The Capelli eigenvalue problem for Lie superalgebras

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1 The Capelli eigenvalue problem for Lie superalgebras Hadi Salmasian Department of Mathematics and Statistics University of Ottawa Joint with S. Sahi and V. Serganova June 24, / 47

2 The TKK Construction Base field: C. J = J 0 J 1 : Jordan superalgebra: a b := ( 1) a b b a. ( 1) a c [L a b, L c] + ( 1) b a [L b c, L a] + ( 1) c b [L c a, L b ] = 0. L x : J J, L x(y) := xy. TKK Lie superalgebra (Kantor s functor) g J := g J( 1) g J(0) g J(1) g J( 1) := J. g J(0) := L a, [L a, L b ] : a, b J End C (J). g J(1) := P, [L a, P ] : a J Hom C (S 2 (J), J), where P : S 2 (J) J, P (x, y) := x y. 2 / 47

3 The TKK Construction Base field: C. J = J 0 J 1 : Jordan superalgebra: a b := ( 1) a b b a. ( 1) a c [L a b, L c] + ( 1) b a [L b c, L a] + ( 1) c b [L c a, L b ] = 0. L x : J J, L x(y) := xy. TKK Lie superalgebra (Kantor s functor) g J := g J( 1) g J(0) g J(1) g J( 1) := J. g J(0) := L a, [L a, L b ] : a, b J End C (J). g J(1) := P, [L a, P ] : a J Hom C (S 2 (J), J), where P : S 2 (J) J, P (x, y) := x y. 3 / 47

4 The TKK Construction Base field: C. J = J 0 J 1 : Jordan superalgebra: a b := ( 1) a b b a. ( 1) a c [L a b, L c] + ( 1) b a [L b c, L a] + ( 1) c b [L c a, L b ] = 0. L x : J J, L x(y) := xy. TKK Lie superalgebra (Kantor s functor) g J := g J( 1) g J(0) g J(1) g J( 1) := J. g J(0) := L a, [L a, L b ] : a, b J End C (J). g J(1) := P, [L a, P ] : a J Hom C (S 2 (J), J), where P : S 2 (J) J, P (x, y) := x y. 4 / 47

5 The TKK Construction Kantor, Kac, Cantarini-Kac Assume that J is unital. Then g J is simple if and only if J is simple. Remark It is better to work with gl(m n) rather than psl(m n), and with q(n) rather than sq(n). Classification of unital simple Lie superalgebras J gl(m, n) + gl(2m 2n) osp(n, 2m) + osp(4n 2m) p(n) + p(2n) q(n) + q(2n) (m, 2n) + osp(m + 3 2n) D t, t 1 D(2 1, t) F F (3 1) JP(0, n) H(n + 3) g 5 / 47

6 The TKK Construction Kantor, Kac, Cantarini-Kac Assume that J is unital. Then g J is simple if and only if J is simple. Remark It is better to work with gl(m n) rather than psl(m n), and with q(n) rather than sq(n). Classification of unital simple Lie superalgebras J gl(m, n) + gl(2m 2n) osp(n, 2m) + osp(4n 2m) p(n) + p(2n) q(n) + q(2n) (m, 2n) + osp(m + 3 2n) D t, t 1 D(2 1, t) F F (3 1) JP(0, n) H(n + 3) g 6 / 47

7 The TKK Construction Kantor, Kac, Cantarini-Kac Assume that J is unital. Then g J is simple if and only if J is simple. Remark It is better to work with gl(m n) rather than psl(m n), and with q(n) rather than sq(n). Classification of unital simple Lie superalgebras J gl(m, n) + gl(2m 2n) osp(n, 2m) + osp(4n 2m) p(n) + p(2n) q(n) + q(2n) (m, 2n) + osp(m + 3 2n) D t, t 1 D(2 1, t) F F (3 1) JP(0, n) H(n + 3) g 7 / 47

8 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 8 / 47

9 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 9 / 47

10 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 10 / 47

11 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 11 / 47

12 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 12 / 47

13 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 13 / 47

14 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 14 / 47

15 The Quadruple (g, g, k, V ) Short subalgebra: e := 1 J g ( 1), h := L 1J g, f := P g (1). [h, e] = e,, [e, f] = h, [h, f] = f. g (κ) := {x g : [h, x] = κx}. g := g (0). Action of s := Span C {e, f, h} on g is by trivial and adjoint modules. k := g e := {x g : [e, x] = 0}. Note that k = {x g : w.x = x} where W := {1, w} is the Weyl group of s. (g, k) is a supersymmetric pair. V := J is naturally a g-module. We can consider P(V ) := S(V ) as a g-module. 15 / 47

16 The Quadruple (g, g, k, V ) J g g k V gl(m, n) + gl(2m 2n) gl(m n) gl(m n) gl(m n) (C m n ) C m n osp(n, 2m) + osp(4n 2m) gl(m 2n) osp(m 2n) S 2 (C m 2n ) p(n) + p(2n) gl(n n) p(n) Π(Λ 2 C n n ) q(n) + q(2n) q(n) q(n) q(n) ((C n n ) C n n ) Π Π (m, 2n) + osp(m + 3 2n) osp(m + 1 2n) C osp(m 2n) C m+1,2n D t, t 1 D(2 1, t) gl(1 2) osp(1 2) C 2 2 t F F (3 1) osp(2 4) C osp(1 2) osp(1 2) C 6 4 JP(0, n) H(n + 3) H(n + 1) C H(n) C 2n 2 n Natural Questions Is P(V ) a completely reducible g-module? Is it also multiplicity-free? Are the g-modules that arise k-spherical? 16 / 47

17 The Quadruple (g, g, k, V ) J g g k V gl(m, n) + gl(2m 2n) gl(m n) gl(m n) gl(m n) (C m n ) C m n osp(n, 2m) + osp(4n 2m) gl(m 2n) osp(m 2n) S 2 (C m 2n ) p(n) + p(2n) gl(n n) p(n) Π(Λ 2 C n n ) q(n) + q(2n) q(n) q(n) q(n) ((C n n ) C n n ) Π Π (m, 2n) + osp(m + 3 2n) osp(m + 1 2n) C osp(m 2n) C m+1,2n D t, t 1 D(2 1, t) gl(1 2) osp(1 2) C 2 2 t F F (3 1) osp(2 4) C osp(1 2) osp(1 2) C 6 4 JP(0, n) H(n + 3) H(n + 1) C H(n) C 2n 2 n Natural Questions Is P(V ) a completely reducible g-module? Is it also multiplicity-free? Are the g-modules that arise k-spherical? 17 / 47

18 Partitions and Representations Partitions λ = (λ 1, λ 2,, λ l(λ) ) such that λ i Z +, λ i λ i+1 for all i. λ = i λi l(λ) : Number of parts of λ. H m,n,d := {λ : λ = d and λ m+1 n}. DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. D : DP n,d H n,n,2d, λ D(λ) (4, 2, 1) / 47

19 Partitions and Representations Partitions λ = (λ 1, λ 2,, λ l(λ) ) such that λ i Z +, λ i λ i+1 for all i. λ = i λi l(λ) : Number of parts of λ. H m,n,d := {λ : λ = d and λ m+1 n}. DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. D : DP n,d H n,n,2d, λ D(λ) (4, 2, 1) / 47

20 Partitions and Representations Partitions λ = (λ 1, λ 2,, λ l(λ) ) such that λ i Z +, λ i λ i+1 for all i. λ = i λi l(λ) : Number of parts of λ. H m,n,d := {λ : λ = d and λ m+1 n}. DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. D : DP n,d H n,n,2d, λ D(λ) (4, 2, 1) / 47

21 Partitions and Representations Partitions λ = (λ 1, λ 2,, λ l(λ) ) such that λ i Z +, λ i λ i+1 for all i. λ = i λi l(λ) : Number of parts of λ. H m,n,d := {λ : λ = d and λ m+1 n}. DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. D : DP n,d H n,n,2d, λ D(λ) (4, 2, 1) / 47

22 Partitions and Representations Irreducible representations from partitions g = gl(m n). Fundamental roots: {ε i ε i+1 : 1 i m 1} {δ i δ i+1 : 1 i n 1}. µ H m,n,d V µ with h.w. µ := µ 1 ε µ m ε m + µ m+1 δ µ m+n δ n. g = q(n). Fundamental roots: {ε i ε i+1 : 1 i n 1} µ DP n,d V µ with h.w. µ := µ 1 ε µ n ε n 22 / 47

23 Partitions and Representations Irreducible representations from partitions g = gl(m n). Fundamental roots: {ε i ε i+1 : 1 i m 1} {δ i δ i+1 : 1 i n 1}. µ H m,n,d V µ with h.w. µ := µ 1 ε µ m ε m + µ m+1 δ µ m+n δ n. g = q(n). Fundamental roots: {ε i ε i+1 : 1 i n 1} µ DP n,d V µ with h.w. µ := µ 1 ε µ n ε n 23 / 47

24 Theorem (Sahi, S., Serganova + Cheng Wang 00, 01,...) The g-module P(V ) is completely reducible in the following cases. The irreducible components of P(V ) are parametrized as shown. J g P d (V ) gl(m, n) + gl(m n) gl(m n) λ H V m,n,d λ Vλ osp(n, 2m) + gl(m 2n) λ H V m,n,d 2λ p(n) + gl(n n) λ DP V n,d D(λ) New q(n) + q(n) q(n) 1 λ DP n,d 2 δ(l(λ)) U λ U λ (m, 2n) +, m 2 N or m 2 n osp(m + 1 2n) C λ H V 2,0,d λ New D t, t t+1 Q [0, 1] gl(1 2) λ H V 1,1,d λ New F osp(2 4) C λ H V 2,1,d λ New In addition, the irreducible components of P(V ) are k-spherical. For (m, 2n) +, the highest weight corresponding to λ = (λ 1, λ 2 ) H 2,0,d is: (λ 1 λ 2 )ε 1 + (λ 1 + λ 2 )ζ. For D t, the highest weight corresponding to λ = (λ 1, 1,..., 1) H 1,1,d is: }{{} (d λ 1 ) (2d 1+t t 4d + 2λ 1)ε 1 + ( d 1+t t + 3d λ 1)(δ 1 + δ 2 ). For F, the highest weight corresponding to λ = (λ 1, 2,..., 2, 1,..., 1 ) H 1,2,d is }{{}}{{} r d λ 1 2r (d + 2λ 1 4)ε 1 + (d 2r λ 1 )(δ 1 + δ 2 ) + dζ. The fundamental system is { ε 1 δ 1, δ 1 δ 2, 2δ 2 } 24 / 47

25 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) = P(V ) S(V ) = Hom g(v µ, V λ ) := λ,µ E V V λ V µ { C if λ = µ, {0} if λ µ. = Hom(V µ, V λ ) λ,µ E V D λ 1 Hom g(v λ, V λ ) λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 25 / 47

26 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) = P(V ) S(V ) = Hom g(v µ, V λ ) := λ,µ E V V λ V µ { C if λ = µ, {0} if λ µ. = Hom(V µ, V λ ) λ,µ E V D λ 1 Hom g(v λ, V λ ) λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 26 / 47

27 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) = P(V ) S(V ) = Hom g(v µ, V λ ) := λ,µ E V V λ V µ { C if λ = µ, {0} if λ µ. = Hom(V µ, V λ ) λ,µ E V D λ 1 Hom g(v λ, V λ ) λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 27 / 47

28 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) = P(V ) S(V ) = Hom g(v µ, V λ ) := λ,µ E V V λ V µ { C if λ = µ, {0} if λ µ. = Hom C (V µ, V λ ) λ,µ E V D λ 1 Hom g(v λ, V λ ) λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 28 / 47

29 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) g = ( P(V ) S(V ) ) g = Hom g(v µ, V λ ) := ( Vλ Vµ ) g = λ,µ E V { C if λ = µ, {0} if λ µ. λ,µ E V Hom g (V µ, V λ ) D λ 1 Hom g(v λ, V λ ) λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 29 / 47

30 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) g = ( P(V ) S(V ) ) g = Hom g(v µ, V λ ) := ( Vλ Vµ ) g = λ,µ E V { C if λ = µ, {0} if λ µ. λ,µ E V Hom g (V µ, V λ ) D λ 1 Hom g(v λ, V λ ) λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 30 / 47

31 The Capelli Basis We remark that P(V ) is a multiplicity-free g-module. P(V ) = V λ D(V ) = S(V ) = Vλ. λ E V λ E V PD(V ) g = ( P(V ) S(V ) ) g = Hom g(v µ, V λ ) := ( Vλ Vµ ) g = λ,µ E V { C if λ = µ, {0} if λ µ. λ,µ E V Hom g (V µ, V λ ) D λ 1 Hom g(v λ, V λ ) The basis {D λ } λ EV is called the Capelli basis for PD(V ) g. λ, µ E V D λ : V µ V µ acts by c λ (µ) C. Problem (Kostant): Give an explicit description of c λ (µ). 31 / 47

32 Interpolation Jack Polynomials Theorem (Sahi 94, Okounkov Olshanski 97) Assume that θ Q 0. Fix an integer N 1. Then for every partition λ such that l(λ) N, there exists a unique (up to scaling) polynomial P λ(z 1,..., z N ; θ) such that: deg(p λ) λ. P λ(z 1,..., z N ; θ) is symmetric in z i + θ(1 i). P λ(µ, θ) = 0 for all µ satisfying l(µ) N and µ λ and µ λ. P λ(λ; θ) 0. The P λ s form a basis for the algebra Λ N,θ of shifted symmetric polynomials in N-variables. (Knop Sahi 96) They are deformations of Jack polynomials in a precise sense. 32 / 47

33 Interpolation Jack Polynomials Theorem (Sahi 94, Okounkov Olshanski 97) Assume that θ Q 0. Fix an integer N 1. Then for every partition λ such that l(λ) N, there exists a unique (up to scaling) polynomial P λ(z 1,..., z N ; θ) such that: deg(p λ) λ. P λ(z 1,..., z N ; θ) is symmetric in z i + θ(1 i). P λ(µ, θ) = 0 for all µ satisfying l(µ) N and µ λ and µ λ. P λ(λ; θ) 0. The P λ s form a basis for the algebra Λ N,θ of shifted symmetric polynomials in N-variables. (Knop Sahi 96) They are deformations of Jack polynomials in a precise sense. 33 / 47

34 Interpolation Jack Polynomials Theorem (Sahi 94, Okounkov Olshanski 97) Assume that θ Q 0. Fix an integer N 1. Then for every partition λ such that l(λ) N, there exists a unique (up to scaling) polynomial P λ(z 1,..., z N ; θ) such that: deg(p λ) λ. P λ(z 1,..., z N ; θ) is symmetric in z i + θ(1 i). P λ(µ, θ) = 0 for all µ satisfying l(µ) N and µ λ and µ λ. P λ(λ; θ) 0. The P λ s form a basis for the algebra Λ N,θ of shifted symmetric polynomials in N-variables. (Knop Sahi 96) They are deformations of Jack polynomials in a precise sense. 34 / 47

35 Sergeev Veselov Polynomials Remark The polynomials P λ(z; θ) are compatible with the inverse system Λ N+1,θ Λ N,θ Λ 0,θ. Therefore we can consider P λ(z; θ) as elements of Λ θ := lim N Λ N,θ in infinitely many variables z 1, z 2, z 3,.... The Algebra Λ m,n,θ (Sergeev Veselov, 2005) Λ m,n,θ : algebra of polynomials f(x; y) in m + n variables x 1,..., x m, y 1,... y n such that f is separately symmetric in the x i s and the y j s. f(x + 1 ei, y 1 ej) = f(x 1 ei, y + 1 ej) on the hyperplane x i + θy j = 0. Bernoulli sums: b k (z; θ) := i 1 B ( k zi θ( 1 i)) ( B k + θ( 1 i)). 2 2 Twisted Kerov map: ϕ : Λ θ Λ m,n,θ, ϕ (b k (z; θ)) :=twisted Bernoulli sum SP λ := ϕ (P λ). Actually, if λ H m,n, λ then SP λ = / 47

36 Sergeev Veselov Polynomials Remark The polynomials P λ(z; θ) are compatible with the inverse system Λ N+1,θ Λ N,θ Λ 0,θ. Therefore we can consider P λ(z; θ) as elements of Λ θ := lim N Λ N,θ in infinitely many variables z 1, z 2, z 3,.... The Algebra Λ m,n,θ (Sergeev Veselov, 2005) Λ m,n,θ : algebra of polynomials f(x; y) in m + n variables x 1,..., x m, y 1,... y n such that f is separately symmetric in the x i s and the y j s. f(x + 1 ei, y 1 ej) = f(x 1 ei, y + 1 ej) on the hyperplane x i + θy j = 0. Bernoulli sums: b k (z; θ) := i 1 B ( k zi θ( 1 i)) ( B k + θ( 1 i)). 2 2 Twisted Kerov map: ϕ : Λ θ Λ m,n,θ, ϕ (b k (z; θ)) :=twisted Bernoulli sum SP λ := ϕ (P λ). Actually, if λ H m,n, λ then SP λ = / 47

37 Sergeev Veselov Polynomials Remark The polynomials P λ(z; θ) are compatible with the inverse system Λ N+1,θ Λ N,θ Λ 0,θ. Therefore we can consider P λ(z; θ) as elements of Λ θ := lim N Λ N,θ in infinitely many variables z 1, z 2, z 3,.... The Algebra Λ m,n,θ (Sergeev Veselov, 2005) Λ m,n,θ : algebra of polynomials f(x; y) in m + n variables x 1,..., x m, y 1,... y n such that f is separately symmetric in the x i s and the y j s. f(x + 1 ei, y 1 ej) = f(x 1 ei, y + 1 ej) on the hyperplane x i + θy j = 0. Bernoulli sums: b k (z; θ) := i 1 B ( k zi θ( 1 i)) ( B k + θ( 1 i)). 2 2 Twisted Kerov map: ϕ : Λ θ Λ m,n,θ, ϕ (b k (z; θ)) :=twisted Bernoulli sum SP λ := ϕ (P λ). Actually, if λ H m,n, λ then SP λ = / 47

38 Sergeev Veselov Polynomials Remark The polynomials P λ(z; θ) are compatible with the inverse system Λ N+1,θ Λ N,θ Λ 0,θ. Therefore we can consider P λ(z; θ) as elements of Λ θ := lim N Λ N,θ in infinitely many variables z 1, z 2, z 3,.... The Algebra Λ m,n,θ (Sergeev Veselov, 2005) Λ m,n,θ : algebra of polynomials f(x; y) in m + n variables x 1,..., x m, y 1,... y n such that f is separately symmetric in the x i s and the y j s. f(x + 1 ei, y 1 ej) = f(x 1 ei, y + 1 ej) on the hyperplane x i + θy j = 0. Bernoulli sums: b k (z; θ) := i 1 B ( k zi θ( 1 i)) ( B k + θ( 1 i)). 2 2 Twisted Kerov map: ϕ : Λ θ Λ m,n,θ, ϕ (b k (z; θ)) :=twisted Bernoulli sum SP λ := ϕ (P λ). Actually, if λ H m,n, λ then SP λ = / 47

39 Factorial Schur Q-functions Recall that: DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. Let Γ N be the algebra of N-variable symmetric polynomials f(z 1,..., z N ) such that f(t, t, z 3,..., z N ) is independent of t. Theorem (Ivanov 01) For every λ DP N := d DP N,d, there exists a unique (up to scalar) polynomial Q λ Γ N such that deg(q λ) = λ. Q λ(µ) = 0 if µ DP N, µ λ, and µ λ. Q λ(λ) 0. Further, the top homogeneous part of Q λ is the Schur Q-function Q λ. 39 / 47

40 Factorial Schur Q-functions Recall that: DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. Let Γ N be the algebra of N-variable symmetric polynomials f(z 1,..., z N ) such that f(t, t, z 3,..., z N ) is independent of t. Theorem (Ivanov 01) For every λ DP N := d DP N,d, there exists a unique (up to scalar) polynomial Q λ Γ N such that deg(q λ) = λ. Q λ(µ) = 0 if µ DP N, µ λ, and µ λ. Q λ(λ) 0. Further, the top homogeneous part of Q λ is the Schur Q-function Q λ. 40 / 47

41 Factorial Schur Q-functions Recall that: DP n,d := {λ : l(λ) n, λ = d and λ i > λ i+1 for every i}. Let Γ N be the algebra of N-variable symmetric polynomials f(z 1,..., z N ) such that f(t, t, z 3,..., z N ) is independent of t. Theorem (Ivanov 01) For every λ DP N := d DP N,d, there exists a unique (up to scalar) polynomial Q λ Γ N such that deg(q λ) = λ. Q λ(µ) = 0 if µ DP N, µ λ, and µ λ. Q λ(λ) 0. Further, the top homogeneous part of Q λ is the Schur Q-function Q λ. 41 / 47

42 The Capelli Eigenvalue Problem Let g, V be as before. Also, let E V be the set of partitions that parametrize the irreducible summands of P(V ). λ E V λ Set a := Zariski closure of ÊV in h. Example ÊV h a := ker(a ) where is w.r.t. Killing form. (Dual to a.) Upon restriction to a, the natural ε i and δ j coordinates of h give rise to coordinates on a. Thus, a = C N, where N := N g. J := osp(n 2m) +, g = gl(m 2n), V = S 2 (C m 2n ). Ê V = { m i=1 2λ iε i + n i=1 λ i m (δ 2i 1 + δ 2i ) }. a = Span{ε 1,..., ε m, δ 1 + δ 2,..., δ 2n 1 + δ 2n }. a = Span {E i,i : 1 i m} {E m+2i 1,m+2i 1 + E m+2i,m+2i : 1 i n}. Basis of a : {ε 1,..., ε m, δ 1,..., δ n} where ε i := ε i a and δ i := δ 2i 1 a = δ 2i a λ ÊV λ a = m i=1 x iε i + n j=1 y jδ j (x 1,..., x m, y 1,..., y n). 42 / 47

43 The Capelli Eigenvalue Problem Let g, V be as before. Also, let E V be the set of partitions that parametrize the irreducible summands of P(V ). λ E V λ Set a := Zariski closure of ÊV in h. Example ÊV h a := ker(a ) where is w.r.t. Killing form. (Dual to a.) Upon restriction to a, the natural ε i and δ j coordinates of h give rise to coordinates on a. Thus, a = C N, where N := N g. J := osp(n 2m) +, g = gl(m 2n), V = S 2 (C m 2n ). Ê V = { m i=1 2λ iε i + n i=1 λ i m (δ 2i 1 + δ 2i ) }. a = Span{ε 1,..., ε m, δ 1 + δ 2,..., δ 2n 1 + δ 2n }. a = Span {E i,i : 1 i m} {E m+2i 1,m+2i 1 + E m+2i,m+2i : 1 i n}. Basis of a : {ε 1,..., ε m, δ 1,..., δ n} where ε i := ε i a and δ i := δ 2i 1 a = δ 2i a λ ÊV λ a = m i=1 x iε i + n j=1 y jδ j (x 1,..., x m, y 1,..., y n). 43 / 47

44 The Capelli Eigenvalue Problem Let g, V be as before. Also, let E V be the set of partitions that parametrize the irreducible summands of P(V ). λ E V λ Set a := Zariski closure of ÊV in h. Example ÊV h a := ker(a ) where is w.r.t. Killing form. (Dual to a.) Upon restriction to a, the natural ε i and δ j coordinates of h give rise to coordinates on a. Thus, a = C N, where N := N g. J := osp(n 2m) +, g = gl(m 2n), V = S 2 (C m 2n ). Ê V = { m i=1 2λ iε i + n i=1 λ i m (δ 2i 1 + δ 2i ) }. a = Span{ε 1,..., ε m, δ 1 + δ 2,..., δ 2n 1 + δ 2n }. a = Span {E i,i : 1 i m} {E m+2i 1,m+2i 1 + E m+2i,m+2i : 1 i n}. Basis of a : {ε 1,..., ε m, δ 1,..., δ n} where ε i := ε i a and δ i := δ 2i 1 a = δ 2i a λ ÊV λ a = m i=1 x iε i + n j=1 y jδ j (x 1,..., x m, y 1,..., y n). 44 / 47

45 The Capelli Eigenvalue Problem Theorem (Sahi, S. Serganova) Let g and V be as before, and let E V be the set of partitions that parametrize the irreducible components V λ P(V ) and also the basis {D µ} of PD(V ) g. For λ, µ E V, the action of D µ on V λ is by the scalar F µ η(λ a ), where F µ( ) = SPµ(, θ) or F µ( ) = Q µ( ), according to the table below. η : a C N is an affine linear transformation, where N := N g. λ ÊV corresponds to λ. J g k gl m n, εε, εδ, δδ θ gl(m, n) + gl(m n) gl(m n) gl(m n) m n, 2, 2, 2 1. osp(n, 2m) + gl(m 2n) osp(m 2n) m n, 1, 2, D t, t 1 gl(1 2) osp(1 2) 1 1,, 2, F osp(2 4) C osp(1 2) osp(1 2) 1 2,, 2, 3 1 t 2 3 J g k Q n θ p(n) + gl(n n) p(n) n, 2 2 q(n) + q(n) q(n) q(n) n, 2 2 (m, 2n) + osp(m + 1 2n) C osp(m 2n) 2, m 1 2n m 1 2 n Furthermore, identifying a with a.v V where v V is a spherical vector, we have top homog.(f µ η) = p µ a, where p µ V µ is a suitably normalized spherical vector. 45 / 47

46 The Capelli Eigenvalue Problem Theorem (Sahi, S. Serganova) Let g and V be as before, and let E V be the set of partitions that parametrize the irreducible components V λ P(V ) and also the basis {D µ} of PD(V ) g. For λ, µ E V, the action of D µ on V λ is by the scalar F µ η(λ a ), where F µ( ) = SPµ(, θ) or F µ( ) = Q µ( ), according to the table below. η : a C N is an affine linear transformation, where N := N g. λ ÊV corresponds to λ. J g k gl m n, εε, εδ, δδ θ gl(m, n) + gl(m n) gl(m n) gl(m n) m n, 2, 2, 2 1. osp(n, 2m) + gl(m 2n) osp(m 2n) m n, 1, 2, D t, t 1 gl(1 2) osp(1 2) 1 1,, 2, F osp(2 4) C osp(1 2) osp(1 2) 1 2,, 2, 3 1 t 2 3 J g k Q n θ p(n) + gl(n n) p(n) n, 2 2 q(n) + q(n) q(n) q(n) n, 2 2 (m, 2n) + osp(m + 1 2n) C osp(m 2n) 2, m 1 2n m 1 2 n Furthermore, identifying a with a.v V where v V is a spherical vector, we have top homog.(f µ η) = p µ a, where p µ V µ is a suitably normalized spherical vector. 46 / 47

47 The Capelli Eigenvalue Problem Theorem (Sahi, S. Serganova) Let g and V be as before, and let E V be the set of partitions that parametrize the irreducible components V λ P(V ) and also the basis {D µ} of PD(V ) g. For λ, µ E V, the action of D µ on V λ is by the scalar F µ η(λ a ), where F µ( ) = SPµ(, θ) or F µ( ) = Q µ( ), according to the table below. η : a C N is an affine linear transformation, where N := N g. λ ÊV corresponds to λ. J g k gl m n, εε, εδ, δδ θ gl(m, n) + gl(m n) gl(m n) gl(m n) m n, 2, 2, 2 1. osp(n, 2m) + gl(m 2n) osp(m 2n) m n, 1, 2, D t, t 1 gl(1 2) osp(1 2) 1 1,, 2, F osp(2 4) C osp(1 2) osp(1 2) 1 2,, 2, 3 1 t 2 3 J g k Q n θ p(n) + gl(n n) p(n) n, 2 2 q(n) + q(n) q(n) q(n) n, 2 2 (m, 2n) + osp(m + 1 2n) C osp(m 2n) 2, m 1 2n m 1 2 n Furthermore, identifying a with a.v V where v V is a spherical vector, we have top homog.(f µ η) = p µ a, where p µ V µ is a suitably normalized spherical vector. 47 / 47

Combinatorial bases for representations. of the Lie superalgebra gl m n

Combinatorial bases for representations of the Lie superalgebra gl m n Alexander Molev University of Sydney Gelfand Tsetlin bases for gln Gelfand Tsetlin bases for gl n Finite-dimensional irreducible representations