The Capelli eigenvalue problem for Lie superalgebras Hadi Salmasian - PowerPoint PPT Presentation

The Capelli eigenvalue problem for Lie superalgebras Hadi Salmasian Department of Mathematics and Statistics University of Ottawa arXiv:1807.07340 July 26, 2018 1 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q – P p V q b D p V q à V λ b V ˚ à – – Hom p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 2 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q – P p V q b D p V q à V λ b V ˚ à – – Hom p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 3 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q – P p V q b D p V q à V λ b V ˚ à – – Hom p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 4 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q – P p V q b D p V q à V λ b V ˚ à – – Hom p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 5 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q – P p V q b D p V q à V λ b V ˚ à – – Hom C p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 6 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q g – ˘ g ` P p V q b D p V q ˘ g – à V λ b V ˚ à ` – Hom g p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 7 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q g – ˘ g ` P p V q b D p V q ˘ g – à V λ b V ˚ à ` – Hom g p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 8 / 91

Capelli Operators g : Reductive Lie algebra/Classical Lie superalgebra. V : irreducible finite dimensional g -module such that P p V q is completely reducible and multiplicity-free. P p V q – S p V ˚ q – S p V ˚ 0 q b Λ p V ˚ V – V 0 ‘ V 1 1 q . ù V ˚ à à P p V q – V λ , D p V q – S p V q – λ . λ P E V λ P Ω PD p V q g – ˘ g ` P p V q b D p V q ˘ g – à V λ b V ˚ à ` – Hom g p V µ , V λ q µ λ,µ P Ω λ,µ P Ω # if λ “ µ, C Hom g p V µ , V λ q : “ D λ Ø 1 P Hom g p V λ , V λ q t 0 u if λ ‰ µ. 9 / 91

Capelli Operators Example g : “ gl n p C q ˆ gl n p C q , V : “ Mat n ˆ n p C q . V ˚ à P p V q – λ b V λ . ℓ p λ qď n For λ “ p 1 q “ p 1 , 0 , . . . q we obtain B ÿ D p 1 q “ (degree operator). x i,j B x i,j 1 ď i,j ď n For λ : “ p 1 n q we obtain B D p 1 n q “ det p x i,j q det p B x i,j q (Capelli operator). ‚ The basis t D λ u λ P E V for PD p V q g is called the Capelli basis . 10 / 91

Capelli Operators Example g : “ gl n p C q ˆ gl n p C q , V : “ Mat n ˆ n p C q . V ˚ à P p V q – λ b V λ . ℓ p λ qď n For λ “ p 1 q “ p 1 , 0 , . . . q we obtain B ÿ D p 1 q “ (degree operator). x i,j B x i,j 1 ď i,j ď n For λ : “ p 1 n q we obtain B D p 1 n q “ det p x i,j q det p B x i,j q (Capelli operator). ‚ The basis t D λ u λ P E V for PD p V q g is called the Capelli basis . 11 / 91

Capelli Operators Example g : “ gl n p C q ˆ gl n p C q , V : “ Mat n ˆ n p C q . V ˚ à P p V q – λ b V λ . ℓ p λ qď n For λ “ p 1 q “ p 1 , 0 , . . . q we obtain B ÿ D p 1 q “ (degree operator). x i,j B x i,j 1 ď i,j ď n For λ : “ p 1 n q we obtain B D p 1 n q “ det p x i,j q det p B x i,j q (Capelli operator). ‚ The basis t D λ u λ P E V for PD p V q g is called the Capelli basis . 12 / 91

Capelli Operators Example g : “ gl n p C q ˆ gl n p C q , V : “ Mat n ˆ n p C q . V ˚ à P p V q – λ b V λ . ℓ p λ qď n For λ “ p 1 q “ p 1 , 0 , . . . q we obtain B ÿ D p 1 q “ (degree operator). x i,j B x i,j 1 ď i,j ď n For λ : “ p 1 n q we obtain B D p 1 n q “ det p x i,j q det p B x i,j q (Capelli operator). ‚ The basis t D λ u λ P E V for PD p V q g is called the Capelli basis . 13 / 91

The Capelli Eigenvalue Problem D λ : P p V q Ñ P p V q is a g -module homomorphism ( D λ is g -invariant). ˇ P p V q multiplicity-free ñ D λ V µ “ c λ p µ q I V µ for all λ, µ . ˇ Problem (Kostant): Compute c λ p µ q . Example F : real division algebra, d : “ dim R p F q P t 1 , 2 , 4 u . g R : “ gl n p F q , V R : “ Herm n ˆ n p F q , g : “ g R b R C , V : “ V R b R C . à P p V q – V λ . ℓ p λ qď n $ λ : “ ř n d “ 1 ñ g – gl n p C q i “ 1 2 λ i ε i , ’ & λ : “ ř n d “ 2 ñ g – gl n p C q ‘ gl n p C q i “ 1 λ i ε i , ’ λ : “ ř n d “ 4 ñ g – gl 2 n p C q i “ 1 λ i p ε 2 i ´ 1 ` ε 2 i q . % 14 / 91

The Capelli Eigenvalue Problem D λ : P p V q Ñ P p V q is a g -module homomorphism ( D λ is g -invariant). ˇ P p V q multiplicity-free ñ D λ V µ “ c λ p µ q I V µ for all λ, µ . ˇ Problem (Kostant): Compute c λ p µ q . Example F : real division algebra, d : “ dim R p F q P t 1 , 2 , 4 u . g R : “ gl n p F q , V R : “ Herm n ˆ n p F q , g : “ g R b R C , V : “ V R b R C . à P p V q – V λ . ℓ p λ qď n $ λ : “ ř n d “ 1 ñ g – gl n p C q i “ 1 2 λ i ε i , ’ & λ : “ ř n d “ 2 ñ g – gl n p C q ‘ gl n p C q i “ 1 λ i ε i , ’ λ : “ ř n d “ 4 ñ g – gl 2 n p C q i “ 1 λ i p ε 2 i ´ 1 ` ε 2 i q . % 15 / 91

The Capelli Eigenvalue Problem D λ : P p V q Ñ P p V q is a g -module homomorphism ( D λ is g -invariant). ˇ P p V q multiplicity-free ñ D λ V µ “ c λ p µ q I V µ for all λ, µ . ˇ Problem (Kostant): Compute c λ p µ q . Example F : real division algebra, d : “ dim R p F q P t 1 , 2 , 4 u . g R : “ gl n p F q , V R : “ Herm n ˆ n p F q , g : “ g R b R C , V : “ V R b R C . à P p V q – V λ . ℓ p λ qď n $ λ : “ ř n d “ 1 ñ g – gl n p C q i “ 1 2 λ i ε i , ’ & λ : “ ř n d “ 2 ñ g – gl n p C q ‘ gl n p C q i “ 1 λ i ε i , ’ λ : “ ř n d “ 4 ñ g – gl 2 n p C q i “ 1 λ i p ε 2 i ´ 1 ` ε 2 i q . % 16 / 91

The Capelli Eigenvalue Problem D λ : P p V q Ñ P p V q is a g -module homomorphism ( D λ is g -invariant). ˇ P p V q multiplicity-free ñ D λ V µ “ c λ p µ q I V µ for all λ, µ . ˇ Problem (Kostant): Compute c λ p µ q . Example F : real division algebra, d : “ dim R p F q P t 1 , 2 , 4 u . g R : “ gl n p F q , V R : “ Herm n ˆ n p F q , g : “ g R b R C , V : “ V R b R C . à P p V q – V λ . ℓ p λ qď n $ λ : “ ř n d “ 1 ñ g – gl n p C q i “ 1 2 λ i ε i , ’ & λ : “ ř n d “ 2 ñ g – gl n p C q ‘ gl n p C q i “ 1 λ i ε i , ’ λ : “ ř n d “ 4 ñ g – gl 2 n p C q i “ 1 λ i p ε 2 i ´ 1 ` ε 2 i q . % 17 / 91