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Spectrum of invariant differential operators for the - - PowerPoint PPT Presentation

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem Spectrum of invariant differential operators for the supersymmetric pair gl m | 2 n , osp m | 2 n Hadi Salmasian Department of Mathematics and Statistics

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SLIDE 1

The spectrum and interpolation polynomials Lie superalgebras Main Theorem

Spectrum of invariant differential operators for the supersymmetric pair

  • glm|2n, ospm|2n
  • Hadi Salmasian

Department of Mathematics and Statistics University of Ottawa University of Hamburg, February 19, 2015

1 / 104

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SLIDE 2

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

Quantum CMS System

Calogero–Moser–Sutherland operators L = −

N

  • i=1

∂2 ∂x2

i

+

  • i<j

k(k − 1) sin2(xi − xj) , k = 1, 1 2. L = −

N

  • i=1
  • xi ∂

∂xi 2 + k(k − 1)

  • 1≤i=j≤n

xixj xi − xj ∂ ∂xi . Radial part of Laplacian corrected by the Euler operator. Olshanetsky–Perelomov (1980) L = −∆ +

  • α∈Φ+

mα(mα + 2m2α + 1)(α, α) sin2(α, x) . Eigenstates are expressible as Jack symmetric functions.

λ := (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0 ⇒ Jλ = mλ + lower order terms

2 / 104

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SLIDE 3

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

Quantum CMS System

Calogero–Moser–Sutherland operators L = −

N

  • i=1

∂2 ∂x2

i

+

  • i<j

k(k − 1) sin2(xi − xj) , k = 1, 1 2. L = −

N

  • i=1
  • xi ∂

∂xi 2 + k(k − 1)

  • 1≤i=j≤n

xixj xi − xj ∂ ∂xi . Radial part of Laplacian corrected by the Euler operator. Olshanetsky–Perelomov (1980) L = −∆ +

  • α∈Φ+

mα(mα + 2m2α + 1)(α, α) sin2(α, x) . Eigenstates are expressible as Jack symmetric functions.

λ := (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0 ⇒ Jλ = mλ + lower order terms

3 / 104

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SLIDE 4

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

Quantum CMS System

Calogero–Moser–Sutherland operators L = −

N

  • i=1

∂2 ∂x2

i

+

  • i<j

k(k − 1) sin2(xi − xj) , k = 1, 1 2. L = −

N

  • i=1
  • xi ∂

∂xi 2 + k(k − 1)

  • 1≤i=j≤n

xixj xi − xj ∂ ∂xi . Radial part of Laplacian corrected by the Euler operator. Olshanetsky–Perelomov (1980) L = −∆ +

  • α∈Φ+

mα(mα + 2m2α + 1)(α, α) sin2(α, x) . Eigenstates are expressible as Jack symmetric functions.

λ := (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0 ⇒ Jλ = mλ + lower order terms

4 / 104

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SLIDE 5

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

Quantum CMS System

Calogero–Moser–Sutherland operators L = −

N

  • i=1

∂2 ∂x2

i

+

  • i<j

k(k − 1) sin2(xi − xj) , k = 1, 1 2. L = −

N

  • i=1
  • xi ∂

∂xi 2 + k(k − 1)

  • 1≤i=j≤n

xixj xi − xj ∂ ∂xi . Radial part of Laplacian corrected by the Euler operator. Olshanetsky–Perelomov (1980) L = −∆ +

  • α∈Φ+

mα(mα + 2m2α + 1)(α, α) sin2(α, x) . Eigenstates are expressible as Jack symmetric functions.

λ := (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0 ⇒ Jλ = mλ + lower order terms

5 / 104

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SLIDE 6

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

Quantum CMS System

Calogero–Moser–Sutherland operators L = −

N

  • i=1

∂2 ∂x2

i

+

  • i<j

k(k − 1) sin2(xi − xj) , k = 1, 1 2. L = −

N

  • i=1
  • xi ∂

∂xi 2 + k(k − 1)

  • 1≤i=j≤n

xixj xi − xj ∂ ∂xi . Radial part of Laplacian corrected by the Euler operator. Olshanetsky–Perelomov (1980) L = −∆ +

  • α∈Φ+

mα(mα + 2m2α + 1)(α, α) sin2(α, x) . Eigenstates are expressible as Jack symmetric functions.

λ := (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0 ⇒ Jλ = mλ + lower order terms

6 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

7 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

8 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

9 / 104

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SLIDE 10

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

10 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

11 / 104

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SLIDE 12

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

12 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Polynomial coefficient differential operators P(W) , GL(W) × P(W) → P(W) , g · p(w) := p(g−1 · w). D(W) ∂w1 · · · ∂wr ∂vp(w) := lim

t→0

1 t p(w + tv) − p(w)

  • D(W) ∼

= S(W) as GL(W)-modules. P(W) ⊗ D(W) ∼ = PD(W) , p ⊗ ∂v → p∂v g ∈ GL(W), D ∈ PD(W) ⇒ (g · D)p := g ·

  • D(g−1 · p)
  • Problem.

Study PD(W)G for a reductive subgroup G ⊆ GL(W).

13 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Assume W is G-multiplicity-free: S(W) ∼ =

  • λ∈

G mλVλ where mλ ≤ 1.

S(W) ∼ =

  • λ∈IW

Vλ ⇒ P(W) ∼ =

  • λ∈IW

V ∗

λ .

PD(W) ∼ = P(W) ⊗ S(W) ∼ =

  • λ,µ∈IW

V ∗

λ ⊗ Vµ

∼ =

  • λ,µ∈IW

Hom(V ∗

µ , V ∗ λ )

HomG(V ∗

µ , V ∗ λ ) :=

  • C

if λ = µ, {0} if λ = µ. Dλ ↔ 1 ∈ HomG(V ∗

λ , V ∗ λ ) 14 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Assume W is G-multiplicity-free: S(W) ∼ =

  • λ∈

G mλVλ where mλ ≤ 1.

S(W) ∼ =

  • λ∈IW

Vλ ⇒ P(W) ∼ =

  • λ∈IW

V ∗

λ .

PD(W) ∼ = P(W) ⊗ S(W) ∼ =

  • λ,µ∈IW

V ∗

λ ⊗ Vµ

∼ =

  • λ,µ∈IW

HomC(V ∗

µ , V ∗ λ )

HomG(V ∗

µ , V ∗ λ ) :=

  • C

if λ = µ, {0} if λ = µ. Dλ ↔ 1 ∈ HomG(V ∗

λ , V ∗ λ ) 15 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Assume W is G-multiplicity-free: S(W) ∼ =

  • λ∈

G mλVλ where mλ ≤ 1.

S(W) ∼ =

  • λ∈IW

Vλ ⇒ P(W) ∼ =

  • λ∈IW

V ∗

λ .

PD(W) ∼ =

  • P(W) ⊗ S(W)

G ∼ =

  • λ,µ∈IW
  • V ∗

λ ⊗ Vµ

G ∼ =

  • λ,µ∈IW

HomG(V ∗

µ , V ∗ λ )

HomG(V ∗

µ , V ∗ λ ) :=

  • C

if λ = µ, {0} if λ = µ. Dλ ↔ 1 ∈ HomG(V ∗

λ , V ∗ λ ) 16 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Assume W is G-multiplicity-free: S(W) ∼ =

  • λ∈

G mλVλ where mλ ≤ 1.

S(W) ∼ =

  • λ∈IW

Vλ ⇒ P(W) ∼ =

  • λ∈IW

V ∗

λ .

PD(W)G ∼ =

  • P(W) ⊗ S(W)

G ∼ =

  • λ,µ∈IW
  • V ∗

λ ⊗ Vµ

G ∼ =

  • λ,µ∈IW

HomG(V ∗

µ , V ∗ λ )

HomG(V ∗

µ , V ∗ λ ) :=

  • C

if λ = µ, {0} if λ = µ. Dλ ↔ 1 ∈ HomG(V ∗

λ , V ∗ λ ) 17 / 104

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SLIDE 18

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Assume W is G-multiplicity-free: S(W) ∼ =

  • λ∈

G mλVλ where mλ ≤ 1.

S(W) ∼ =

  • λ∈IW

Vλ ⇒ P(W) ∼ =

  • λ∈IW

V ∗

λ .

PD(W)G ∼ =

  • P(W) ⊗ S(W)

G ∼ =

  • λ,µ∈IW
  • V ∗

λ ⊗ Vµ

G ∼ =

  • λ,µ∈IW

HomG(V ∗

µ , V ∗ λ )

HomG(V ∗

µ , V ∗ λ ) :=

  • C

if λ = µ, {0} if λ = µ. Dλ ↔ 1 ∈ HomG(V ∗

λ , V ∗ λ )

  • The basis {Dλ}λ∈IW is called the Capelli basis for PD(W)G.

18 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

multiplicity-free spaces and the Capelli basis

Assume W is G-multiplicity-free: S(W) ∼ =

  • λ∈

G mλVλ where mλ ≤ 1.

S(W) ∼ =

  • λ∈IW

Vλ ⇒ P(W) ∼ =

  • λ∈IW

V ∗

λ .

PD(W)G ∼ =

  • P(W) ⊗ S(W)

G ∼ =

  • λ,µ∈IW
  • V ∗

λ ⊗ Vµ

G ∼ =

  • λ,µ∈IW

HomG(V ∗

µ , V ∗ λ )

HomG(V ∗

µ , V ∗ λ ) :=

  • C

if λ = µ, {0} if λ = µ. Dλ ↔ 1 ∈ HomG(V ∗

λ , V ∗ λ )

  • The basis {Dλ}λ∈IW is called the Capelli basis for PD(W)G.

λ, µ ∈ IW ⇒ Dλ : V ∗

µ → V ∗ µ acts by cλ(µ) ∈ C.

Problem (Kostant). Give an explicit description of cλ(µ).

19 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

20 / 104

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

21 / 104

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SLIDE 22

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

22 / 104

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SLIDE 23

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

23 / 104

slide-24
SLIDE 24

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

24 / 104

slide-25
SLIDE 25

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

25 / 104

slide-26
SLIDE 26

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Example

V = Cn , W = S2(V ) , G = GL(V ) ∼ = GLn(C) , K = O(V ) ∼ = On(C). P(W) ∼ =

  • λ

V ∗

λ

where λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. λ = h.w. of V ∗

λ = (−2λn)ε1 + · · · + (−2λ1)εn.

Every V ∗

λ ⊂ P(W) contains a K-invariant vector 0 = zλ ∈ V ∗ λ .

w◦ ∈ W a K-invariant vector ι : G/K ֒ → W , g → g · w◦. a :=

  • diag(a1, . . . , an) | a1, . . . , an ∈ C
  • ⊂ gln ∼

= gl(V ). a ֒ → gl(V ) dι − → W

zλ

− − → C

  • Jλ := zλ
  • a ∈ P(a)

.

26 / 104

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SLIDE 27

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. (a) There exists a polynomial J⋆

λ ∈ P(a∗)Sn such that

deg(J⋆

λ) = |λ| = λ1 + · · · + λn and cλ(µ) = J⋆ λ(µ + ρ)

where ρ = n−1

2 ε1 + · · · + 1−n 2 εn and µ = h.w. of V ∗ µ .

(b) J⋆

λ is determined up to scalar by the following conditions:

J⋆

λ ∈ P(a∗)Sn,

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 = Jλ + 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

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The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. (a) There exists a polynomial J⋆

λ ∈ P(a∗)Sn such that

deg(J⋆

λ) = |λ| = λ1 + · · · + λn and cλ(µ) = J⋆ λ(µ + ρ)

where ρ = n−1

2 ε1 + · · · + 1−n 2 εn and µ = h.w. of V ∗ µ .

(b) J⋆

λ is determined up to scalar by the following conditions:

J⋆

λ ∈ P(a∗)Sn,

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 = Jλ + 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

slide-29
SLIDE 29

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. (a) There exists a polynomial J⋆

λ ∈ P(a∗)Sn such that

deg(J⋆

λ) = |λ| = λ1 + · · · + λn and cλ(µ) = J⋆ λ(µ + ρ)

where ρ = n−1

2 ε1 + · · · + 1−n 2 εn and µ = h.w. of V ∗ µ .

(b) J⋆

λ is determined up to scalar by the following conditions:

J⋆

λ ∈ P(a∗)Sn,

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 = Jλ + 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

slide-30
SLIDE 30

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. (a) There exists a polynomial J⋆

λ ∈ P(a∗)Sn such that

deg(J⋆

λ) = |λ| = λ1 + · · · + λn and cλ(µ) = J⋆ λ(µ + ρ)

where ρ = n−1

2 ε1 + · · · + 1−n 2 εn and µ = h.w. of V ∗ µ .

(b) J⋆

λ is determined up to scalar by the following conditions:

J⋆

λ ∈ P(a∗)Sn,

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 = Jλ + 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

slide-31
SLIDE 31

The spectrum and interpolation polynomials Lie superalgebras Main Theorem multiplicity-free spaces and the Capelli basis The spectrum cλ(µ)

The spectrum cλ(µ)

Theorem (Sahi ’94, Knop–Sahi ’96) Fix λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn ≥ 0. (a) There exists a polynomial J⋆

λ ∈ P(a∗)Sn such that

deg(J⋆

λ) = |λ| = λ1 + · · · + λn and cλ(µ) = J⋆ λ(µ + ρ)

where ρ = n−1

2 ε1 + · · · + 1−n 2 εn and µ = h.w. of V ∗ µ .

(b) J⋆

λ is determined up to scalar by the following conditions:

J⋆

λ ∈ P(a∗)Sn,

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 = Jλ + 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

slide-32
SLIDE 32

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Deformed CMS operators

The deformation Lm,n,θ (Sergeev–Veselov, 2005)

Lm,n,θ = −

m

  • i=1

∂2 ∂x2

i

+ θ

n

  • i=1

∂2 ∂y2

i

+

  • 1≤i<j≤m

2θ(θ − 1) sin2(xi − xj) −

  • 1≤i<j≤n

2(θ−1 + 1) sin2(yi − yj) −

m

  • i=1

n

  • j=1

2(θ − 1) sin2(xi − yj)

θ-supersymmetric functions

Let Λm,n,θ be the subalgebra of all f ∈ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn such that

  • ∂

∂xi + θ ∂ ∂yj

  • f = 0 on the hyperplane xi − yj = 0.

Λn := C[x1, . . . , xn]Sn , Λ = lim ← − Λn, ϕ : Λ → Λm,n,θ ,

i xr i → m i=1 xr i − 1 θ

n

j=1 yr j

Λ

L

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

Lm,n,θ Λm,n,θ

λ = (λ1, λ2, . . .) ⇒ sJλ := ϕ(Jλ) super Jack polynomials 32 / 104

slide-33
SLIDE 33

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Deformed CMS operators

The deformation Lm,n,θ (Sergeev–Veselov, 2005)

Lm,n,θ = −

m

  • i=1

∂2 ∂x2

i

+ θ

n

  • i=1

∂2 ∂y2

i

+

  • 1≤i<j≤m

2θ(θ − 1) sin2(xi − xj) −

  • 1≤i<j≤n

2(θ−1 + 1) sin2(yi − yj) −

m

  • i=1

n

  • j=1

2(θ − 1) sin2(xi − yj)

θ-supersymmetric functions

Let Λm,n,θ be the subalgebra of all f ∈ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn such that

  • ∂

∂xi + θ ∂ ∂yj

  • f = 0 on the hyperplane xi − yj = 0.

Λn := C[x1, . . . , xn]Sn , Λ = lim ← − Λn, ϕ : Λ → Λm,n,θ ,

i xr i → m i=1 xr i − 1 θ

n

j=1 yr j

Λ

L

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

Lm,n,θ Λm,n,θ

λ = (λ1, λ2, . . .) ⇒ sJλ := ϕ(Jλ) super Jack polynomials 33 / 104

slide-34
SLIDE 34

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Deformed CMS operators

The deformation Lm,n,θ (Sergeev–Veselov, 2005)

Lm,n,θ = −

m

  • i=1

∂2 ∂x2

i

+ θ

n

  • i=1

∂2 ∂y2

i

+

  • 1≤i<j≤m

2θ(θ − 1) sin2(xi − xj) −

  • 1≤i<j≤n

2(θ−1 + 1) sin2(yi − yj) −

m

  • i=1

n

  • j=1

2(θ − 1) sin2(xi − yj)

θ-supersymmetric functions

Let Λm,n,θ be the subalgebra of all f ∈ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn such that

  • ∂

∂xi + θ ∂ ∂yj

  • f = 0 on the hyperplane xi − yj = 0.

Λn := C[x1, . . . , xn]Sn , Λ = lim ← − Λn, ϕ : Λ → Λm,n,θ ,

i xr i → m i=1 xr i − 1 θ

n

j=1 yr j

Λ

L

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

Lm,n,θ Λm,n,θ

λ = (λ1, λ2, . . .) ⇒ sJλ := ϕ(Jλ) super Jack polynomials 34 / 104

slide-35
SLIDE 35

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Deformed CMS operators

The deformation Lm,n,θ (Sergeev–Veselov, 2005)

Lm,n,θ = −

m

  • i=1

∂2 ∂x2

i

+ θ

n

  • i=1

∂2 ∂y2

i

+

  • 1≤i<j≤m

2θ(θ − 1) sin2(xi − xj) −

  • 1≤i<j≤n

2(θ−1 + 1) sin2(yi − yj) −

m

  • i=1

n

  • j=1

2(θ − 1) sin2(xi − yj)

θ-supersymmetric functions

Let Λm,n,θ be the subalgebra of all f ∈ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn such that

  • ∂

∂xi + θ ∂ ∂yj

  • f = 0 on the hyperplane xi − yj = 0.

Λn := C[x1, . . . , xn]Sn , Λ = lim ← − Λn, ϕ : Λ → Λm,n,θ ,

i xr i → m i=1 xr i − 1 θ

n

j=1 yr j

Λ

L

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

Lm,n,θ Λm,n,θ

λ = (λ1, λ2, . . .) ⇒ sJλ := ϕ(Jλ) super Jack polynomials 35 / 104

slide-36
SLIDE 36

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Deformed CMS operators

The deformation Lm,n,θ (Sergeev–Veselov, 2005)

Lm,n,θ = −

m

  • i=1

∂2 ∂x2

i

+ θ

n

  • i=1

∂2 ∂y2

i

+

  • 1≤i<j≤m

2θ(θ − 1) sin2(xi − xj) −

  • 1≤i<j≤n

2(θ−1 + 1) sin2(yi − yj) −

m

  • i=1

n

  • j=1

2(θ − 1) sin2(xi − yj)

θ-supersymmetric functions

Let Λm,n,θ be the subalgebra of all f ∈ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn such that

  • ∂

∂xi + θ ∂ ∂yj

  • f = 0 on the hyperplane xi − yj = 0.

Λn := C[x1, . . . , xn]Sn , Λ = lim ← − Λn, ϕ : Λ → Λm,n,θ ,

i xr i → m i=1 xr i − 1 θ

n

j=1 yr j

Λ

L

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

Lm,n,θ Λm,n,θ

λ = (λ1, λ2, . . .) ⇒ sJλ := ϕ(Jλ) super Jack polynomials 36 / 104

slide-37
SLIDE 37

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Deformed CMS operators

The deformation Lm,n,θ (Sergeev–Veselov, 2005)

Lm,n,θ = −

m

  • i=1

∂2 ∂x2

i

+ θ

n

  • i=1

∂2 ∂y2

i

+

  • 1≤i<j≤m

2θ(θ − 1) sin2(xi − xj) −

  • 1≤i<j≤n

2(θ−1 + 1) sin2(yi − yj) −

m

  • i=1

n

  • j=1

2(θ − 1) sin2(xi − yj)

θ-supersymmetric functions

Let Λm,n,θ be the subalgebra of all f ∈ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn such that

  • ∂

∂xi + θ ∂ ∂yj

  • f = 0 on the hyperplane xi − yj = 0.

Λn := C[x1, . . . , xn]Sn , Λ = lim ← − Λn, ϕ : Λ → Λm,n,θ ,

i xr i → m i=1 xr i − 1 θ

n

j=1 yr j

Λ

L

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

Lm,n,θ Λm,n,θ

λ = (λ1, λ2, . . .) ⇒ sJλ := ϕ(Jλ) super Jack polynomials 37 / 104

slide-38
SLIDE 38

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

Quantum integrals Λn,θ := C[x1, . . . , xi + θ(1 − i), . . . , xn + θ(1 − n)]Sn Λθ := lim ← − Λn,θ L L f

θ

, Lm,n,θ L f

m,n,θ for every f ∈ Λθ

Λ

L f

θ

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

L f

m,n,θ

Λm,n,θ

L f

m,n,θ sJλ = f(λ)sJλ.

λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ ϕ(Jλ) = 0.

38 / 104

slide-39
SLIDE 39

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

Quantum integrals Λn,θ := C[x1, . . . , xi + θ(1 − i), . . . , xn + θ(1 − n)]Sn Λθ := lim ← − Λn,θ L L f

θ

, Lm,n,θ L f

m,n,θ for every f ∈ Λθ

Λ

L f

θ

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

L f

m,n,θ

Λm,n,θ

L f

m,n,θ sJλ = f(λ)sJλ.

λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ ϕ(Jλ) = 0.

39 / 104

slide-40
SLIDE 40

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

Quantum integrals Λn,θ := C[x1, . . . , xi + θ(1 − i), . . . , xn + θ(1 − n)]Sn Λθ := lim ← − Λn,θ L L f

θ

, Lm,n,θ L f

m,n,θ for every f ∈ Λθ

Λ

L f

θ

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

L f

m,n,θ

Λm,n,θ

L f

m,n,θ sJλ = f(λ)sJλ.

λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ ϕ(Jλ) = 0.

40 / 104

slide-41
SLIDE 41

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

Quantum integrals Λn,θ := C[x1, . . . , xi + θ(1 − i), . . . , xn + θ(1 − n)]Sn Λθ := lim ← − Λn,θ L L f

θ

, Lm,n,θ L f

m,n,θ for every f ∈ Λθ

Λ

L f

θ

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

L f

m,n,θ

Λm,n,θ

L f

m,n,θ sJλ = f(λ)sJλ.

λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ ϕ(Jλ) = 0.

41 / 104

slide-42
SLIDE 42

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

Quantum integrals Λn,θ := C[x1, . . . , xi + θ(1 − i), . . . , xn + θ(1 − n)]Sn Λθ := lim ← − Λn,θ L L f

θ

, Lm,n,θ L f

m,n,θ for every f ∈ Λθ

Λ

L f

θ

  • ϕ
  • Λ

ϕ

  • Λm,n,θ

L f

m,n,θ

Λm,n,θ

L f

m,n,θ sJλ = f(λ)sJλ.

λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ ϕ(Jλ) = 0.

42 / 104

slide-43
SLIDE 43

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

The algebra Λ♮

m,n,θ and the polynomials sJ⋆ λ

Let Λ♮

m,n,θ ⊂ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn be defined as follows: f ∈ Λ♮

m,n,θ

iff f

  • xi +

1 2 , yj − 1 2

  • = f
  • xi −

1 2 , yj + 1 2

  • n the hyperplane xi + θyj = 0.

Set ϕ♮ : Λθ → Λ♮

m,n,θ ,

ϕ♮(f)(p, q) := f(F−1(p, q)) where the map F : {(λ1, . . . , λm, µ1, . . . , µn)} → Cm+n is given by “Frobenius coordinates”:    pi = λi − θ(i − 1

2 ) − 1 2 (n − θm)

1 ≤ i ≤ m, qj = µ′

j − θ−1(j − 1 2 ) + 1 2 (θ−1n + m)

1 ≤ j ≤ n. sJ⋆

λ := ϕ♮(J⋆ λ)

shifted super Jack polynomials λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ sJ⋆

λ = 0. 43 / 104

slide-44
SLIDE 44

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

The algebra Λ♮

m,n,θ and the polynomials sJ⋆ λ

Let Λ♮

m,n,θ ⊂ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn be defined as follows: f ∈ Λ♮

m,n,θ

iff f

  • xi +

1 2 , yj − 1 2

  • = f
  • xi −

1 2 , yj + 1 2

  • n the hyperplane xi + θyj = 0.

Set ϕ♮ : Λθ → Λ♮

m,n,θ ,

ϕ♮(f)(p, q) := f(F−1(p, q)) where the map F : {(λ1, . . . , λm, µ1, . . . , µn)} → Cm+n is given by “Frobenius coordinates”:    pi = λi − θ(i − 1

2 ) − 1 2 (n − θm)

1 ≤ i ≤ m, qj = µ′

j − θ−1(j − 1 2 ) + 1 2 (θ−1n + m)

1 ≤ j ≤ n. sJ⋆

λ := ϕ♮(J⋆ λ)

shifted super Jack polynomials λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ sJ⋆

λ = 0. 44 / 104

slide-45
SLIDE 45

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

The algebra Λ♮

m,n,θ and the polynomials sJ⋆ λ

Let Λ♮

m,n,θ ⊂ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn be defined as follows: f ∈ Λ♮

m,n,θ

iff f

  • xi +

1 2 , yj − 1 2

  • = f
  • xi −

1 2 , yj + 1 2

  • n the hyperplane xi + θyj = 0.

Set ϕ♮ : Λθ → Λ♮

m,n,θ ,

ϕ♮(f)(p, q) := f(F−1(p, q)) where the map F : {(λ1, . . . , λm, µ1, . . . , µn)} → Cm+n is given by “Frobenius coordinates”:    pi = λi − θ(i − 1

2 ) − 1 2 (n − θm)

1 ≤ i ≤ m, qj = µ′

j − θ−1(j − 1 2 ) + 1 2 (θ−1n + m)

1 ≤ j ≤ n. sJ⋆

λ := ϕ♮(J⋆ λ)

shifted super Jack polynomials λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ sJ⋆

λ = 0. 45 / 104

slide-46
SLIDE 46

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

The algebra Λ♮

m,n,θ and the polynomials sJ⋆ λ

Let Λ♮

m,n,θ ⊂ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn be defined as follows: f ∈ Λ♮

m,n,θ

iff f

  • xi +

1 2 , yj − 1 2

  • = f
  • xi −

1 2 , yj + 1 2

  • n the hyperplane xi + θyj = 0.

Set ϕ♮ : Λθ → Λ♮

m,n,θ ,

ϕ♮(f)(p, q) := f(F−1(p, q)) where the map F : {(λ1, . . . , λm, µ1, . . . , µn)} → Cm+n is given by “Frobenius coordinates”:    pi = λi − θ(i − 1

2 ) − 1 2 (n − θm)

1 ≤ i ≤ m, qj = µ′

j − θ−1(j − 1 2 ) + 1 2 (θ−1n + m)

1 ≤ j ≤ n. sJ⋆

λ := ϕ♮(J⋆ λ)

shifted super Jack polynomials λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ sJ⋆

λ = 0. 46 / 104

slide-47
SLIDE 47

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Shifted super Jack polynomials

The algebra Λ♮

m,n,θ and the polynomials sJ⋆ λ

Let Λ♮

m,n,θ ⊂ C[x1, . . . , xm, y1, . . . , yn]Sm×Sn be defined as follows: f ∈ Λ♮

m,n,θ

iff f

  • xi +

1 2 , yj − 1 2

  • = f
  • xi −

1 2 , yj + 1 2

  • n the hyperplane xi + θyj = 0.

Set ϕ♮ : Λθ → Λ♮

m,n,θ ,

ϕ♮(f)(p, q) := f(F−1(p, q)) where the map F : {(λ1, . . . , λm, µ1, . . . , µn)} → Cm+n is given by “Frobenius coordinates”:    pi = λi − θ(i − 1

2 ) − 1 2 (n − θm)

1 ≤ i ≤ m, qj = µ′

j − θ−1(j − 1 2 ) + 1 2 (θ−1n + m)

1 ≤ j ≤ n. sJ⋆

λ := ϕ♮(J⋆ λ)

shifted super Jack polynomials λ = (λ1, λ2, . . .) such that λm+1 > n ⇒ sJ⋆

λ = 0. 47 / 104

slide-48
SLIDE 48

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Generalities on Lie superalgebras

SVec: symmetric monoidal category SVec of Z/2-graded vector spaces. Z/2 = {0, 1} V ∈ objSVec V = V0 ⊕ V1. MorSVec(V, W) =

  • T ∈ HomC(V, W) : TV0 ⊂ W0 and TV1 ⊂ W1
  • V ⊗ W → W ⊗ V , v ⊗ w → (−1)|v|·|w|w ⊗ v.

S(V ) ∼ = S(V0) ⊗ Λ(V1) , P(V ) = S(V ∗) Lie superalgebra: g = g0 ⊕ g1 such that (−1)|x|·|z|[x, [y, z]] + (−1)|y|·|x|[y, [z, x]] + (−1)|z|·|y|[z, [x, y]] = 0. V = V0 ⊕V1 EndC(V ) = End(V )0 ⊕End(V )1 is a Lie superalgebra: [S, T] = ST − (−1)|S|·|T |TS.

48 / 104

slide-49
SLIDE 49

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Generalities on Lie superalgebras

SVec: symmetric monoidal category SVec of Z/2-graded vector spaces. Z/2 = {0, 1} V ∈ objSVec V = V0 ⊕ V1. MorSVec(V, W) =

  • T ∈ HomC(V, W) : TV0 ⊂ W0 and TV1 ⊂ W1
  • V ⊗ W → W ⊗ V , v ⊗ w → (−1)|v|·|w|w ⊗ v.

S(V ) ∼ = S(V0) ⊗ Λ(V1) , P(V ) = S(V ∗) Lie superalgebra: g = g0 ⊕ g1 such that (−1)|x|·|z|[x, [y, z]] + (−1)|y|·|x|[y, [z, x]] + (−1)|z|·|y|[z, [x, y]] = 0. V = V0 ⊕V1 EndC(V ) = End(V )0 ⊕End(V )1 is a Lie superalgebra: [S, T] = ST − (−1)|S|·|T |TS.

49 / 104

slide-50
SLIDE 50

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Generalities on Lie superalgebras

SVec: symmetric monoidal category SVec of Z/2-graded vector spaces. Z/2 = {0, 1} V ∈ objSVec V = V0 ⊕ V1. MorSVec(V, W) =

  • T ∈ HomC(V, W) : TV0 ⊂ W0 and TV1 ⊂ W1
  • V ⊗ W → W ⊗ V , v ⊗ w → (−1)|v|·|w|w ⊗ v.

S(V ) ∼ = S(V0) ⊗ Λ(V1) , P(V ) = S(V ∗) Lie superalgebra: g = g0 ⊕ g1 such that (−1)|x|·|z|[x, [y, z]] + (−1)|y|·|x|[y, [z, x]] + (−1)|z|·|y|[z, [x, y]] = 0. V = V0 ⊕V1 EndC(V ) = End(V )0 ⊕End(V )1 is a Lie superalgebra: [S, T] = ST − (−1)|S|·|T |TS.

50 / 104

slide-51
SLIDE 51

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Generalities on Lie superalgebras

SVec: symmetric monoidal category SVec of Z/2-graded vector spaces. Z/2 = {0, 1} V ∈ objSVec V = V0 ⊕ V1. MorSVec(V, W) =

  • T ∈ HomC(V, W) : TV0 ⊂ W0 and TV1 ⊂ W1
  • V ⊗ W → W ⊗ V , v ⊗ w → (−1)|v|·|w|w ⊗ v.

S(V ) ∼ = S(V0) ⊗ Λ(V1) , P(V ) = S(V ∗) Lie superalgebra: g = g0 ⊕ g1 such that (−1)|x|·|z|[x, [y, z]] + (−1)|y|·|x|[y, [z, x]] + (−1)|z|·|y|[z, [x, y]] = 0. V = V0 ⊕V1 EndC(V ) = End(V )0 ⊕End(V )1 is a Lie superalgebra: [S, T] = ST − (−1)|S|·|T |TS.

51 / 104

slide-52
SLIDE 52

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Generalities on Lie superalgebras

SVec: symmetric monoidal category SVec of Z/2-graded vector spaces. Z/2 = {0, 1} V ∈ objSVec V = V0 ⊕ V1. MorSVec(V, W) =

  • T ∈ HomC(V, W) : TV0 ⊂ W0 and TV1 ⊂ W1
  • V ⊗ W → W ⊗ V , v ⊗ w → (−1)|v|·|w|w ⊗ v.

S(V ) ∼ = S(V0) ⊗ Λ(V1) , P(V ) = S(V ∗) Lie superalgebra: g = g0 ⊕ g1 such that (−1)|x|·|z|[x, [y, z]] + (−1)|y|·|x|[y, [z, x]] + (−1)|z|·|y|[z, [x, y]] = 0. V = V0 ⊕V1 EndC(V ) = End(V )0 ⊕End(V )1 is a Lie superalgebra: [S, T] = ST − (−1)|S|·|T |TS.

52 / 104

slide-53
SLIDE 53

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

Generalities on Lie superalgebras

SVec: symmetric monoidal category SVec of Z/2-graded vector spaces. Z/2 = {0, 1} V ∈ objSVec V = V0 ⊕ V1. MorSVec(V, W) =

  • T ∈ HomC(V, W) : TV0 ⊂ W0 and TV1 ⊂ W1
  • V ⊗ W → W ⊗ V , v ⊗ w → (−1)|v|·|w|w ⊗ v.

S(V ) ∼ = S(V0) ⊗ Λ(V1) , P(V ) = S(V ∗) Lie superalgebra: g = g0 ⊕ g1 such that (−1)|x|·|z|[x, [y, z]] + (−1)|y|·|x|[y, [z, x]] + (−1)|z|·|y|[z, [x, y]] = 0. V = V0 ⊕V1 EndC(V ) = End(V )0 ⊕End(V )1 is a Lie superalgebra: [S, T] = ST − (−1)|S|·|T |TS.

53 / 104

slide-54
SLIDE 54

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Root system V = Cm|n glm|n = End(Cm|n). g = glm|n ⇒ g = n− ⊕ h ⊕ n+, where n± =

  • α∈Φ±

gα for Φ± = Φ±

0 ∪ Φ± 1 .

Φ+

0 = {εi − εj : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n},

Φ+

1 = {εi − εj : 1 ≤ i ≤ m < j ≤ m + n}.

Φ−

0 = −Φ+ 0 , Φ− 1 = −Φ+ 1 .

Invariant form X = A B C D

  • ⇒

str(X) = tr(A) − tr(D). X, Y ∈ glm|n ⇒ κ(X, Y ) = str(XY ) is a nondegenerate invariant form.

54 / 104

slide-55
SLIDE 55

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Root system V = Cm|n glm|n = End(Cm|n). g = glm|n ⇒ g = n− ⊕ h ⊕ n+, where n± =

  • α∈Φ±

gα for Φ± = Φ±

0 ∪ Φ± 1 .

Φ+

0 = {εi − εj : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n},

Φ+

1 = {εi − εj : 1 ≤ i ≤ m < j ≤ m + n}.

Φ−

0 = −Φ+ 0 , Φ− 1 = −Φ+ 1 .

Invariant form X = A B C D

  • ⇒

str(X) = tr(A) − tr(D). X, Y ∈ glm|n ⇒ κ(X, Y ) = str(XY ) is a nondegenerate invariant form.

55 / 104

slide-56
SLIDE 56

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Root system V = Cm|n glm|n = End(Cm|n). g = glm|n ⇒ g = n− ⊕ h ⊕ n+, where n± =

  • α∈Φ±

gα for Φ± = Φ±

0 ∪ Φ± 1 .

Φ+

0 = {εi − εj : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n},

Φ+

1 = {εi − εj : 1 ≤ i ≤ m < j ≤ m + n}.

Φ−

0 = −Φ+ 0 , Φ− 1 = −Φ+ 1 .

Invariant form X = A B C D

  • ⇒

str(X) = tr(A) − tr(D). X, Y ∈ glm|n ⇒ κ(X, Y ) = str(XY ) is a nondegenerate invariant form.

56 / 104

slide-57
SLIDE 57

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Root system V = Cm|n glm|n = End(Cm|n). g = glm|n ⇒ g = n− ⊕ h ⊕ n+, where n± =

  • α∈Φ±

gα for Φ± = Φ±

0 ∪ Φ± 1 .

Φ+

0 = {εi − εj : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n},

Φ+

1 = {εi − εj : 1 ≤ i ≤ m < j ≤ m + n}.

Φ−

0 = −Φ+ 0 , Φ− 1 = −Φ+ 1 .

Invariant form X = A B C D

  • ⇒

str(X) = tr(A) − tr(D). X, Y ∈ glm|n ⇒ κ(X, Y ) = str(XY ) is a nondegenerate invariant form.

57 / 104

slide-58
SLIDE 58

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Root system V = Cm|n glm|n = End(Cm|n). g = glm|n ⇒ g = n− ⊕ h ⊕ n+, where n± =

  • α∈Φ±

gα for Φ± = Φ±

0 ∪ Φ± 1 .

Φ+

0 = {εi − εj : 1 ≤ i < j ≤ m or m + 1 ≤ i < j ≤ m + n},

Φ+

1 = {εi − εj : 1 ≤ i ≤ m < j ≤ m + n}.

Φ−

0 = −Φ+ 0 , Φ− 1 = −Φ+ 1 .

Invariant form X = A B C D

  • ⇒

str(X) = tr(A) − tr(D). X, Y ∈ glm|n ⇒ κ(X, Y ) = str(XY ) is a nondegenerate invariant form.

58 / 104

slide-59
SLIDE 59

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Highest weight modules of glm|n Every irreducible finite dimensional representation of glm|n is a highest weight module Vλ where λ = λ1ε1 + · · · + λm+nεm+n satisfies λi ∈ Z and λ1 ≥ · · · ≥ λm and λm+1 ≥ · · · ≥ λm+n. Signed Sd-action on V ⊗d σ · (v1 ⊗ · · · ⊗ vd) = (−1)ǫ(σ−1;v1,...,vd)vσ−1(1) ⊗ · · · ⊗ vσ−1(d) where ǫ(σ; v1, . . . , vd) =

  • 1≤r<s≤d

σ(r)>σ(s)

|vσ(r)| · |vσ(s)|.

59 / 104

slide-60
SLIDE 60

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|n

Highest weight modules of glm|n Every irreducible finite dimensional representation of glm|n is a highest weight module Vλ where λ = λ1ε1 + · · · + λm+nεm+n satisfies λi ∈ Z and λ1 ≥ · · · ≥ λm and λm+1 ≥ · · · ≥ λm+n. Signed Sd-action on V ⊗d σ · (v1 ⊗ · · · ⊗ vd) = (−1)ǫ(σ−1;v1,...,vd)vσ−1(1) ⊗ · · · ⊗ vσ−1(d) where ǫ(σ; v1, . . . , vd) =

  • 1≤r<s≤d

σ(r)>σ(s)

|vσ(r)| · |vσ(s)|.

60 / 104

slide-61
SLIDE 61

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|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 = Cm|n. (Sergeev ’84, Berele–Regev ’87) As glm|n × Sd-module, V ⊗d ∼ =

  • D∈H(m,n,d)

VD ⊗ UD h.w. of VD = ♭1ε1 + · · · + ♭mεm + ♭′

1 − mεm+1 + · · · + ♭′ n − mεm+n

where ♭′

i − m := max{♭′ i − m, 0}.

∈ H(2, 3, 16) 7ε1 + 5ε2 + 2ε3 + ε4 + ε5

61 / 104

slide-62
SLIDE 62

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|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 = Cm|n. (Sergeev ’84, Berele–Regev ’87) As glm|n × Sd-module, V ⊗d ∼ =

  • D∈H(m,n,d)

VD ⊗ UD h.w. of VD = ♭1ε1 + · · · + ♭mεm + ♭′

1 − mεm+1 + · · · + ♭′ n − mεm+n

where ♭′

i − m := max{♭′ i − m, 0}.

∈ H(2, 3, 16) 7ε1 + 5ε2 + 2ε3 + ε4 + ε5

62 / 104

slide-63
SLIDE 63

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|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 = Cm|n. (Sergeev ’84, Berele–Regev ’87) As glm|n × Sd-module, V ⊗d ∼ =

  • D∈H(m,n,d)

VD ⊗ UD h.w. of VD = ♭1ε1 + · · · + ♭mεm + ♭′

1 − mεm+1 + · · · + ♭′ n − mεm+n

where ♭′

i − m := max{♭′ i − m, 0}.

∈ H(2, 3, 16) 7ε1 + 5ε2 + 2ε3 + ε4 + ε5

63 / 104

slide-64
SLIDE 64

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|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 = Cm|n. (Sergeev ’84, Berele–Regev ’87) As glm|n × Sd-module, V ⊗d ∼ =

  • D∈H(m,n,d)

VD ⊗ UD h.w. of VD = ♭1ε1 + · · · + ♭mεm + ♭′

1 − mεm+1 + · · · + ♭′ n − mεm+n

where ♭′

i − m := max{♭′ i − m, 0}.

∈ H(2, 3, 16) 7ε1 + 5ε2 + 2ε3 + ε4 + ε5

64 / 104

slide-65
SLIDE 65

The spectrum and interpolation polynomials Lie superalgebras Main Theorem Deformed CMS operators Generalities on Lie superalgebras The Lie superalgebra glm|n

The Lie superalgebra glm|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 = Cm|n. (Sergeev ’84, Berele–Regev ’87) As glm|n × Sd-module, V ⊗d ∼ =

  • D∈H(m,n,d)

VD ⊗ UD h.w. of VD = ♭1ε1 + · · · + ♭mεm + ♭′

1 − mεm+1 + · · · + ♭′ n − mεm+n

where ♭′

i − m := max{♭′ i − m, 0}.

∈ H(2, 3, 16) 7ε1 + 5ε2 + 2ε3 + ε4 + ε5

65 / 104

slide-66
SLIDE 66

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

66 / 104

slide-67
SLIDE 67

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

67 / 104

slide-68
SLIDE 68

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

68 / 104

slide-69
SLIDE 69

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

69 / 104

slide-70
SLIDE 70

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

70 / 104

slide-71
SLIDE 71

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

71 / 104

slide-72
SLIDE 72

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

72 / 104

slide-73
SLIDE 73

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The supersymmetric pair
  • glm|2n, ospm|2n
  • The Lie superalgebra ospm|2n

J2 := 1 −1

  • , J2n = diag(J2, . . . , J2
  • n times

) Θ : glm|2n → glm|2n , A B C D

  • →
  • −At

−CtJ2n −J2nBt J2nDtJ2n

  • spm|2n =
  • X ∈ glm|2n : Θ(X) = X
  • g := glm|2n, k := ospm|2n ⇒ g = k ⊕ a ⊕ n.

Proposition (Sahi–S.) For every D ∈ H(m, 2n, d), the irreducible g-module V ∗

2λ ⊂ P(W) contains a

unique (up to scalar) k-fixed vector 0 = zD. (In fact zD

  • a = sJD for θ = 1

2 .) (Alldridge–Schmitter ’13) Cartan–Helgason (assuming the highest weight is “high enough”).

  • Does not imply the above proposition.

73 / 104

slide-74
SLIDE 74

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The eigenvalue problem

Decomposing S(S2(W)) where W = S2(V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a glm|n-module, Sd(W) =

  • D∈H(m,n,d)

V2D. The Capelli basis D ∈ H(m, n, d)

  • V2D ⊂ S(W)

h.w. = λ V ∗

2D ⊂ P(W)

h.w. = λ∗ λ Dλ ∈ PD(W)glm|2n Capelli basis. Vµ∗ ⊂ P(W) ⇒ Dλ : Vµ∗ → Vµ∗ acts by a scalar cλ(µ∗). cλ( · ) ∈ P(a∗) ∼ = P(a). Theorem (Sahi–S.) cλ = zD

  • a + lower degree terms.

74 / 104

slide-75
SLIDE 75

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The eigenvalue problem

Decomposing S(S2(W)) where W = S2(V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a glm|n-module, Sd(W) =

  • D∈H(m,n,d)

V2D. The Capelli basis D ∈ H(m, n, d)

  • V2D ⊂ S(W)

h.w. = λ V ∗

2D ⊂ P(W)

h.w. = λ∗ λ Dλ ∈ PD(W)glm|2n Capelli basis. Vµ∗ ⊂ P(W) ⇒ Dλ : Vµ∗ → Vµ∗ acts by a scalar cλ(µ∗). cλ( · ) ∈ P(a∗) ∼ = P(a). Theorem (Sahi–S.) cλ = zD

  • a + lower degree terms.

75 / 104

slide-76
SLIDE 76

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The eigenvalue problem

Decomposing S(S2(W)) where W = S2(V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a glm|n-module, Sd(W) =

  • D∈H(m,n,d)

V2D. The Capelli basis D ∈ H(m, n, d)

  • V2D ⊂ S(W)

h.w. = λ V ∗

2D ⊂ P(W)

h.w. = λ∗ λ Dλ ∈ PD(W)glm|2n Capelli basis. Vµ∗ ⊂ P(W) ⇒ Dλ : Vµ∗ → Vµ∗ acts by a scalar cλ(µ∗). cλ( · ) ∈ P(a∗) ∼ = P(a). Theorem (Sahi–S.) cλ = zD

  • a + lower degree terms.

76 / 104

slide-77
SLIDE 77

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The eigenvalue problem

Decomposing S(S2(W)) where W = S2(V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a glm|n-module, Sd(W) =

  • D∈H(m,n,d)

V2D. The Capelli basis D ∈ H(m, n, d)

  • V2D ⊂ S(W)

h.w. = λ V ∗

2D ⊂ P(W)

h.w. = λ∗ λ Dλ ∈ PD(W)glm|2n Capelli basis. Vµ∗ ⊂ P(W) ⇒ Dλ : Vµ∗ → Vµ∗ acts by a scalar cλ(µ∗). cλ( · ) ∈ P(a∗) ∼ = P(a). Theorem (Sahi–S.) cλ = zD

  • a + lower degree terms.

77 / 104

slide-78
SLIDE 78

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The eigenvalue problem

Decomposing S(S2(W)) where W = S2(V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a glm|n-module, Sd(W) =

  • D∈H(m,n,d)

V2D. The Capelli basis D ∈ H(m, n, d)

  • V2D ⊂ S(W)

h.w. = λ V ∗

2D ⊂ P(W)

h.w. = λ∗ λ Dλ ∈ PD(W)glm|2n Capelli basis. Vµ∗ ⊂ P(W) ⇒ Dλ : Vµ∗ → Vµ∗ acts by a scalar cλ(µ∗). cλ( · ) ∈ P(a∗) ∼ = P(a). Theorem (Sahi–S.) cλ = zD

  • a + lower degree terms.

78 / 104

slide-79
SLIDE 79

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The eigenvalue problem

Decomposing S(S2(W)) where W = S2(V ) Theorem (Brini–Huang–Teolis ’92, Cheng–Wang ’01) As a glm|n-module, Sd(W) =

  • D∈H(m,n,d)

V2D. The Capelli basis D ∈ H(m, n, d)

  • V2D ⊂ S(W)

h.w. = λ V ∗

2D ⊂ P(W)

h.w. = λ∗ λ Dλ ∈ PD(W)glm|2n Capelli basis. Vµ∗ ⊂ P(W) ⇒ Dλ : Vµ∗ → Vµ∗ acts by a scalar cλ(µ∗). cλ( · ) ∈ P(a∗) ∼ = P(a). Theorem (Sahi–S.) cλ = zD

  • a + lower degree terms.

79 / 104

slide-80
SLIDE 80

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • Relation with shifted super Jack polynomials

µ∗ D = Dµ∗ = (♭1, ♭2, . . .) Theorem (Sahi–S.) cλ(µ∗) is a polynomial in (♭1, . . . , ♭m, ♭′

1, . . . , ♭′ n).

Up to the Frobenius coordinates, cλ = sJ⋆

λ for θ = 1 2.

   pi = ♭i − θ(i − 1

2) − 1 2(n − θm)

1 ≤ i ≤ m, qj = ♭′

j − θ−1(j − 1 2) + 1 2(θ−1n + m)

1 ≤ j ≤ n.

80 / 104

slide-81
SLIDE 81

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • Relation with shifted super Jack polynomials

µ∗ D = Dµ∗ = (♭1, ♭2, . . .) Theorem (Sahi–S.) cλ(µ∗) is a polynomial in (♭1, . . . , ♭m, ♭′

1, . . . , ♭′ n).

Up to the Frobenius coordinates, cλ = sJ⋆

λ for θ = 1 2.

   pi = ♭i − θ(i − 1

2) − 1 2(n − θm)

1 ≤ i ≤ m, qj = ♭′

j − θ−1(j − 1 2) + 1 2(θ−1n + m)

1 ≤ j ≤ n.

81 / 104

slide-82
SLIDE 82

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • Relation with shifted super Jack polynomials

µ∗ D = Dµ∗ = (♭1, ♭2, . . .) Theorem (Sahi–S.) cλ(µ∗) is a polynomial in (♭1, . . . , ♭m, ♭′

1, . . . , ♭′ n).

Up to the Frobenius coordinates, cλ = sJ⋆

λ for θ = 1 2.

   pi = ♭i − θ(i − 1

2) − 1 2(n − θm)

1 ≤ i ≤ m, qj = ♭′

j − θ−1(j − 1 2) + 1 2(θ−1n + m)

1 ≤ j ≤ n.

82 / 104

slide-83
SLIDE 83

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Action of glm|n on P(W): Ei,j ∈ glm|n

  • r

(−1)|i|+|i|·|j|yr,i∂r,j ∈ PD(W). ρ : glm|n → PD(W)

  • ρ : U(glm|n) → PD(W).

ρ

  • Z(glm|n)
  • ⊂ PD(W)glm|n.
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|n?

83 / 104

slide-84
SLIDE 84

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Action of glm|n on P(W): Ei,j ∈ glm|n

  • r

(−1)|i|+|i|·|j|yr,i∂r,j ∈ PD(W). ρ : glm|n → PD(W)

  • ρ : U(glm|n) → PD(W).

ρ

  • Z(glm|n)
  • ⊂ PD(W)glm|n.
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|n?

84 / 104

slide-85
SLIDE 85

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Action of glm|n on P(W): Ei,j ∈ glm|n

  • r

(−1)|i|+|i|·|j|yr,i∂r,j ∈ PD(W). ρ : glm|n → PD(W)

  • ρ : U(glm|n) → PD(W).

ρ

  • Z(glm|n)
  • ⊂ PD(W)glm|n.
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|n?

85 / 104

slide-86
SLIDE 86

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Action of glm|n on P(W): Ei,j ∈ glm|n

  • r

(−1)|i|+|i|·|j|yr,i∂r,j ∈ PD(W). ρ : glm|n → PD(W)

  • ρ : U(glm|n) → PD(W).

ρ

  • Z(glm|n)
  • ⊂ PD(W)glm|n.
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|n?

86 / 104

slide-87
SLIDE 87

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|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.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0, where

Zd(glm|n) = Z(glm|n) ∩ Ud(glm|n). Difference with Howe–Umeda: Use of the Capelli basis vs. generators

  • f the algebra PD(W)glm|n.

87 / 104

slide-88
SLIDE 88

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|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.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0, where

Zd(glm|n) = Z(glm|n) ∩ Ud(glm|n). Difference with Howe–Umeda: Use of the Capelli basis vs. generators

  • f the algebra PD(W)glm|n.

88 / 104

slide-89
SLIDE 89

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|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.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0, where

Zd(glm|n) = Z(glm|n) ∩ Ud(glm|n). Difference with Howe–Umeda: Use of the Capelli basis vs. generators

  • f the algebra PD(W)glm|n.

89 / 104

slide-90
SLIDE 90

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem
  • Question. Is it true that ρ
  • Z(glm|n)
  • = PD(W)glm|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.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0, where

Zd(glm|n) = Z(glm|n) ∩ Ud(glm|n). Difference with Howe–Umeda: Use of the Capelli basis vs. generators

  • f the algebra PD(W)glm|n.

90 / 104

slide-91
SLIDE 91

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

Passage to grading using PD(W) ∼ = P(W) ⊗ S(W). S(W) is generated by {xi,j}, satisfying xj,i = (−1)|i|·|j|xi,j. P(W) is generated by {yi,j}, satisfying yj,i = (−1)|i|·|j|yi,j. Schur–Weyl duality ⇒ the glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd, where ϕi,j =

r∈Im,n(−1)|r|+|i|·|j|yr,jxr,i and

ˇ ǫ(σ; t1, . . . , td) =

  • 1≤r<s≤d

|tr| · |tσ(s)| +

  • 1≤r<s≤d

σ(r)<σ(s)

|tσ(r)| · |tσ(s)|.

91 / 104

slide-92
SLIDE 92

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

Passage to grading using PD(W) ∼ = P(W) ⊗ S(W). S(W) is generated by {xi,j}, satisfying xj,i = (−1)|i|·|j|xi,j. P(W) is generated by {yi,j}, satisfying yj,i = (−1)|i|·|j|yi,j. Schur–Weyl duality ⇒ the glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd, where ϕi,j =

r∈Im,n(−1)|r|+|i|·|j|yr,jxr,i and

ˇ ǫ(σ; t1, . . . , td) =

  • 1≤r<s≤d

|tr| · |tσ(s)| +

  • 1≤r<s≤d

σ(r)<σ(s)

|tσ(r)| · |tσ(s)|.

92 / 104

slide-93
SLIDE 93

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

Passage to grading using PD(W) ∼ = P(W) ⊗ S(W). S(W) is generated by {xi,j}, satisfying xj,i = (−1)|i|·|j|xi,j. P(W) is generated by {yi,j}, satisfying yj,i = (−1)|i|·|j|yi,j. Schur–Weyl duality ⇒ the glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd, where ϕi,j =

r∈Im,n(−1)|r|+|i|·|j|yr,jxr,i and

ˇ ǫ(σ; t1, . . . , td) =

  • 1≤r<s≤d

|tr| · |tσ(s)| +

  • 1≤r<s≤d

σ(r)<σ(s)

|tσ(r)| · |tσ(s)|.

93 / 104

slide-94
SLIDE 94

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

Passage to grading using PD(W) ∼ = P(W) ⊗ S(W). S(W) is generated by {xi,j}, satisfying xj,i = (−1)|i|·|j|xi,j. P(W) is generated by {yi,j}, satisfying yj,i = (−1)|i|·|j|yi,j. Schur–Weyl duality ⇒ the glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd, where ϕi,j =

r∈Im,n(−1)|r|+|i|·|j|yr,jxr,i and

ˇ ǫ(σ; t1, . . . , td) =

  • 1≤r<s≤d

|tr| · |tσ(s)| +

  • 1≤r<s≤d

σ(r)<σ(s)

|tσ(r)| · |tσ(s)|.

94 / 104

slide-95
SLIDE 95

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

Passage to grading using PD(W) ∼ = P(W) ⊗ S(W). S(W) is generated by {xi,j}, satisfying xj,i = (−1)|i|·|j|xi,j. P(W) is generated by {yi,j}, satisfying yj,i = (−1)|i|·|j|yi,j. Schur–Weyl duality ⇒ the glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd, where ϕi,j =

r∈Im,n(−1)|r|+|i|·|j|yr,jxr,i and

ˇ ǫ(σ; t1, . . . , td) =

  • 1≤r<s≤d

|tr| · |tσ(s)| +

  • 1≤r<s≤d

σ(r)<σ(s)

|tσ(r)| · |tσ(s)|.

95 / 104

slide-96
SLIDE 96

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

96 / 104

slide-97
SLIDE 97

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

97 / 104

slide-98
SLIDE 98

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

98 / 104

slide-99
SLIDE 99

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) ⇒ tσ = tσ1 · · · tσℓ. 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

99 / 104

slide-100
SLIDE 100

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) ⇒ tσ = tσ1 · · · tσℓ. 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

100 / 104

slide-101
SLIDE 101

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) ⇒ tσ = tσ1 · · · tσℓ. 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

101 / 104

slide-102
SLIDE 102

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) ⇒ tσ = tσ1 · · · tσℓ. 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

102 / 104

slide-103
SLIDE 103

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • The Capelli problem

Theorem (Sahi–S.) ρ

  • Zd(glm|n)
  • = PDd(W)glm|n for every d ≥ 0.

Idea of proof

glm|n-invariants of P(W) ⊗ S(W) are spanned by tσ =

  • t1,...,td

(−1)ˇ

ǫ(σ;t1,...,td)ϕtσ(1),t1 · · · ϕtσ(d),td

, σ ∈ Sd. tσ depends only on the conjugacy class of σ ∈ Sd. σ = σ1 . . . σℓ, σk = (dk + 1, . . . , dk+1) ⇒ tσ = tσ1 · · · tσℓ. 0 = d1 < · · · < dℓ+1 = d. Ei,j = (−1)|i|·|j|Ei,j, Zd = str(Ed) ∈ U(glm|n). Gelfand elements : Zd ∈ Zd(glm|n). P(W) ⊗ S(W) m − → PD(W) , p ⊗ ∂ → p∂. σ = (1, . . . , d) ⇒

  • rd
  • ρ(Zd) − m(tσ)
  • < d.

103 / 104

slide-104
SLIDE 104

The spectrum and interpolation polynomials Lie superalgebras Main Theorem The supersymmetric pair

  • glm|2n, ospm|2n
  • Thank you!

104 / 104