊗ Λ( V 1 ) , P ( V ) = S ( V ∗ ) Lie superalgebra: g = g 0 ⊕ g 1 such that ( − 1) | x |·| z | [ x, [ y, z ]] + ( − 1) | y |·| x | [ y, [ z, x ]] + ( − 1) | z |·| y | [ z, [ x, y ]] = 0 . V = V 0 ⊕ V 1 � End C ( V ) = End( V ) 0 ⊕ End( V ) 1 is a Lie superalgebra: [ S, T ] = ST − ( − 1) | S |·| T | TS. 50 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Generalities on Lie superalgebras SVec : symmetric monoidal category SVec of Z / 2-graded vector spaces. Z / 2 = { 0 , 1 } V ∈ obj SVec � V = V 0 ⊕ V 1 . � � Mor SVec ( V, W ) = T ∈ Hom C ( V, W ) : TV 0 ⊂ W 0 and TV 1 ⊂ W 1 V ⊗ W → W ⊗ V , v ⊗ w �→ ( − 1) | v |·| w | w ⊗ v. S ( V ) ∼ = S ( V 0 ) ⊗ Λ( V 1 ) , P ( V ) = S ( V ∗ ) Lie superalgebra: g = g 0 ⊕ g 1 such that ( − 1) | x |·| z | [ x, [ y, z ]] + ( − 1) | y |·| x | [ y, [ z, x ]] + ( − 1) | z |·| y | [ z, [ x, y ]] = 0 . V = V 0 ⊕ V 1 � End C ( V ) = End( V ) 0 ⊕ End( V ) 1 is a Lie superalgebra: [ S, T ] = ST − ( − 1) | S |·| T | TS. 51 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Generalities on Lie superalgebras SVec : symmetric monoidal category SVec of Z / 2-graded vector spaces. Z / 2 = { 0 , 1 } V ∈ obj SVec � V = V 0 ⊕ V 1 . � � Mor SVec ( V, W ) = T ∈ Hom C ( V, W ) : TV 0 ⊂ W 0 and TV 1 ⊂ W 1 V ⊗ W → W ⊗ V , v ⊗ w �→ ( − 1) | v |·| w | w ⊗ v. S ( V ) ∼ = S ( V 0 ) ⊗ Λ( V 1 ) , P ( V ) = S ( V ∗ ) Lie superalgebra: g = g 0 ⊕ g 1 such that ( − 1) | x |·| z | [ x, [ y, z ]] + ( − 1) | y |·| x | [ y, [ z, x ]] + ( − 1) | z |·| y | [ z, [ x, y ]] = 0 . V = V 0 ⊕ V 1 � End C ( V ) = End( V ) 0 ⊕ End( V ) 1 is a Lie superalgebra: [ S, T ] = ST − ( − 1) | S |·| T | TS. 52 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n Generalities on Lie superalgebras SVec : symmetric monoidal category SVec of Z / 2-graded vector spaces. Z / 2 = { 0 , 1 } V ∈ obj SVec � V = V 0 ⊕ V 1 . � � Mor SVec ( V, W ) = T ∈ Hom C ( V, W ) : TV 0 ⊂ W 0 and TV 1 ⊂ W 1 V ⊗ W → W ⊗ V , v ⊗ w �→ ( − 1) | v |·| w | w ⊗ v. S ( V ) ∼ = S ( V 0 ) ⊗ Λ( V 1 ) , P ( V ) = S ( V ∗ ) Lie superalgebra: g = g 0 ⊕ g 1 such that ( − 1) | x |·| z | [ x, [ y, z ]] + ( − 1) | y |·| x | [ y, [ z, x ]] + ( − 1) | z |·| y | [ z, [ x, y ]] = 0 . V = V 0 ⊕ V 1 � End C ( V ) = End( V ) 0 ⊕ End( V ) 1 is a Lie superalgebra: [ S, T ] = ST − ( − 1) | S |·| T | TS. 53 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Root system V = C m | n � gl m | n = End( C m | n ). g = gl m | n ⇒ g = n − ⊕ h ⊕ n + , where n ± = Φ ± = Φ ± � 0 ∪ Φ ± for 1 . g α α ∈ Φ ± Φ + 0 = { ε i − ε j : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n } , Φ + 1 = { ε i − ε j : 1 ≤ i ≤ m < j ≤ m + n } . Φ − 0 = − Φ + 0 , Φ − 1 = − Φ + 1 . Invariant form � A � B X = ⇒ str( X ) = tr( A ) − tr( D ) . C D X, Y ∈ gl m | n ⇒ κ ( X, Y ) = str( XY ) is a nondegenerate invariant form. 54 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Root system V = C m | n � gl m | n = End( C m | n ). g = gl m | n ⇒ g = n − ⊕ h ⊕ n + , where n ± = Φ ± = Φ ± � 0 ∪ Φ ± for 1 . g α α ∈ Φ ± Φ + 0 = { ε i − ε j : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n } , Φ + 1 = { ε i − ε j : 1 ≤ i ≤ m < j ≤ m + n } . Φ − 0 = − Φ + 0 , Φ − 1 = − Φ + 1 . Invariant form � A � B X = ⇒ str( X ) = tr( A ) − tr( D ) . C D X, Y ∈ gl m | n ⇒ κ ( X, Y ) = str( XY ) is a nondegenerate invariant form. 55 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Root system V = C m | n � gl m | n = End( C m | n ). g = gl m | n ⇒ g = n − ⊕ h ⊕ n + , where n ± = Φ ± = Φ ± � 0 ∪ Φ ± for 1 . g α α ∈ Φ ± Φ + 0 = { ε i − ε j : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n } , Φ + 1 = { ε i − ε j : 1 ≤ i ≤ m < j ≤ m + n } . Φ − 0 = − Φ + 0 , Φ − 1 = − Φ + 1 . Invariant form � A � B X = ⇒ str( X ) = tr( A ) − tr( D ) . C D X, Y ∈ gl m | n ⇒ κ ( X, Y ) = str( XY ) is a nondegenerate invariant form. 56 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Root system V = C m | n � gl m | n = End( C m | n ). g = gl m | n ⇒ g = n − ⊕ h ⊕ n + , where n ± = Φ ± = Φ ± � 0 ∪ Φ ± for 1 . g α α ∈ Φ ± Φ + 0 = { ε i − ε j : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n } , Φ + 1 = { ε i − ε j : 1 ≤ i ≤ m < j ≤ m + n } . Φ − 0 = − Φ + 0 , Φ − 1 = − Φ + 1 . Invariant form � A � B X = ⇒ str( X ) = tr( A ) − tr( D ) . C D X, Y ∈ gl m | n ⇒ κ ( X, Y ) = str( XY ) is a nondegenerate invariant form. 57 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Root system V = C m | n � gl m | n = End( C m | n ). g = gl m | n ⇒ g = n − ⊕ h ⊕ n + , where n ± = Φ ± = Φ ± � 0 ∪ Φ ± for 1 . g α α ∈ Φ ± Φ + 0 = { ε i − ε j : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n } , Φ + 1 = { ε i − ε j : 1 ≤ i ≤ m < j ≤ m + n } . Φ − 0 = − Φ + 0 , Φ − 1 = − Φ + 1 . Invariant form � A � B X = ⇒ str( X ) = tr( A ) − tr( D ) . C D X, Y ∈ gl m | n ⇒ κ ( X, Y ) = str( XY ) is a nondegenerate invariant form. 58 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Highest weight modules of gl m | n Every irreducible finite dimensional representation of gl m | n is a highest weight module V λ where λ = λ 1 ε 1 + · · · + λ m + n ε m + n satisfies λ i ∈ Z and λ 1 ≥ · · · ≥ λ m and λ m +1 ≥ · · · ≥ λ m + n . Signed S d -action on V ⊗ d σ · ( v 1 ⊗ · · · ⊗ v d ) = ( − 1) ǫ ( σ − 1 ; v 1 ,...,v d ) v σ − 1 (1) ⊗ · · · ⊗ v σ − 1 ( d ) � where ǫ ( σ ; v 1 , . . . , v d ) = | v σ ( r ) | · | v σ ( s ) | . 1 ≤ r<s ≤ d σ ( r ) >σ ( s ) 59 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Highest weight modules of gl m | n Every irreducible finite dimensional representation of gl m | n is a highest weight module V λ where λ = λ 1 ε 1 + · · · + λ m + n ε m + n satisfies λ i ∈ Z and λ 1 ≥ · · · ≥ λ m and λ m +1 ≥ · · · ≥ λ m + n . Signed S d -action on V ⊗ d σ · ( v 1 ⊗ · · · ⊗ v d ) = ( − 1) ǫ ( σ − 1 ; v 1 ,...,v d ) v σ − 1 (1) ⊗ · · · ⊗ v σ − 1 ( d ) � where ǫ ( σ ; v 1 , . . . , v d ) = | v σ ( r ) | · | v σ ( s ) | . 1 ≤ r<s ≤ d σ ( r ) >σ ( s ) 60 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Schur–Weyl duality ( m, n ) -hook diagram : a Young diagram D = ( ♭ 1 , ♭ 2 , . . . ) that satisfies ♭ m +1 ≤ n . H( m, n, d ) = { ( m, n )-hook diagrams of size d } . Recall: V = C m | n . ( Sergeev ’84, Berele–Regev ’87 ) As gl m | n × S d -module, � V ⊗ d ∼ = V D ⊗ U D D ∈ H( m,n,d ) h.w. of V D = ♭ 1 ε 1 + · · · + ♭ m ε m + � ♭ ′ 1 − m � ε m +1 + · · · + � ♭ ′ n − m � ε m + n where � ♭ ′ i − m � := max { ♭ ′ i − m, 0 } . ∈ H(2 , 3 , 16) � 7 ε 1 + 5 ε 2 + 2 ε 3 + ε 4 + ε 5 61 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Schur–Weyl duality ( m, n ) -hook diagram : a Young diagram D = ( ♭ 1 , ♭ 2 , . . . ) that satisfies ♭ m +1 ≤ n . H( m, n, d ) = { ( m, n )-hook diagrams of size d } . Recall: V = C m | n . ( Sergeev ’84, Berele–Regev ’87 ) As gl m | n × S d -module, � V ⊗ d ∼ = V D ⊗ U D D ∈ H( m,n,d ) h.w. of V D = ♭ 1 ε 1 + · · · + ♭ m ε m + � ♭ ′ 1 − m � ε m +1 + · · · + � ♭ ′ n − m � ε m + n where � ♭ ′ i − m � := max { ♭ ′ i − m, 0 } . ∈ H(2 , 3 , 16) � 7 ε 1 + 5 ε 2 + 2 ε 3 + ε 4 + ε 5 62 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Schur–Weyl duality ( m, n ) -hook diagram : a Young diagram D = ( ♭ 1 , ♭ 2 , . . . ) that satisfies ♭ m +1 ≤ n . H( m, n, d ) = { ( m, n )-hook diagrams of size d } . Recall: V = C m | n . ( Sergeev ’84, Berele–Regev ’87 ) As gl m | n × S d -module, � V ⊗ d ∼ = V D ⊗ U D D ∈ H( m,n,d ) h.w. of V D = ♭ 1 ε 1 + · · · + ♭ m ε m + � ♭ ′ 1 − m � ε m +1 + · · · + � ♭ ′ n − m � ε m + n where � ♭ ′ i − m � := max { ♭ ′ i − m, 0 } . ∈ H(2 , 3 , 16) � 7 ε 1 + 5 ε 2 + 2 ε 3 + ε 4 + ε 5 63 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Schur–Weyl duality ( m, n ) -hook diagram : a Young diagram D = ( ♭ 1 , ♭ 2 , . . . ) that satisfies ♭ m +1 ≤ n . H( m, n, d ) = { ( m, n )-hook diagrams of size d } . Recall: V = C m | n . ( Sergeev ’84, Berele–Regev ’87 ) As gl m | n × S d -module, � V ⊗ d ∼ = V D ⊗ U D D ∈ H( m,n,d ) h.w. of V D = ♭ 1 ε 1 + · · · + ♭ m ε m + � ♭ ′ 1 − m � ε m +1 + · · · + � ♭ ′ n − m � ε m + n where � ♭ ′ i − m � := max { ♭ ′ i − m, 0 } . ∈ H(2 , 3 , 16) � 7 ε 1 + 5 ε 2 + 2 ε 3 + ε 4 + ε 5 64 / 104

The spectrum and interpolation polynomials Deformed CMS operators Lie superalgebras Generalities on Lie superalgebras Main Theorem The Lie superalgebra gl m | n The Lie superalgebra gl m | n Schur–Weyl duality ( m, n ) -hook diagram : a Young diagram D = ( ♭ 1 , ♭ 2 , . . . ) that satisfies ♭ m +1 ≤ n . H( m, n, d ) = { ( m, n )-hook diagrams of size d } . Recall: V = C m | n . ( Sergeev ’84, Berele–Regev ’87 ) As gl m | n × S d -module, � V ⊗ d ∼ = V D ⊗ U D D ∈ H( m,n,d ) h.w. of V D = ♭ 1 ε 1 + · · · + ♭ m ε m + � ♭ ′ 1 − m � ε m +1 + · · · + � ♭ ′ n − m � ε m + n where � ♭ ′ i − m � := max { ♭ ′ i − m, 0 } . ∈ H(2 , 3 , 16) � 7 ε 1 + 5 ε 2 + 2 ε 3 + ε 4 + ε 5 65 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 66 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 67 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 68 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 69 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 70 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 71 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 72 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem � � The supersymmetric pair gl m | 2 n , osp m | 2 n The Lie superalgebra osp m | 2 n � 0 � 1 J 2 := , J 2 n = diag( J 2 , . . . , J 2 ) − 1 0 � �� � n times � A � � � − A t − C t J 2 n B Θ : gl m | 2 n → gl m | 2 n , �→ − J 2 n B t J 2 n D t J 2 n C D � � osp m | 2 n = X ∈ gl m | 2 n : Θ( X ) = X g := gl m | 2 n , k := osp m | 2 n ⇒ g = k ⊕ a ⊕ n . Proposition (Sahi–S.) For every D ∈ H( m, 2 n, d ), the irreducible g -module V ∗ 2 λ ⊂ P ( W ) contains a � a = sJ D for θ = 1 unique (up to scalar) k -fixed vector 0 � = z D . (In fact z D 2 .) � (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”). • Does not imply the above proposition. 73 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The eigenvalue problem Decomposing S ( S 2 ( W )) where W = S 2 ( V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a gl m | n -module, � S d ( W ) = V 2 D . D ∈ H( m,n,d ) The Capelli basis � V 2 D ⊂ S ( W ) h.w. = λ D ∈ H( m, n, d ) � V ∗ h.w. = λ ∗ 2 D ⊂ P ( W ) λ � D λ ∈ PD ( W ) gl m | 2 n Capelli basis. V µ ∗ ⊂ P ( W ) ⇒ D λ : V µ ∗ → V µ ∗ acts by a scalar c λ ( µ ∗ ). c λ ( · ) ∈ P ( a ∗ ) ∼ = P ( a ). Theorem (Sahi–S.) � c λ = z D a + lower degree terms. � 74 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The eigenvalue problem Decomposing S ( S 2 ( W )) where W = S 2 ( V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a gl m | n -module, � S d ( W ) = V 2 D . D ∈ H( m,n,d ) The Capelli basis � V 2 D ⊂ S ( W ) h.w. = λ D ∈ H( m, n, d ) � V ∗ h.w. = λ ∗ 2 D ⊂ P ( W ) λ � D λ ∈ PD ( W ) gl m | 2 n Capelli basis. V µ ∗ ⊂ P ( W ) ⇒ D λ : V µ ∗ → V µ ∗ acts by a scalar c λ ( µ ∗ ). c λ ( · ) ∈ P ( a ∗ ) ∼ = P ( a ). Theorem (Sahi–S.) � c λ = z D a + lower degree terms. � 75 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The eigenvalue problem Decomposing S ( S 2 ( W )) where W = S 2 ( V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a gl m | n -module, � S d ( W ) = V 2 D . D ∈ H( m,n,d ) The Capelli basis � V 2 D ⊂ S ( W ) h.w. = λ D ∈ H( m, n, d ) � V ∗ h.w. = λ ∗ 2 D ⊂ P ( W ) λ � D λ ∈ PD ( W ) gl m | 2 n Capelli basis. V µ ∗ ⊂ P ( W ) ⇒ D λ : V µ ∗ → V µ ∗ acts by a scalar c λ ( µ ∗ ). c λ ( · ) ∈ P ( a ∗ ) ∼ = P ( a ). Theorem (Sahi–S.) � c λ = z D a + lower degree terms. � 76 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The eigenvalue problem Decomposing S ( S 2 ( W )) where W = S 2 ( V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a gl m | n -module, � S d ( W ) = V 2 D . D ∈ H( m,n,d ) The Capelli basis � V 2 D ⊂ S ( W ) h.w. = λ D ∈ H( m, n, d ) � V ∗ h.w. = λ ∗ 2 D ⊂ P ( W ) λ � D λ ∈ PD ( W ) gl m | 2 n Capelli basis. V µ ∗ ⊂ P ( W ) ⇒ D λ : V µ ∗ → V µ ∗ acts by a scalar c λ ( µ ∗ ). c λ ( · ) ∈ P ( a ∗ ) ∼ = P ( a ). Theorem (Sahi–S.) � c λ = z D a + lower degree terms. � 77 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The eigenvalue problem Decomposing S ( S 2 ( W )) where W = S 2 ( V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a gl m | n -module, � S d ( W ) = V 2 D . D ∈ H( m,n,d ) The Capelli basis � V 2 D ⊂ S ( W ) h.w. = λ D ∈ H( m, n, d ) � V ∗ h.w. = λ ∗ 2 D ⊂ P ( W ) λ � D λ ∈ PD ( W ) gl m | 2 n Capelli basis. V µ ∗ ⊂ P ( W ) ⇒ D λ : V µ ∗ → V µ ∗ acts by a scalar c λ ( µ ∗ ). c λ ( · ) ∈ P ( a ∗ ) ∼ = P ( a ). Theorem (Sahi–S.) � c λ = z D a + lower degree terms. � 78 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The eigenvalue problem Decomposing S ( S 2 ( W )) where W = S 2 ( V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a gl m | n -module, � S d ( W ) = V 2 D . D ∈ H( m,n,d ) The Capelli basis � V 2 D ⊂ S ( W ) h.w. = λ D ∈ H( m, n, d ) � V ∗ h.w. = λ ∗ 2 D ⊂ P ( W ) λ � D λ ∈ PD ( W ) gl m | 2 n Capelli basis. V µ ∗ ⊂ P ( W ) ⇒ D λ : V µ ∗ → V µ ∗ acts by a scalar c λ ( µ ∗ ). c λ ( · ) ∈ P ( a ∗ ) ∼ = P ( a ). Theorem (Sahi–S.) � c λ = z D a + lower degree terms. � 79 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem Relation with shifted super Jack polynomials µ ∗ � D = D µ ∗ = ( ♭ 1 , ♭ 2 , . . . ) Theorem (Sahi–S.) c λ ( µ ∗ ) is a polynomial in ( ♭ 1 , . . . , ♭ m , ♭ ′ 1 , . . . , ♭ ′ n ). Up to the Frobenius coordinates, c λ = sJ ⋆ λ for θ = 1 2 .  p i = ♭ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = ♭ ′ j − θ − 1 ( j − 1 2 ( θ − 1 n + m ) 2 ) + 1 1 ≤ j ≤ n.  80 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem Relation with shifted super Jack polynomials µ ∗ � D = D µ ∗ = ( ♭ 1 , ♭ 2 , . . . ) Theorem (Sahi–S.) c λ ( µ ∗ ) is a polynomial in ( ♭ 1 , . . . , ♭ m , ♭ ′ 1 , . . . , ♭ ′ n ). Up to the Frobenius coordinates, c λ = sJ ⋆ λ for θ = 1 2 .  p i = ♭ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = ♭ ′ j − θ − 1 ( j − 1 2 ( θ − 1 n + m ) 2 ) + 1 1 ≤ j ≤ n.  81 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem Relation with shifted super Jack polynomials µ ∗ � D = D µ ∗ = ( ♭ 1 , ♭ 2 , . . . ) Theorem (Sahi–S.) c λ ( µ ∗ ) is a polynomial in ( ♭ 1 , . . . , ♭ m , ♭ ′ 1 , . . . , ♭ ′ n ). Up to the Frobenius coordinates, c λ = sJ ⋆ λ for θ = 1 2 .  p i = ♭ i − θ ( i − 1 2 ) − 1 2 ( n − θm ) 1 ≤ i ≤ m,  q j = ♭ ′ j − θ − 1 ( j − 1 2 ( θ − 1 n + m ) 2 ) + 1 1 ≤ j ≤ n.  82 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Action of gl m | n on P ( W ): � ( − 1) | i | + | i |·| j | y r,i ∂ r,j ∈ PD ( W ) . E i,j ∈ gl m | n � r ρ : gl m | n → PD ( W ) ρ : U ( gl m | n ) → PD ( W ) . � ⊂ PD ( W ) gl m | n . � � ρ Z ( gl m | n ) � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) 83 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Action of gl m | n on P ( W ): � ( − 1) | i | + | i |·| j | y r,i ∂ r,j ∈ PD ( W ) . E i,j ∈ gl m | n � r ρ : gl m | n → PD ( W ) ρ : U ( gl m | n ) → PD ( W ) . � ⊂ PD ( W ) gl m | n . � � ρ Z ( gl m | n ) � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) 84 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Action of gl m | n on P ( W ): � ( − 1) | i | + | i |·| j | y r,i ∂ r,j ∈ PD ( W ) . E i,j ∈ gl m | n � r ρ : gl m | n → PD ( W ) ρ : U ( gl m | n ) → PD ( W ) . � ⊂ PD ( W ) gl m | n . � � ρ Z ( gl m | n ) � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) 85 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Action of gl m | n on P ( W ): � ( − 1) | i | + | i |·| j | y r,i ∂ r,j ∈ PD ( W ) . E i,j ∈ gl m | n � r ρ : gl m | n → PD ( W ) ρ : U ( gl m | n ) → PD ( W ) . � ⊂ PD ( W ) gl m | n . � � ρ Z ( gl m | n ) � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) 86 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) m = 0 or n = 0 : special case of Howe-Umeda ’91. See also Goodman–Wallach ’09, Turnbull ’47, G ˚ arding ’47. Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0, where Z d ( gl m | n ) ρ � � Z d ( gl m | n ) = Z ( gl m | n ) ∩ U d ( gl m | n ) . Difference with Howe–Umeda: Use of the Capelli basis vs. generators of the algebra PD ( W ) gl m | n . 87 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) m = 0 or n = 0 : special case of Howe-Umeda ’91. See also Goodman–Wallach ’09, Turnbull ’47, G ˚ arding ’47. Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0, where Z d ( gl m | n ) ρ � � Z d ( gl m | n ) = Z ( gl m | n ) ∩ U d ( gl m | n ) . Difference with Howe–Umeda: Use of the Capelli basis vs. generators of the algebra PD ( W ) gl m | n . 88 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) m = 0 or n = 0 : special case of Howe-Umeda ’91. See also Goodman–Wallach ’09, Turnbull ’47, G ˚ arding ’47. Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0, where Z d ( gl m | n ) ρ � � Z d ( gl m | n ) = Z ( gl m | n ) ∩ U d ( gl m | n ) . Difference with Howe–Umeda: Use of the Capelli basis vs. generators of the algebra PD ( W ) gl m | n . 89 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem � � = PD ( W ) gl m | n ? Question. Is it true that ρ Z ( gl m | n ) m = 0 or n = 0 : special case of Howe-Umeda ’91. See also Goodman–Wallach ’09, Turnbull ’47, G ˚ arding ’47. Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0, where Z d ( gl m | n ) ρ � � Z d ( gl m | n ) = Z ( gl m | n ) ∩ U d ( gl m | n ) . Difference with Howe–Umeda: Use of the Capelli basis vs. generators of the algebra PD ( W ) gl m | n . 90 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. Z d ( gl m | n ) � � ρ Idea of proof Passage to grading using PD ( W ) ∼ = P ( W ) ⊗ S ( W ). S ( W ) is generated by { x i,j } , satisfying x j,i = ( − 1) | i |·| j | x i,j . P ( W ) is generated by { y i,j } , satisfying y j,i = ( − 1) | i |·| j | y i,j . Schur–Weyl duality ⇒ the gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d , t 1 ,...,t d where ϕ i,j = � r ∈I m,n ( − 1) | r | + | i |·| j | y r,j x r,i and � � ˇ ǫ ( σ ; t 1 , . . . , t d ) = | t r | · | t σ ( s ) | + | t σ ( r ) | · | t σ ( s ) | . 1 ≤ r<s ≤ d 1 ≤ r<s ≤ d σ ( r ) <σ ( s ) 91 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. Z d ( gl m | n ) � � ρ Idea of proof Passage to grading using PD ( W ) ∼ = P ( W ) ⊗ S ( W ). S ( W ) is generated by { x i,j } , satisfying x j,i = ( − 1) | i |·| j | x i,j . P ( W ) is generated by { y i,j } , satisfying y j,i = ( − 1) | i |·| j | y i,j . Schur–Weyl duality ⇒ the gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d , t 1 ,...,t d where ϕ i,j = � r ∈I m,n ( − 1) | r | + | i |·| j | y r,j x r,i and � � ˇ ǫ ( σ ; t 1 , . . . , t d ) = | t r | · | t σ ( s ) | + | t σ ( r ) | · | t σ ( s ) | . 1 ≤ r<s ≤ d 1 ≤ r<s ≤ d σ ( r ) <σ ( s ) 92 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. Z d ( gl m | n ) � � ρ Idea of proof Passage to grading using PD ( W ) ∼ = P ( W ) ⊗ S ( W ). S ( W ) is generated by { x i,j } , satisfying x j,i = ( − 1) | i |·| j | x i,j . P ( W ) is generated by { y i,j } , satisfying y j,i = ( − 1) | i |·| j | y i,j . Schur–Weyl duality ⇒ the gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d , t 1 ,...,t d where ϕ i,j = � r ∈I m,n ( − 1) | r | + | i |·| j | y r,j x r,i and � � ˇ ǫ ( σ ; t 1 , . . . , t d ) = | t r | · | t σ ( s ) | + | t σ ( r ) | · | t σ ( s ) | . 1 ≤ r<s ≤ d 1 ≤ r<s ≤ d σ ( r ) <σ ( s ) 93 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. Z d ( gl m | n ) � � ρ Idea of proof Passage to grading using PD ( W ) ∼ = P ( W ) ⊗ S ( W ). S ( W ) is generated by { x i,j } , satisfying x j,i = ( − 1) | i |·| j | x i,j . P ( W ) is generated by { y i,j } , satisfying y j,i = ( − 1) | i |·| j | y i,j . Schur–Weyl duality ⇒ the gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d , t 1 ,...,t d where ϕ i,j = � r ∈I m,n ( − 1) | r | + | i |·| j | y r,j x r,i and � � ˇ ǫ ( σ ; t 1 , . . . , t d ) = | t r | · | t σ ( s ) | + | t σ ( r ) | · | t σ ( s ) | . 1 ≤ r<s ≤ d 1 ≤ r<s ≤ d σ ( r ) <σ ( s ) 94 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. Z d ( gl m | n ) � � ρ Idea of proof Passage to grading using PD ( W ) ∼ = P ( W ) ⊗ S ( W ). S ( W ) is generated by { x i,j } , satisfying x j,i = ( − 1) | i |·| j | x i,j . P ( W ) is generated by { y i,j } , satisfying y j,i = ( − 1) | i |·| j | y i,j . Schur–Weyl duality ⇒ the gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d , t 1 ,...,t d where ϕ i,j = � r ∈I m,n ( − 1) | r | + | i |·| j | y r,j x r,i and � � ˇ ǫ ( σ ; t 1 , . . . , t d ) = | t r | · | t σ ( s ) | + | t σ ( r ) | · | t σ ( s ) | . 1 ≤ r<s ≤ d 1 ≤ r<s ≤ d σ ( r ) <σ ( s ) 95 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. � Z d ( gl m | n ) � ρ Idea of proof gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d . t 1 ,...,t d t σ depends only on the conjugacy class of σ ∈ S d . σ = σ 1 . . . σ ℓ , σ k = ( d k + 1 , . . . , d k +1 ) 0 = d 1 < · · · < d ℓ +1 = d . E i,j = ( − 1) | i |·| j | E i,j , Z d = str( E d ) ∈ U ( gl m | n ). Gelfand elements : Z d ∈ Z d ( gl m | n ). P ( W ) ⊗ S ( W ) m − → PD ( W ) , p ⊗ ∂ �→ p∂ . � � σ = (1 , . . . , d ) ⇒ ord ρ ( Z d ) − m ( t σ ) < d. 96 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. � Z d ( gl m | n ) � ρ Idea of proof gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d . t 1 ,...,t d t σ depends only on the conjugacy class of σ ∈ S d . σ = σ 1 . . . σ ℓ , σ k = ( d k + 1 , . . . , d k +1 ) 0 = d 1 < · · · < d ℓ +1 = d . E i,j = ( − 1) | i |·| j | E i,j , Z d = str( E d ) ∈ U ( gl m | n ). Gelfand elements : Z d ∈ Z d ( gl m | n ). P ( W ) ⊗ S ( W ) m − → PD ( W ) , p ⊗ ∂ �→ p∂ . � � σ = (1 , . . . , d ) ⇒ ord ρ ( Z d ) − m ( t σ ) < d. 97 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. � Z d ( gl m | n ) � ρ Idea of proof gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d . t 1 ,...,t d t σ depends only on the conjugacy class of σ ∈ S d . σ = σ 1 . . . σ ℓ , σ k = ( d k + 1 , . . . , d k +1 ) 0 = d 1 < · · · < d ℓ +1 = d . E i,j = ( − 1) | i |·| j | E i,j , Z d = str( E d ) ∈ U ( gl m | n ). Gelfand elements : Z d ∈ Z d ( gl m | n ). P ( W ) ⊗ S ( W ) m − → PD ( W ) , p ⊗ ∂ �→ p∂ . � � σ = (1 , . . . , d ) ⇒ ord ρ ( Z d ) − m ( t σ ) < d. 98 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. � Z d ( gl m | n ) � ρ Idea of proof gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d . t 1 ,...,t d t σ depends only on the conjugacy class of σ ∈ S d . σ = σ 1 . . . σ ℓ , σ k = ( d k + 1 , . . . , d k +1 ) ⇒ t σ = t σ 1 · · · t σ ℓ . 0 = d 1 < · · · < d ℓ +1 = d . E i,j = ( − 1) | i |·| j | E i,j , Z d = str( E d ) ∈ U ( gl m | n ). Gelfand elements : Z d ∈ Z d ( gl m | n ). P ( W ) ⊗ S ( W ) m − → PD ( W ) , p ⊗ ∂ �→ p∂ . � � σ = (1 , . . . , d ) ⇒ ord ρ ( Z d ) − m ( t σ ) < d. 99 / 104

The spectrum and interpolation polynomials � � Lie superalgebras The supersymmetric pair gl m | 2 n , osp m | 2 n Main Theorem The Capelli problem Theorem (Sahi–S.) = PD d ( W ) gl m | n for every d ≥ 0. � Z d ( gl m | n ) � ρ Idea of proof gl m | n -invariants of P ( W ) ⊗ S ( W ) are spanned by � ( − 1) ˇ ǫ ( σ ; t 1 ,...,t d ) ϕ t σ (1) ,t 1 · · · ϕ t σ ( d ) ,t d t σ = , σ ∈ S d . t 1 ,...,t d t σ depends only on the conjugacy class of σ ∈ S d . σ = σ 1 . . . σ ℓ , σ k = ( d k + 1 , . . . , d k +1 ) ⇒ t σ = t σ 1 · · · t σ ℓ . 0 = d 1 < · · · < d ℓ +1 = d . E i,j = ( − 1) | i |·| j | E i,j , Z d = str( E d ) ∈ U ( gl m | n ). Gelfand elements : Z d ∈ Z d ( gl m | n ). P ( W ) ⊗ S ( W ) m − → PD ( W ) , p ⊗ ∂ �→ p∂ . � � σ = (1 , . . . , d ) ⇒ ord ρ ( Z d ) − m ( t σ ) < d. 100 / 104