Clebsch-Gordan Coefficients and Principal Series Representations - - PowerPoint PPT Presentation

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Clebsch-Gordan Coefficients and Principal Series Representations

Clebsch-Gordan Coefficients and Principal Series Representations

Zhuohui Zhang, Rutgers University

Workshop on Automorphic Forms, Representations of Lie Groups and Several Complex Variables CAS

July 12, 2018

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Clebsch-Gordan Coefficients and Principal Series Representations Overview

Overview

This talk will be about:

◮ Compact picture of principal series, on SL(2, R);

SLIDE 3

Clebsch-Gordan Coefficients and Principal Series Representations Overview

Overview

This talk will be about:

◮ Compact picture of principal series, on SL(2, R); ◮ On rank 1 groups, SU(2, 1);

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Clebsch-Gordan Coefficients and Principal Series Representations Overview

Overview

This talk will be about:

◮ Compact picture of principal series, on SL(2, R); ◮ On rank 1 groups, SU(2, 1); ◮ On rank 2 groups, Sp(4, R);

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Clebsch-Gordan Coefficients and Principal Series Representations Overview

Overview

This talk will be about:

◮ Compact picture of principal series, on SL(2, R); ◮ On rank 1 groups, SU(2, 1); ◮ On rank 2 groups, Sp(4, R); ◮ Calculating intertwining operators for principal series.

SLIDE 6

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

What is a Compact Picture?

◮ Irreducible (g, K)-modules of a real reductive Lie group G can be found as a

sub/quotient of a principal series representations IP(δ ⊗ λ) IP(δ ⊗ λ) =

• f : G −

→ Vσ|f (gman) = a−λ−ρδ(m)−1f (g)

• Induction data: a parabolic subgroup P = MAN, a representation (δ, Vδ) of M,

and a character λ of A. If δ ⊗ λ is a character, write χδ,λ.

SLIDE 7

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

What is a Compact Picture?

◮ Irreducible (g, K)-modules of a real reductive Lie group G can be found as a

sub/quotient of a principal series representations IP(δ ⊗ λ) IP(δ ⊗ λ) =

• f : G −

→ Vσ|f (gman) = a−λ−ρδ(m)−1f (g)

• Induction data: a parabolic subgroup P = MAN, a representation (δ, Vδ) of M,

and a character λ of A. If δ ⊗ λ is a character, write χδ,λ.

◮ The irreducible K-representations occurring in the subspace of K-finite smooth

vectors of a representation of G are called K-types.

SLIDE 8

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

What is a Compact Picture?

◮ Irreducible (g, K)-modules of a real reductive Lie group G can be found as a

sub/quotient of a principal series representations IP(δ ⊗ λ) IP(δ ⊗ λ) =

• f : G −

→ Vσ|f (gman) = a−λ−ρδ(m)−1f (g)

• Induction data: a parabolic subgroup P = MAN, a representation (δ, Vδ) of M,

and a character λ of A. If δ ⊗ λ is a character, write χδ,λ.

◮ The irreducible K-representations occurring in the subspace of K-finite smooth

vectors of a representation of G are called K-types.

◮ Since G = KMAN, as a (g, K)-module, IP(δ ⊗ λ) can be embedded into the

following space of functions on K C ∞

δ (K) =

• f : K −

→ Vδ|f (km) = δ(m)−1f (k)

SLIDE 9

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

What is a Compact Picture?

◮ Irreducible (g, K)-modules of a real reductive Lie group G can be found as a

sub/quotient of a principal series representations IP(δ ⊗ λ) IP(δ ⊗ λ) =

• f : G −

→ Vσ|f (gman) = a−λ−ρδ(m)−1f (g)

• Induction data: a parabolic subgroup P = MAN, a representation (δ, Vδ) of M,

and a character λ of A. If δ ⊗ λ is a character, write χδ,λ.

◮ The irreducible K-representations occurring in the subspace of K-finite smooth

vectors of a representation of G are called K-types.

◮ Since G = KMAN, as a (g, K)-module, IP(δ ⊗ λ) can be embedded into the

following space of functions on K C ∞

δ (K) =

• f : K −

→ Vδ|f (km) = δ(m)−1f (k)

• ◮ The embedding is given by the restriction of function, this is called the compact

picture of principal series.

SLIDE 10

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

What is a Compact Picture?

◮ Irreducible (g, K)-modules of a real reductive Lie group G can be found as a

sub/quotient of a principal series representations IP(δ ⊗ λ) IP(δ ⊗ λ) =

• f : G −

→ Vσ|f (gman) = a−λ−ρδ(m)−1f (g)

• Induction data: a parabolic subgroup P = MAN, a representation (δ, Vδ) of M,

and a character λ of A. If δ ⊗ λ is a character, write χδ,λ.

◮ The irreducible K-representations occurring in the subspace of K-finite smooth

vectors of a representation of G are called K-types.

◮ Since G = KMAN, as a (g, K)-module, IP(δ ⊗ λ) can be embedded into the

following space of functions on K C ∞

δ (K) =

• f : K −

→ Vδ|f (km) = δ(m)−1f (k)

• ◮ The embedding is given by the restriction of function, this is called the compact

picture of principal series.

◮ Why? Representations of K = U(n), Sp(n) etc. are well studied, easy to work out

the structure explicitly.

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Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ Borel subgroup: B =

• (−1)ǫ √y

1/√y

1 x

0 1

• |x ∈ R, y > 0

SLIDE 12

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ Borel subgroup: B =

• (−1)ǫ √y

1/√y

1 x

0 1

• |x ∈ R, y > 0
• ◮ Characters on the Borel: χδ,λ : (−1)ǫ √y

1/√y

1 x

0 1

• → (−1)ǫδyλ/2

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Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ Borel subgroup: B =

• (−1)ǫ √y

1/√y

1 x

0 1

• |x ∈ R, y > 0
• ◮ Characters on the Borel: χδ,λ : (−1)ǫ √y

1/√y

1 x

0 1

• → (−1)ǫδyλ/2

◮ We define the principal series to be the set of K-finite smooth functions

I(χδ,λ) =

• f : G → C|f (gb) = χδ,λ+1(b)−1f (g)
• where G acts by left translation.

SLIDE 14

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ Borel subgroup: B =

• (−1)ǫ √y

1/√y

1 x

0 1

• |x ∈ R, y > 0
• ◮ Characters on the Borel: χδ,λ : (−1)ǫ √y

1/√y

1 x

0 1

• → (−1)ǫδyλ/2

◮ We define the principal series to be the set of K-finite smooth functions

I(χδ,λ) =

• f : G → C|f (gb) = χδ,λ+1(b)−1f (g)
• where G acts by left translation.

◮ Decomposition into K-types:

I(χδ,λ) =

• k≡δ

mod 2

Ceikθ

SLIDE 15

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ We can find the following irreducible (g, K)-modules in I(χδ,λ) if

λ + 1 ≡ δ mod 2:

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Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ We can find the following irreducible (g, K)-modules in I(χδ,λ) if

λ + 1 ≡ δ mod 2:

◮ The (limit of) discrete series representations D± |λ|:

D±

|λ| =

• k=|λ|+1

k≡δ mod 2

Ce±ikθ.

SLIDE 17

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ We can find the following irreducible (g, K)-modules in I(χδ,λ) if

λ + 1 ≡ δ mod 2:

◮ The (limit of) discrete series representations D± |λ|:

D±

|λ| =

• k=|λ|+1

k≡δ mod 2

Ce±ikθ.

◮ W|λ| of finite dimension |λ|.

SLIDE 18

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ We can find the following irreducible (g, K)-modules in I(χδ,λ) if

λ + 1 ≡ δ mod 2:

◮ The (limit of) discrete series representations D± |λ|:

D±

|λ| =

• k=|λ|+1

k≡δ mod 2

Ce±ikθ.

◮ W|λ| of finite dimension |λ|. ◮ I(χδ,λ) is irreducible in all other cases.

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Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ λ > 0: D+

|λ| ⊕ D− |λ| ֒

→ I(χδ,λ) ։ W|λ|

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Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ λ > 0: D+

|λ| ⊕ D− |λ| ֒

→ I(χδ,λ) ։ W|λ| e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• 1

U+ U−

• e2iθ

U+

. . .

U+ U−

• e2ikθ

U−

SLIDE 21

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ λ > 0: D+

|λ| ⊕ D− |λ| ֒

→ I(χδ,λ) ։ W|λ| e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• 1

U+ U−

• e2iθ

U+

. . .

U+ U−

• e2ikθ

U−

• ◮ λ < 0: W|λ| ֒

→ I(χδ,λ) ։ D+

|λ| ⊕ D− |λ|

e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• U+

1

e2iθ

U+

• U−
• . . .

U+ U−

• e2ikθ

U−

SLIDE 22

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ λ > 0: D+

|λ| ⊕ D− |λ| ֒

→ I(χδ,λ) ։ W|λ| e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• 1

U+ U−

• e2iθ

U+

. . .

U+ U−

• e2ikθ

U−

• ◮ λ < 0: W|λ| ֒

→ I(χδ,λ) ։ D+

|λ| ⊕ D− |λ|

e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• U+

1

e2iθ

U+

• U−
• . . .

U+ U−

• e2ikθ

U−

• ◮ I(χ−1,0) = D+

0 ⊕ D−

(limit of discrete series)

SLIDE 23

Clebsch-Gordan Coefficients and Principal Series Representations Compact picture of Principal Series

SL(2, R) Example from Textbooks(Bargmann, 1940’s)

◮ λ > 0: D+

|λ| ⊕ D− |λ| ֒

→ I(χδ,λ) ։ W|λ| e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• 1

U+ U−

• e2iθ

U+

. . .

U+ U−

• e2ikθ

U−

• ◮ λ < 0: W|λ| ֒

→ I(χδ,λ) ։ D+

|λ| ⊕ D− |λ|

e−2ikθ

U+

. . .

U+ U−

• e−2iθ

U−

• U+

1

e2iθ

U+

• U−
• . . .

U+ U−

• e2ikθ

U−

• ◮ I(χ−1,0) = D+

0 ⊕ D−

(limit of discrete series) e−ikθ

U+

. . .

U+ U−

• e−iθ

U−

• eiθ

U+

. . .

U+ U−

• eikθ

U−

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Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ The Lie algebra u(2) is generated by Pauli matrices

γ0 = i 2 1 0

0 1

• , γ1 = i

2 0 1

1 0

• , γ2 = i

2 0 −i

i

• , γ3 = i

2 1

0 −1

• The matrices γ1, γ2, γ3 can be interpreted as rotations in R3 around x, y, z axis

respectively.

SLIDE 25

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ The Lie algebra u(2) is generated by Pauli matrices

γ0 = i 2 1 0

0 1

• , γ1 = i

2 0 1

1 0

• , γ2 = i

2 0 −i

i

• , γ3 = i

2 1

0 −1

• The matrices γ1, γ2, γ3 can be interpreted as rotations in R3 around x, y, z axis

respectively.

◮ Euler angles on U(2): a general element in U(2) can be expressed as

eζγ0eφγ3eθγ2eψγ3

SLIDE 26

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Representations of U(2): π(j,n) = Pol2j(C2) ⊗ detj+n. The action on

f ∈ Pol2j(C2) is given by f (z) → f (g−1z).

SLIDE 27

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Representations of U(2): π(j,n) = Pol2j(C2) ⊗ detj+n. The action on

f ∈ Pol2j(C2) is given by f (z) → f (g−1z).

◮ Highest weight (j, n), j, n half integers, j ≥ 0; weight vectors

v(j,n)

m

=

zj−m

1

zj+m

2

√

(j−m)!(j+m)! with −j ≤ m ≤ j.

SLIDE 28

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Representations of U(2): π(j,n) = Pol2j(C2) ⊗ detj+n. The action on

f ∈ Pol2j(C2) is given by f (z) → f (g−1z).

◮ Highest weight (j, n), j, n half integers, j ≥ 0; weight vectors

v(j,n)

m

=

zj−m

1

zj+m

2

√

(j−m)!(j+m)! with −j ≤ m ≤ j.

◮ Action of Pauli matrices: weight vectors are normalized such that

γ0v(j,n)

m

= inv(j,n)

m

, γ3v(j,n)

m

= imv(j,n)

m

(γ1 ± iγ2)vj

m = −i

• (j ∓ m)(j ± m + 1)vj

m±1

SLIDE 29

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Representations of U(2): π(j,n) = Pol2j(C2) ⊗ detj+n. The action on

f ∈ Pol2j(C2) is given by f (z) → f (g−1z).

◮ Highest weight (j, n), j, n half integers, j ≥ 0; weight vectors

v(j,n)

m

=

zj−m

1

zj+m

2

√

(j−m)!(j+m)! with −j ≤ m ≤ j.

◮ Action of Pauli matrices: weight vectors are normalized such that

γ0v(j,n)

m

= inv(j,n)

m

, γ3v(j,n)

m

= imv(j,n)

m

(γ1 ± iγ2)vj

m = −i

• (j ∓ m)(j ± m + 1)vj

m±1

◮ The raising and lowering operators

SLIDE 30

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Representations of U(2): π(j,n) = Pol2j(C2) ⊗ detj+n. The action on

f ∈ Pol2j(C2) is given by f (z) → f (g−1z).

◮ Highest weight (j, n), j, n half integers, j ≥ 0; weight vectors

v(j,n)

m

=

zj−m

1

zj+m

2

√

(j−m)!(j+m)! with −j ≤ m ≤ j.

◮ Action of Pauli matrices: weight vectors are normalized such that

γ0v(j,n)

m

= inv(j,n)

m

, γ3v(j,n)

m

= imv(j,n)

m

(γ1 ± iγ2)vj

m = −i

• (j ∓ m)(j ± m + 1)vj

m±1

◮ The raising and lowering operators ◮ 2j ≡ 0 mod 2:

−j

γ1+iγ2 . . . γ1+iγ2 γ1−iγ2

• −1

γ1+iγ2 γ1−iγ2

• γ1−iγ2

γ1−iγ2

• 1

γ1−iγ2

• γ1+iγ2 . . .

γ1+iγ2 γ1−iγ2

• j

γ1−iγ2

SLIDE 31

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Representations of U(2): π(j,n) = Pol2j(C2) ⊗ detj+n. The action on

f ∈ Pol2j(C2) is given by f (z) → f (g−1z).

◮ Highest weight (j, n), j, n half integers, j ≥ 0; weight vectors

v(j,n)

m

=

zj−m

1

zj+m

2

√

(j−m)!(j+m)! with −j ≤ m ≤ j.

◮ Action of Pauli matrices: weight vectors are normalized such that

γ0v(j,n)

m

= inv(j,n)

m

, γ3v(j,n)

m

= imv(j,n)

m

(γ1 ± iγ2)vj

m = −i

• (j ∓ m)(j ± m + 1)vj

m±1

◮ The raising and lowering operators ◮ 2j ≡ 0 mod 2:

−j

γ1+iγ2 . . . γ1+iγ2 γ1−iγ2

• −1

γ1+iγ2 γ1−iγ2

• γ1−iγ2

γ1−iγ2

• 1

γ1−iγ2

• γ1+iγ2 . . .

γ1+iγ2 γ1−iγ2

• j

γ1−iγ2

• ◮ 2j ≡ 1 mod 2:

−j

γ1+iγ2 . . . γ1+iγ2 γ1−iγ2

• − 1

2 γ1+iγ2 γ1−iγ2

• 1

2 γ1−iγ2

• γ1+iγ2 . . .

γ1+iγ2 γ1−iγ2

• j

γ1−iγ2

SLIDE 32

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Matrix coefficients: Wigner D-functions

D(j,n)

m1,m2(ζ, φ, θ, ψ) = vj m2, eζγ0eφγ3eθγ2eψγ3vj m1K

= cj

m1cj m2einζei(m1ψ+m2φ)d(j,n) m1,m2(θ).

in which

SLIDE 33

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Matrix coefficients: Wigner D-functions

D(j,n)

m1,m2(ζ, φ, θ, ψ) = vj m2, eζγ0eφγ3eθγ2eψγ3vj m1K

= cj

m1cj m2einζei(m1ψ+m2φ)d(j,n) m1,m2(θ).

in which

◮ cj m =

• (j + m)!(j − m)! is a normalization factor

SLIDE 34

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ Matrix coefficients: Wigner D-functions

D(j,n)

m1,m2(ζ, φ, θ, ψ) = vj m2, eζγ0eφγ3eθγ2eψγ3vj m1K

= cj

m1cj m2einζei(m1ψ+m2φ)d(j,n) m1,m2(θ).

in which

◮ cj m =

• (j + m)!(j − m)! is a normalization factor

◮ d(j,n) m1,m2(θ) is given by the hypergeometric sum:

d(j,n)

m1,m2(θ) = min(j−m2,j+m1)

• p=max(0,m1−m2)

(−1)m2−m1+p (j + m1 − p)!p!(m2 − m1 + p)!(j − m2 − p)! sinm2−m1+2p θ 2

• cos2j+m1−m2−2p

θ 2

• .

SLIDE 35

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ The Jacobi polynomials are defined as in Wolfram Function Site, NIST,

Abramovich-Stegun e.g. Pα,β

n

(x) = n + α n x + 1 2 n

2F1

• −n, −n − β, α + 1; x − 1

x + 1

• .

where n ≥ 0.

SLIDE 36

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

◮ The Jacobi polynomials are defined as in Wolfram Function Site, NIST,

Abramovich-Stegun e.g. Pα,β

n

(x) = n + α n x + 1 2 n

2F1

• −n, −n − β, α + 1; x − 1

x + 1

• .

where n ≥ 0.

◮ d(j,n)

m1,m2(θ) can be expressed in terms of Jacobi polynomials

d(j,n)

m1,m2(θ) =

• sin θ

2

m1−m2 cos θ

2

m1+m2 (j + m2)!(j − m2)! P(m1−m2,m1+m2)

j−m1

(cos θ)

SLIDE 37

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

Why we do this?

◮ Generating function for Jacobi polynomials

∞

• n=0

P(α−n,β−n)

n

(x)tn =

• 1 + 1

2 (x + 1)t α 1 + 1 2 (x − 1)t β

SLIDE 38

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The Compact Group U(2)

Why we do this?

◮ Generating function for Jacobi polynomials

∞

• n=0

P(α−n,β−n)

n

(x)tn =

• 1 + 1

2 (x + 1)t α 1 + 1 2 (x − 1)t β

◮ Product of Wigner D-functions: Clebsch-Gordan coefficients

D(j1,n1)

m11,m12D(j2,n2) m21,m22 =

• |j1−j2|≤J≤j1+j2

J−|j1−j2|∈Z

• J,M1

j1,m11,j2,m21 J,M2 j1,m12,j2,m22

• D(J,n1+n2)

M1,M2

The Clebsch-Gordan coefficients can be computed following a recursive procedure.

SLIDE 39

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The (g, K)-Module Structure Explicitly

◮ The (g, K) action can be written as differential operators on C ∞(K) acting on

Wigner D-functions.

SLIDE 40

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The (g, K)-Module Structure Explicitly

◮ The (g, K) action can be written as differential operators on C ∞(K) acting on

Wigner D-functions.

◮ For a K-type τ (j,n), calculate the homomorphism of K-representations

Te(g/k) ⊗ τ (j,n) ∼ = pC ⊗ τ (j,n) − → C ∞(K) which sends X ⊗ v to X · v. (compare SL(3, R) Buttcane-Miller ’17)

SLIDE 41

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2)

The (g, K)-Module Structure Explicitly

◮ The (g, K) action can be written as differential operators on C ∞(K) acting on

Wigner D-functions.

◮ For a K-type τ (j,n), calculate the homomorphism of K-representations

Te(g/k) ⊗ τ (j,n) ∼ = pC ⊗ τ (j,n) − → C ∞(K) which sends X ⊗ v to X · v. (compare SL(3, R) Buttcane-Miller ’17)

◮ The Cartan decomposition g = k ⊕ p puts a Z/2-grading on g. If rank of G and K

are equal, the roots fall into two subsets ∆ = ∆c ∪ ∆nc compact/noncompact roots respectively, depending on whether the root subspace gα is a subset of k or p.

SLIDE 42

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1)

◮ The group G = SU(n, 1) has real rank 1, and maximal compact subgroup

K = S(U(n) ⊗ U(1)).

SLIDE 43

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1)

◮ The group G = SU(n, 1) has real rank 1, and maximal compact subgroup

K = S(U(n) ⊗ U(1)).

◮ SU(2, 1) = {g ∈ SL(3, C)|¯

gtJg = J}, J = 12 −1

• ;

SLIDE 44

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1)

◮ The group G = SU(n, 1) has real rank 1, and maximal compact subgroup

K = S(U(n) ⊗ U(1)).

◮ SU(2, 1) = {g ∈ SL(3, C)|¯

gtJg = J}, J = 12 −1

• ;

◮ σ(g) = J(¯

g t)−1J−1 an antiholomorphic involution, SU(2, 1) = SL(3, C)σ;

SLIDE 45

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1)

◮ The group G = SU(n, 1) has real rank 1, and maximal compact subgroup

K = S(U(n) ⊗ U(1)).

◮ SU(2, 1) = {g ∈ SL(3, C)|¯

gtJg = J}, J = 12 −1

• ;

◮ σ(g) = J(¯

g t)−1J−1 an antiholomorphic involution, SU(2, 1) = SL(3, C)σ;

◮ θ(g) = (¯

g t)−1 a Cartan involution, K = U(2) (det)−1

• = G θ;

SLIDE 46

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1)

◮ The group G = SU(n, 1) has real rank 1, and maximal compact subgroup

K = S(U(n) ⊗ U(1)).

◮ SU(2, 1) = {g ∈ SL(3, C)|¯

gtJg = J}, J = 12 −1

• ;

◮ σ(g) = J(¯

g t)−1J−1 an antiholomorphic involution, SU(2, 1) = SL(3, C)σ;

◮ θ(g) = (¯

g t)−1 a Cartan involution, K = U(2) (det)−1

• = G θ;

◮ For SU(2, 1), ∆+

c = {α1}, ∆+ nc = {α2, α1 + α2}

α1 α2 α1 + α2 ̟1 ̟2

Green: ∆nc; Blue: ∆c

SLIDE 47

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1)

◮ The group G = SU(n, 1) has real rank 1, and maximal compact subgroup

K = S(U(n) ⊗ U(1)).

◮ SU(2, 1) = {g ∈ SL(3, C)|¯

gtJg = J}, J = 12 −1

• ;

◮ σ(g) = J(¯

g t)−1J−1 an antiholomorphic involution, SU(2, 1) = SL(3, C)σ;

◮ θ(g) = (¯

g t)−1 a Cartan involution, K = U(2) (det)−1

• = G θ;

◮ For SU(2, 1), ∆+

c = {α1}, ∆+ nc = {α2, α1 + α2}

α1 α2 α1 + α2 ̟1 ̟2

Green: ∆nc; Blue: ∆c

◮ The two irreducible pieces of pC are p±

C = span{u(1/2,±3/2) m

}m∈{−1/2,1/2}

SLIDE 48

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1) Induction Data

◮ Minimal parabolic P = MAN,

• eis

0 e−2is eis

cosh t 0 sinh t

sinh t 0 cosh t

• exp

iw z −iw

−¯ z 0 ¯ z iw z −iw

SLIDE 49

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1) Induction Data

◮ Minimal parabolic P = MAN,

• eis

0 e−2is eis

cosh t 0 sinh t

sinh t 0 cosh t

• exp

iw z −iw

−¯ z 0 ¯ z iw z −iw

• ◮ M ∼

= S1, character on M given by δ ∈ Z

SLIDE 50

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1) Induction Data

◮ Minimal parabolic P = MAN,

• eis

0 e−2is eis

cosh t 0 sinh t

sinh t 0 cosh t

• exp

iw z −iw

−¯ z 0 ¯ z iw z −iw

• ◮ M ∼

= S1, character on M given by δ ∈ Z

◮ A ∼

= R>0, character on A given by a λ ∈ C

SLIDE 51

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

Example: SU(2, 1) Induction Data

◮ Minimal parabolic P = MAN,

• eis

0 e−2is eis

cosh t 0 sinh t

sinh t 0 cosh t

• exp

iw z −iw

−¯ z 0 ¯ z iw z −iw

• ◮ M ∼

= S1, character on M given by δ ∈ Z

◮ A ∼

= R>0, character on A given by a λ ∈ C

◮ They define a character χδ,λ sending the above element to eiδt+λt.

SLIDE 52

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

◮ The principal series IP(χδ,λ) decompose into K-isotypic spaces:

IP(χδ,λ) =

• −3j+δ≤n≤3j+δ

τ (j,n)

SLIDE 53

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

◮ The principal series IP(χδ,λ) decompose into K-isotypic spaces:

IP(χδ,λ) =

• −3j+δ≤n≤3j+δ

τ (j,n)

◮ Each τ (j,n) has decomposition:

τ (j,n) =

• m2∈{−j,−j+1,...,j}

m1=(n−δ)/3

CD(j,n)

m1,m2

SLIDE 54

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

◮ The principal series IP(χδ,λ) decompose into K-isotypic spaces:

IP(χδ,λ) =

• −3j+δ≤n≤3j+δ

τ (j,n)

◮ Each τ (j,n) has decomposition:

τ (j,n) =

• m2∈{−j,−j+1,...,j}

m1=(n−δ)/3

CD(j,n)

m1,m2

◮ A better picture to draw: let n = δ + 3

2 l, j = k 2 , we can depict all the K-types on

a quadrant: {(k, l) ∈ Z≥0 × Z| − k ≤ l ≤ k and k ≡ l mod 2}.

SLIDE 55

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

◮ The homomorphism dl : p ⊗ τ (j,n) −

→ C ∞(K) is(compare Buttcane-Miller ’17): dl(u

( 1

2 ,± 3 2 )

m

)D(j,n)

m1,m2 =

1 2√2j + 1

• j0∈{± 1

2 }

j+j0,m2−m

J,m2, 1

2 ,−m

• qj0,∓κj0,∓(j, n, m1; λ)D

(j+j0,n∓ 3

2 )

m1∓ 1

2 ,m2−m

SLIDE 56

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

◮ The homomorphism dl : p ⊗ τ (j,n) −

→ C ∞(K) is(compare Buttcane-Miller ’17): dl(u

( 1

2 ,± 3 2 )

m

)D(j,n)

m1,m2 =

1 2√2j + 1

• j0∈{± 1

2 }

j+j0,m2−m

J,m2, 1

2 ,−m

• qj0,∓κj0,∓(j, n, m1; λ)D

(j+j0,n∓ 3

2 )

m1∓ 1

2 ,m2−m ◮ The coefficients qj0,± are in the following tables:

qj0,∓

• +

j0 = − 1

2

√j + m1 √j − m1 j0 = 1

2

√j − m1 + 1 √j + m1 + 1

SLIDE 57

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

◮ The homomorphism dl : p ⊗ τ (j,n) −

→ C ∞(K) is(compare Buttcane-Miller ’17): dl(u

( 1

2 ,± 3 2 )

m

)D(j,n)

m1,m2 =

1 2√2j + 1

• j0∈{± 1

2 }

j+j0,m2−m

J,m2, 1

2 ,−m

• qj0,∓κj0,∓(j, n, m1; λ)D

(j+j0,n∓ 3

2 )

m1∓ 1

2 ,m2−m ◮ The coefficients qj0,± are in the following tables:

qj0,∓

• +

j0 = − 1

2

√j + m1 √j − m1 j0 = 1

2

√j − m1 + 1 √j + m1 + 1

◮ The κj0,∓’s are affine linear functions in λ:

κj0,∓

• +

j0 = − 1

2

2j − m1 + n − (λ + ρ0) + 2 2j + m1 − n − (λ + ρ0) + 2 j0 = 1

2

−2j − m1 + n − (λ + ρ0) 2j − m1 + n + (λ + ρ0)

SLIDE 58

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

Legend: V W means there exists a submodule V ⊂ IP(χδ,λ) such that IP(χδ,λ)/V ∼ = W . I1

λ + δ λ − δ k l (0, δ) ( λ−δ 4 , 3λ+δ 4 ) ( λ+δ 4 , −3λ+δ 4 ) ( λ 2 , − δ 2 )

The composition series when χδ,λ ∈ Weyl Chamber I1: VH W1 W2 ⊕ Vfin

SLIDE 59

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

Legend: V W means there exists a submodule V ⊂ IP(χδ,λ) such that IP(χδ,λ)/V ∼ = W . II1

λ + δ λ − δ (0, δ) ( −λ−δ 4 , −3λ+δ 4 ) ( λ−δ 4 , 3λ+δ 4 )

The composition series when χδ,λ ∈ Weyl Chamber II1: V2 VH W2 ⊕

SLIDE 60

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

Legend: V W means there exists a submodule V ⊂ IP(χδ,λ) such that IP(χδ,λ)/V ∼ = W . II2

λ + δ λ − δ (0, δ) ( −λ−δ 4 , −3λ+δ 4 ) ( λ−δ 4 , 3λ+δ 4 )

The composition series when χδ,λ ∈ Weyl Chamber II2: V2 VH W2 ⊕

SLIDE 61

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

Legend: V W means there exists a submodule V ⊂ IP(χδ,λ) such that IP(χδ,λ)/V ∼ = W . I2

λ + δ λ − δ (0, δ) ( λ−δ 4 , 3λ+δ 4 ) ( λ+δ 4 , −3λ+δ 4 ) ( λ 2 , − δ 2 )

The composition series when χδ,λ ∈ Weyl Chamber I2: Vfin W1 W2 ⊕ VH

SLIDE 62

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

Legend: V W means there exists a submodule V ⊂ IP(χδ,λ) such that IP(χδ,λ)/V ∼ = W . III1

λ + δ λ − δ (0, δ) ( λ+δ 4 , −3λ+δ 4 ) ( −λ+δ 4 , 3λ+δ 4 )

The composition series when χδ,λ ∈ Weyl Chamber III1: V1 VH W1 ⊕

SLIDE 63

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU(n, 1)

SU(2, 1) Principal Series (Bars-Teng ’90, Ishikawa ’91 or earlier)

Legend: V W means there exists a submodule V ⊂ IP(χδ,λ) such that IP(χδ,λ)/V ∼ = W . III2

λ + δ λ − δ (0, δ) ( −λ+δ 4 , 3λ+δ 4 ) ( λ+δ 4 , −3λ+δ 4 )

The composition series when χδ,λ ∈ Weyl Chamber III2: V1 VH W1 ⊕

SLIDE 64

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2).

SLIDE 65

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2). ◮ Sp(4, R) = {g ∈ SL(4, C)|gtJg = J}, J =

02 12 −12 02

SLIDE 66

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2). ◮ Sp(4, R) = {g ∈ SL(4, C)|gtJg = J}, J =

02 12 −12 02

• ◮ σ(g) = ¯

g an antiholomorphic involution, Sp(4, R) = Sp(4, C)σ;

SLIDE 67

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2). ◮ Sp(4, R) = {g ∈ SL(4, C)|gtJg = J}, J =

02 12 −12 02

• ◮ σ(g) = ¯

g an antiholomorphic involution, Sp(4, R) = Sp(4, C)σ;

◮ θ(g) = (g t)−1 a Cartan involution, k =

• A

B −B A

• = gθ;

SLIDE 68

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2). ◮ Sp(4, R) = {g ∈ SL(4, C)|gtJg = J}, J =

02 12 −12 02

• ◮ σ(g) = ¯

g an antiholomorphic involution, Sp(4, R) = Sp(4, C)σ;

◮ θ(g) = (g t)−1 a Cartan involution, k =

• A

B −B A

• = gθ;

◮ k ∼

= u(2) via

• A

B −B A

• → A + iB.

SLIDE 69

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2). ◮ Sp(4, R) = {g ∈ SL(4, C)|gtJg = J}, J =

02 12 −12 02

• ◮ σ(g) = ¯

g an antiholomorphic involution, Sp(4, R) = Sp(4, C)σ;

◮ θ(g) = (g t)−1 a Cartan involution, k =

• A

B −B A

• = gθ;

◮ k ∼

= u(2) via

• A

B −B A

• → A + iB.

◮ ∆+

c = {α1}, ∆+ nc = {α2, α1 + α2, 2α1 + α2}

α1 α2 α1 + α2 2α1 + α2 ̟1 ̟2

Green: ∆nc; Blue: ∆c

SLIDE 70

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R)

◮ The group G = Sp(4, R) is rank 2, with maximal compact K = U(2). ◮ Sp(4, R) = {g ∈ SL(4, C)|gtJg = J}, J =

02 12 −12 02

• ◮ σ(g) = ¯

g an antiholomorphic involution, Sp(4, R) = Sp(4, C)σ;

◮ θ(g) = (g t)−1 a Cartan involution, k =

• A

B −B A

• = gθ;

◮ k ∼

= u(2) via

• A

B −B A

• → A + iB.

◮ ∆+

c = {α1}, ∆+ nc = {α2, α1 + α2, 2α1 + α2}

α1 α2 α1 + α2 2α1 + α2 ̟1 ̟2

Green: ∆nc; Blue: ∆c

◮ The two irreducible pieces of pC are p±

C = span{u(1,±1) m

}m∈{−1,0,1}

SLIDE 71

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R) Induction Data

◮ Minimal parabolic: P = MAN

• 1

0 −1 0 1 0 −1

ǫ1

−1 0 1 0 −1 0 1

ǫ2 t1 0

0 t2 0 0 1/t1 0 0 1/t2

• N

SLIDE 72

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R) Induction Data

◮ Minimal parabolic: P = MAN

• 1

0 −1 0 1 0 −1

ǫ1

−1 0 1 0 −1 0 1

ǫ2 t1 0

0 t2 0 0 1/t1 0 0 1/t2

• N

◮ M ∼

= Z2 × Z2, character on M given by (δ1, δ2), δi ∈ {0, 1}

SLIDE 73

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R) Induction Data

◮ Minimal parabolic: P = MAN

• 1

0 −1 0 1 0 −1

ǫ1

−1 0 1 0 −1 0 1

ǫ2 t1 0

0 t2 0 0 1/t1 0 0 1/t2

• N

◮ M ∼

= Z2 × Z2, character on M given by (δ1, δ2), δi ∈ {0, 1}

◮ A ∼

= (R>0)2, character on A given by (λ1, λ2) ∈ C2

SLIDE 74

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Example: Sp(4, R) Induction Data

◮ Minimal parabolic: P = MAN

• 1

0 −1 0 1 0 −1

ǫ1

−1 0 1 0 −1 0 1

ǫ2 t1 0

0 t2 0 0 1/t1 0 0 1/t2

• N

◮ M ∼

= Z2 × Z2, character on M given by (δ1, δ2), δi ∈ {0, 1}

◮ A ∼

= (R>0)2, character on A given by (λ1, λ2) ∈ C2

◮ They define a character χδ,λ sending the above element to (−1)δ1ǫ1+δ2ǫ2tλ1

1 tλ2 2

SLIDE 75

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Sp(4, R) Principal Series

◮ The restriction of the principal series IP(χδ,λ) to K can be decomposed as a

direct sum of K-isotypic spaces τ (j,n): IP(χδ,λ) =

• (j,n)∈KTypes(δ1,δ2)

τ (j,n).

SLIDE 76

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Sp(4, R) Principal Series

◮ The restriction of the principal series IP(χδ,λ) to K can be decomposed as a

direct sum of K-isotypic spaces τ (j,n): IP(χδ,λ) =

• (j,n)∈KTypes(δ1,δ2)

τ (j,n).

◮ Each space τ (j,n) is

τ (j,n) =

• m2∈{−j,−j+1,...,j}

m1∈M(j,n;δ1,δ2)

CD(j,n)

m1,m2

SLIDE 77

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Sp(4, R) Principal Series

◮ The restriction of the principal series IP(χδ,λ) to K can be decomposed as a

direct sum of K-isotypic spaces τ (j,n): IP(χδ,λ) =

• (j,n)∈KTypes(δ1,δ2)

τ (j,n).

◮ Each space τ (j,n) is

τ (j,n) =

• m2∈{−j,−j+1,...,j}

m1∈M(j,n;δ1,δ2)

CD(j,n)

m1,m2

◮ The two sets of admissible j, n, m1, m2’s are defined as:

KTypes(δ1, δ2) = {(j, n) ∈ 1 2 Z+ × 1 2 Z|2j ≡ δ2 − δ1 and 2n ≡ δ2 + δ1 mod 2} M(j, n; δ1, δ2) = {m1 ∈ {−j, −j + 1, . . . , j − 1, j}|n − m1 ≡ δ1 and n + m1 ≡ δ2 mod 2}.

SLIDE 78

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Sp(4, R) Principal Series

The homomorphism dl : p ⊗ τ (j,n) − → C ∞(K) is dl(u(1,±1)

m

)D(j,n)

m1,m2 = (−1)m1i

2

• j0∈{−1,0,1}
• j+j0,m2−m

j,m2,1,−m ε=±1

Cj+j0qj0,εκ±,j0,ε(j, n, m1; λ)D(j+j0,n∓1)

m1+ε,m2−m

with the coefficients given the tables in the next frame.

SLIDE 79

Clebsch-Gordan Coefficients and Principal Series Representations Sp(4, R) Example

Sp(4, R) Principal Series

Cj+j0 j0 = −1 j− 1

2 (2j + 1)− 1 2

j0 = 0 j− 1

2 (j + 1)− 1 2

j0 = 1 (j + 1)− 1

2 (2j + 1)− 1 2

qj0,ε ε = −1 ε = 1 j0 = −1

• (j + m1 − 1)(j + m1)
• (j − m1 − 1)(j − m1)

j0 = 0

• (j + m1)(j − m1 + 1)
• (j − m1)(j + m1 + 1)

j0 = 1

• (j − m1 + 1)(j − m1 + 2)
• (j + m1 + 1)(j + m1 + 2)

κ+,j0,ε ε = −1 ε = 1 j0 = −1 2j − m1 + n − ˇ α1, λ + ρ0 + 2 n − m1 − ˇ α2, λ + ρ0 j0 = 0 n − m1 − ˇ α1, λ + ρ0 + 2 m1 − n + ˇ α2, λ + ρ0 j0 = 1 −2j − m1 + n − ˇ α1, λ + ρ0 n − m1 − ˇ α2, λ + ρ0 κ−,j0,ε ε = −1 ε = 1 j0 = −1 −n + m1 − ˇ α2, λ + ρ0 2j + m1 − n − ˇ α1, λ + ρ0 + 2 j0 = 0 −n + m1 − ˇ α2, λ + ρ0 −m1 + n + ˇ α1, λ + ρ0 − 2 j0 = 1 −n + m1 − ˇ α2, λ + ρ0 2j + m1 − n − ˇ α1, λ + ρ0 + 2

SLIDE 80

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ The Bargmann’s classification of unitary representations is done by introducing

the intertwining operators between principal series: A(λ) : I(χδ,λ) − → I(χδ,−λ)

SLIDE 81

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ The Bargmann’s classification of unitary representations is done by introducing

the intertwining operators between principal series: A(λ) : I(χδ,λ) − → I(χδ,−λ)

◮ It is defines as the integral operator:

(A(λ)f )(g) = ∞

−∞

f ( −1 1 1 x 1

• g)

SLIDE 82

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ The Bargmann’s classification of unitary representations is done by introducing

the intertwining operators between principal series: A(λ) : I(χδ,λ) − → I(χδ,−λ)

◮ It is defines as the integral operator:

(A(λ)f )(g) = ∞

−∞

f ( −1 1 1 x 1

• g)

◮ By computing the Iwasawa decompositions and use the vector space structure of

the principal series, denote the function eikθ ∈ IP(χδ,λ) by φλ,k A(λ)φλ,k = (−i)k√π Γ( λ+1

2 )Γ( λ 2 )

Γ( λ+1+k

2

)Γ( λ−k+1

2

) φ−λ,k When δ = 0, λ ∈ R, then all k ∈ Z are even. The intertwining operator defines a Hermitian pairing on I(χδ,λ).

SLIDE 83

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

SLIDE 84

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ For w ∈ W , f ∈ IP(χδ,λ), define the intertwining operator

A(w, χδ,λ)f (g) =

• ¯

N∩Nw f (gw ¯

n)d ¯ n such that it intertwines IP(χδ,λ) and IP(wχδ,λ) A(w, χδ,λ)πP(χδ,λ)(g) = πP(wχδ,λ)(g)A(w, χδ,λ)

SLIDE 85

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ For w ∈ W , f ∈ IP(χδ,λ), define the intertwining operator

A(w, χδ,λ)f (g) =

• ¯

N∩Nw f (gw ¯

n)d ¯ n such that it intertwines IP(χδ,λ) and IP(wχδ,λ) A(w, χδ,λ)πP(χδ,λ)(g) = πP(wχδ,λ)(g)A(w, χδ,λ)

◮ It can be meromorphically continuated to all λ

SLIDE 86

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ For w ∈ W , f ∈ IP(χδ,λ), define the intertwining operator

A(w, χδ,λ)f (g) =

• ¯

N∩Nw f (gw ¯

n)d ¯ n such that it intertwines IP(χδ,λ) and IP(wχδ,λ) A(w, χδ,λ)πP(χδ,λ)(g) = πP(wχδ,λ)(g)A(w, χδ,λ)

◮ It can be meromorphically continuated to all λ ◮ For dominant λ, A(w0) defines the Langlands quotient of the principal series by

quotienting out its kernel.

SLIDE 87

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

What are Intertwining Operators

◮ For w ∈ W , f ∈ IP(χδ,λ), define the intertwining operator

A(w, χδ,λ)f (g) =

• ¯

N∩Nw f (gw ¯

n)d ¯ n such that it intertwines IP(χδ,λ) and IP(wχδ,λ) A(w, χδ,λ)πP(χδ,λ)(g) = πP(wχδ,λ)(g)A(w, χδ,λ)

◮ It can be meromorphically continuated to all λ ◮ For dominant λ, A(w0) defines the Langlands quotient of the principal series by

quotienting out its kernel.

◮ (Knapp, Wallach, Barbasch)If

w0δ ∼ = δ, w0λ = −¯ λ Choose an isomorphism τ : w0δ ∼ = δ, an hermitian form on the Langlands quotient is given by v1, v2 = v1, τ ◦ A′(w0, χδ,λ)v2

SLIDE 88

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for SU(2, 1)

◮ We can evaluate the intertwining integral explicitly by setting f = D(j,n)

m1,m2,

SLIDE 89

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for SU(2, 1)

◮ We can evaluate the intertwining integral explicitly by setting f = D(j,n)

m1,m2,

◮ (Kostant ’62 Johnson-Wallach ’75, Fabec ’91) The long intertwining operator

A(w, χδ,λ) acts on each D(j,n)

m1,m2 as a scalar

• A(w, χδ,λ)
• m1, with a closed form

formula:

• A(w, χδ,λ)
• m1 =

−π22−λ−2(−1)λ+δΓ(λ) Γ

• 1 − λ−δ

2

• Γ
• 1 − λ+δ

2

• Γ
• j − m1 − λ+δ

2

+ 1

• Γ
• j + m1 − λ−δ

2

+ 1

• Γ
• j − m1 + λ−δ

2

+ 1

• Γ
• j + m1 + λ+δ

2

+ 1

SLIDE 90

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for SU(2, 1)

◮ We can evaluate the intertwining integral explicitly by setting f = D(j,n)

m1,m2,

◮ (Kostant ’62 Johnson-Wallach ’75, Fabec ’91) The long intertwining operator

A(w, χδ,λ) acts on each D(j,n)

m1,m2 as a scalar

• A(w, χδ,λ)
• m1, with a closed form

formula:

• A(w, χδ,λ)
• m1 =

−π22−λ−2(−1)λ+δΓ(λ) Γ

• 1 − λ−δ

2

• Γ
• 1 − λ+δ

2

• Γ
• j − m1 − λ+δ

2

+ 1

• Γ
• j + m1 − λ−δ

2

+ 1

• Γ
• j − m1 + λ−δ

2

+ 1

• Γ
• j + m1 + λ+δ

2

+ 1

• ◮ Normalized intertwining operator: divide by the Harish-Chandra c-function, the

normalized intertwining operator becomes a rational function

• A(w, χδ,λ)
• m1 =
• 2−λ−δ

2

(j−m1)

2−λ+δ 2

(j+m1)

• 2+λ−δ

2

(j−m1)

2+λ+δ 2

(j+m1)

SLIDE 91

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

◮ A(w0, λ) can be factorized into 4 intertwining operators corresponding to simple

reflections: A(w0, λ) = A(wα2, wα1wα2wα1λ)A(wα1, wα2wα1λ)A(wα2, wα1λ)A(wα1, λ).

SLIDE 92

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

◮ A(w0, λ) can be factorized into 4 intertwining operators corresponding to simple

reflections: A(w0, λ) = A(wα2, wα1wα2wα1λ)A(wα1, wα2wα1λ)A(wα2, wα1λ)A(wα1, λ).

◮ (λ1, λ2) wα1

− − − → (λ2, λ1): [A(wα1, λ)]j,n

m1,m2 = Sj,n m1,m2

λ1−λ2+1

2

SLIDE 93

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

◮ A(w0, λ) can be factorized into 4 intertwining operators corresponding to simple

reflections: A(w0, λ) = A(wα2, wα1wα2wα1λ)A(wα1, wα2wα1λ)A(wα2, wα1λ)A(wα1, λ).

◮ (λ1, λ2) wα1

− − − → (λ2, λ1): [A(wα1, λ)]j,n

m1,m2 = Sj,n m1,m2

λ1−λ2+1

2

• ◮ (λ2, λ1)

wα2

− − − → (λ2, −λ1): [A(wα2, wα1λ)]j,n

m1,m2 = T n m1

λ1+1

2

• δm1,m2

SLIDE 94

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

◮ A(w0, λ) can be factorized into 4 intertwining operators corresponding to simple

reflections: A(w0, λ) = A(wα2, wα1wα2wα1λ)A(wα1, wα2wα1λ)A(wα2, wα1λ)A(wα1, λ).

◮ (λ1, λ2) wα1

− − − → (λ2, λ1): [A(wα1, λ)]j,n

m1,m2 = Sj,n m1,m2

λ1−λ2+1

2

• ◮ (λ2, λ1)

wα2

− − − → (λ2, −λ1): [A(wα2, wα1λ)]j,n

m1,m2 = T n m1

λ1+1

2

• δm1,m2

◮ (λ2, −λ1) wα1

− − − → (−λ1, λ2): [A(wα1, wα2wα1λ)]j,n

m1,m2 = Sj,n m1,m2

λ1+λ2+1

2

SLIDE 95

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

◮ A(w0, λ) can be factorized into 4 intertwining operators corresponding to simple

reflections: A(w0, λ) = A(wα2, wα1wα2wα1λ)A(wα1, wα2wα1λ)A(wα2, wα1λ)A(wα1, λ).

◮ (λ1, λ2) wα1

− − − → (λ2, λ1): [A(wα1, λ)]j,n

m1,m2 = Sj,n m1,m2

λ1−λ2+1

2

• ◮ (λ2, λ1)

wα2

− − − → (λ2, −λ1): [A(wα2, wα1λ)]j,n

m1,m2 = T n m1

λ1+1

2

• δm1,m2

◮ (λ2, −λ1) wα1

− − − → (−λ1, λ2): [A(wα1, wα2wα1λ)]j,n

m1,m2 = Sj,n m1,m2

λ1+λ2+1

2

• ◮ (−λ1, λ2)

wα2

− − − → (−λ1, −λ2): [A(wα2, wα1wα2wα1λ)]j,n

m1,m2 = T n m1

λ2+1

2

• δm1,m2

◮ Normalization: divide each simple intertwining operator Sj,n

m1,m2(z), T n m1(z) by √πΓ(z−1/2) Γ(z)

, get rational functions Sj,n

m1,m2(z), T n m1(z) such that the corresponding

normalized simple intertwining operator satisfies A(−λ)A(λ) = 1.

SLIDE 96

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

◮ A(w0, λ) can be factorized into 4 intertwining operators corresponding to simple

reflections: A(w0, λ) = A(wα2, wα1wα2wα1λ)A(wα1, wα2wα1λ)A(wα2, wα1λ)A(wα1, λ).

◮ (λ1, λ2) wα1

− − − → (λ2, λ1): [A(wα1, λ)]j,n

m1,m2 = Sj,n m1,m2

λ1−λ2+1

2

• ◮ (λ2, λ1)

wα2

− − − → (λ2, −λ1): [A(wα2, wα1λ)]j,n

m1,m2 = T n m1

λ1+1

2

• δm1,m2

◮ (λ2, −λ1) wα1

− − − → (−λ1, λ2): [A(wα1, wα2wα1λ)]j,n

m1,m2 = Sj,n m1,m2

λ1+λ2+1

2

• ◮ (−λ1, λ2)

wα2

− − − → (−λ1, −λ2): [A(wα2, wα1wα2wα1λ)]j,n

m1,m2 = T n m1

λ2+1

2

• δm1,m2

◮ Normalization: divide each simple intertwining operator Sj,n

m1,m2(z), T n m1(z) by √πΓ(z−1/2) Γ(z)

, get rational functions Sj,n

m1,m2(z), T n m1(z) such that the corresponding

normalized simple intertwining operator satisfies A(−λ)A(λ) = 1.

◮ For the convenience of calculation, our principal series is induced from the

character

• 1

0 −1 0 1 0 −1

ǫ1

−1 0 1 0 −1 0 1

ǫ2 t1 0

0 t2 0 0 1/t1 0 0 1/t2

• N → (−1)(ǫ1+ǫ2)δtλ1

1 tλ2 2

SLIDE 97

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

(j, n) A(λ) (0,0) ( 1 ) (0,2)

• (λ1−1)(λ2−1)

(λ1+1)(λ2+1)

• (1,-2)
• (λ1−1)(λ1−λ2−1)(λ2−1)(λ1+λ2−1)

(λ1+1)(λ1−λ2+1)(λ2+1)(λ1+λ2+1)

• (1,-1)

 

−

λ3 1−λ2 1−λ2 2λ1−λ1+λ2 2−1

(λ1+1)(λ1−λ2+1)(λ1+λ2+1) −

2λ1(λ2−1)

(λ1+1)(λ1−λ2+1)(λ1+λ2+1) −

2λ1(λ2−1)

(λ1+1)(λ1−λ2+1)(λ1+λ2+1) − (λ2−1)(λ3

1+λ2 1−λ2 2λ1−λ1−λ2 2+1)

(λ1+1)(λ1−λ2+1)(λ2+1)(λ1+λ2+1)

  (1,1)  

− (λ2−1)(λ3

1+λ2 1−λ2 2λ1−λ1−λ2 2+1)

(λ1+1)(λ1−λ2+1)(λ2+1)(λ1+λ2+1) −

2λ1(λ2−1)

(λ1+1)(λ1−λ2+1)(λ1+λ2+1) −

2λ1(λ2−1)

(λ1+1)(λ1−λ2+1)(λ1+λ2+1) −

λ3 1−λ2 1−λ2 2λ1−λ1+λ2 2−1

(λ1+1)(λ1−λ2+1)(λ1+λ2+1)

 

SLIDE 98

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

(j, n) Sj,n

m1,m2(z)

(0,0) ( 1 ) (1,0) ( − z−1

z

) (2,0)      

3 4z(z+1) 3 2 (2z−1) 2z(z+1) (2z−1)(2z+1) 4z(z+1) 3 2 (2z−1) 2z(z+1) 2z2−4z+3 2z(z+1) 3 2 (2z−1) 2z(z+1) (2z−1)(2z+1) 4z(z+1) 3 2 (2z−1) 2z(z+1) 3 4z(z+1)

      (3,0)      

−

15(z−1) 4z(z+1)(z+2)

−

15 2 (z−1)(2z−1) 2z(z+1)(z+2)

− (z−1)(2z−1)(2z+1)

4z(z+1)(z+2)

−

15 2 (z−1)(2z−1) 2z(z+1)(z+2)

−

(z−1)(2z2−4z+9) 2z(z+1)(z+2)

−

15 2 (z−1)(2z−1) 2z(z+1)(z+2)

− (z−1)(2z−1)(2z+1)

4z(z+1)(z+2)

−

15 2 (z−1)(2z−1) 2z(z+1)(z+2)

−

15(z−1) 4z(z+1)(z+2)

     

SLIDE 99

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

(ZZ)The matrices for simple intertwining operators are given by: Sj,n

m1,m2(z) =

i−m1−m2(2j)!√π cj

m1cj m2Γ

• 1−2j−m1+m2

2

(z − 1/2)(− m1−m2

2

)

(z)(j)

3F2

• −j+z−1,−j−m1,−j+m2

−2j,−j− m1−m2

2

+ 1

2

; 1

• T n

m1(z) =

in−m1 (z)( m1−n

2

)(z)( −m1+n

2

) .

SLIDE 100

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Intertwining Operator for Sp(4, R)

(ZZ)The matrices for A(λ) for δ ∈ {(0, 0), (1, 1)} is the constant Laurent series coefficient of [A(λ)]j,n

m1,m2(t1, t2) =

(−1)j+n−ǫj,n

δ ((2j)!)2

• λ1+1

2

j+n−ǫj,n

δ 2

• λ1+1

2

−

j+n−ǫj,n δ 2

• λ1−λ2+1

2

(j)

λ1+λ2+1 2

(j) i−m1−m2 cj

m1cj m2

• λ1+λ2

2

(

j+m2−ǫj,n δ 2

) λ1−λ2 2

(

−j−m1+ǫj,n δ 2

)

• λ2+1

2

m2−n

2

• λ2+1

2

− m2−n

2

Γ

• 1−ǫj,n

δ +j−m2

2

• Γ
• 1−ǫj,n

δ +j−m1

2

• (1 − t1)

−1−ǫj,n δ +j−m1 2

(1 − t2)

−1+ǫj,n δ −j+m2 2

t

ǫj,n

δ −2j

1

t

−ǫj,n

δ

2 2F1

• −j−m1, λ1−λ2−2j−1

2

−2j

; t1

• 2F1
• −j+m2, λ1+λ2−2j−1

2

−2j

; t2

• 4F3

  1,

−j+m1+1+ǫj,n δ 2

,

−j−n−λ1+1+ǫj,n δ 2

,

−j−m1+λ1−λ2+ǫj,n δ 2 −j+m2+1+ǫj,n δ 2

,

−j−n+λ1+1+ǫj,n δ 2

,

−j−m2−λ1−λ2+ǫj,n δ 2

+1

; t2

1 (1 − t2)

(1 − t1) t2

2

  where ǫj,n

δ

=

• 0 j−n≡δi mod 2

1 j−n≡δi mod 2 .

SLIDE 101

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ Kostant proof: using the (g, K) module action and the intertwining property to

find the correct formula for the matrix entries of the intertwining operator.

SLIDE 102

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ Kostant proof: using the (g, K) module action and the intertwining property to

find the correct formula for the matrix entries of the intertwining operator.

◮ Fabec proof: writing the matrix coefficient in terms of Laguerre polynomials and

use special function identities.

SLIDE 103

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ Kostant proof: using the (g, K) module action and the intertwining property to

find the correct formula for the matrix entries of the intertwining operator.

◮ Fabec proof: writing the matrix coefficient in terms of Laguerre polynomials and

use special function identities.

◮ Johnson-Wallach: use spherical harmonics. ◮ ZZ’s exercise: contour integral and brute force: ◮ The intertwining operator acts on D(j,n) m1,m2 as

• C×R
• (|z|2 + 1)2 + 4w 2− λ+2

2

D(j,n)

m1,m2

     

|z|2−2iw−1

√

(|z|2+1)2+4w2 2¯ z |z|2−2iw+1 − 2z

√

(|z|2+1)2+4w2 |z|2+2iw−1 |z|2−2iw+1 |z|2−2iw+1

√

(|z|2+1)2+4w2

      dxdydw

SLIDE 104

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ Kostant proof: using the (g, K) module action and the intertwining property to

find the correct formula for the matrix entries of the intertwining operator.

◮ Fabec proof: writing the matrix coefficient in terms of Laguerre polynomials and

use special function identities.

◮ Johnson-Wallach: use spherical harmonics. ◮ ZZ’s exercise: contour integral and brute force: ◮ The intertwining operator acts on D(j,n) m1,m2 as

• C×R
• (|z|2 + 1)2 + 4w 2− λ+2

2

D(j,n)

m1,m2

     

|z|2−2iw−1

√

(|z|2+1)2+4w2 2¯ z |z|2−2iw+1 − 2z

√

(|z|2+1)2+4w2 |z|2+2iw−1 |z|2−2iw+1 |z|2−2iw+1

√

(|z|2+1)2+4w2

      dxdydw

◮ It suffices to calculate the contour integral using residue theorem

• R

(−1 + r 2 + 2iw)

k−l 2 −p(−1 + r 2 − 2iw) k+l 2 −p(1 + r 2 − 2iw)− k+l+(λ+δ+2) 2

(1 + r 2 + 2iw)− k−l+(λ−δ+2)

2

dw

SLIDE 105

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ The integral yields a sum of binomial coefficients, the matrix entries for the

intertwining operator can be written as [A(w, δ, λ)]m1 = 2−λ−2(−1)1+k+l+ λ+δ

2 π2

Γ 1

2 (k − l + 2)

• Γ

1

2 (k + l + 2)

• Γ(λ)

Γ 1

2 (k − l − δ + λ) + 1

• Γ

1

2 (k + l + δ + λ) + 1

• K1,K2,K3≥0

K1+K2+K3= k+l+λ+δ

2

p≥0

Γ

• k+l+δ+λ+2

2

• Γ(K1 + 1)Γ(K2 + 1)Γ(K3 + 1)
• −K1+ k+l

2 +λ−1

λ−1

k+l

2 −K1

p

k−l

2

−K2 p

• (−1)K3

SLIDE 106

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ The integral yields a sum of binomial coefficients, the matrix entries for the

intertwining operator can be written as [A(w, δ, λ)]m1 = 2−λ−2(−1)1+k+l+ λ+δ

2 π2

Γ 1

2 (k − l + 2)

• Γ

1

2 (k + l + 2)

• Γ(λ)

Γ 1

2 (k − l − δ + λ) + 1

• Γ

1

2 (k + l + δ + λ) + 1

• K1,K2,K3≥0

K1+K2+K3= k+l+λ+δ

2

p≥0

Γ

• k+l+δ+λ+2

2

• Γ(K1 + 1)Γ(K2 + 1)Γ(K3 + 1)
• −K1+ k+l

2 +λ−1

λ−1

k+l

2 −K1

p

k−l

2

−K2 p

• (−1)K3

◮ We are very lucky that the sum inside is the Taylor coefficient for yλ−1 of the

function (−1)

1 2 (k+l+δ+λ)x− 1 2 (k−l)(1 + x) 1 2 (k−l−δ−λ)y 1 2 (k+l+δ+λ)(1 + y) λ−δ 2

−1

SLIDE 107

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for SU(2, 1)

◮ The integral yields a sum of binomial coefficients, the matrix entries for the

intertwining operator can be written as [A(w, δ, λ)]m1 = 2−λ−2(−1)1+k+l+ λ+δ

2 π2

Γ 1

2 (k − l + 2)

• Γ

1

2 (k + l + 2)

• Γ(λ)

Γ 1

2 (k − l − δ + λ) + 1

• Γ

1

2 (k + l + δ + λ) + 1

• K1,K2,K3≥0

K1+K2+K3= k+l+λ+δ

2

p≥0

Γ

• k+l+δ+λ+2

2

• Γ(K1 + 1)Γ(K2 + 1)Γ(K3 + 1)
• −K1+ k+l

2 +λ−1

λ−1

k+l

2 −K1

p

k−l

2

−K2 p

• (−1)K3

◮ We are very lucky that the sum inside is the Taylor coefficient for yλ−1 of the

function (−1)

1 2 (k+l+δ+λ)x− 1 2 (k−l)(1 + x) 1 2 (k−l−δ−λ)y 1 2 (k+l+δ+λ)(1 + y) λ−δ 2

−1

◮ The Taylor coefficient for yλ−1 is

(−1)

1 2 (k+l+δ+λ)Γ

• 1 + k−l

2

− λ+δ

2

• Γ
• λ−δ

2

• Γ
• k−l

2

+ 1

• Γ
• k+l

2

+ 1

• Γ
• 1 − λ+δ

2

• Γ
• − 1

2 (k + l) + λ−δ 2

SLIDE 108

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ The Mellin transform of a function f (x) is defined as

M(f (x))(z) = ∞ f (x)xz−1dx

SLIDE 109

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ The Mellin transform of a function f (x) is defined as

M(f (x))(z) = ∞ f (x)xz−1dx

◮ Consider a function F(z) of one complex variable z = σ + iτ, such that

• 1. F(z) is regular on the strip S = {z ∈ C|a < σ < b}, such that F(z) → 0 uniformly in

the strip Sǫ = {z ∈ C|a + ǫ < σ < b − ǫ} for arbitrarily small ǫ > 0, 2. ∞

−∞ |F(σ + iτ)|dτ < ∞.

We define the inverse Mellin transform of the function F(z) as M−1(F(z))(x) = 1 2πi γ+i∞

γ−i∞

F(z)x−zdz, for x > 0 and some fixed γ ∈ (a, b). It satisfies the property that M(M−1(F))(z) = F(z)

SLIDE 110

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ Which simple intertwining operators are diagonal?

wα1 = eπγ2, wα2 = eπ(γ0−γ3) Under the Wigner D-matrix basis, exponentials of γ0, γ3 are diagonal, all others are not. Use adjoint action by K to diagonalise Sj,n

m1,m4(z) =

• −j≤m3≤j

(−1)m3Mj,n

m1,m3Nj,n m3,m4Q(z, m3)

SLIDE 111

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ Which simple intertwining operators are diagonal?

wα1 = eπγ2, wα2 = eπ(γ0−γ3) Under the Wigner D-matrix basis, exponentials of γ0, γ3 are diagonal, all others are not. Use adjoint action by K to diagonalise Sj,n

m1,m4(z) =

• −j≤m3≤j

(−1)m3Mj,n

m1,m3Nj,n m3,m4Q(z, m3)

◮ We can diagonalize the simple intertwining operators and calculate their inverse

Mellin transform M−1

• π22−2zΓ(2z − 1)

Γ(z + m3)Γ(z − m3)

• (x) =

1

• x(1 − x)

cos(2m3 arcsin(√1−x)) |x|<1

|x|>1 .

The trigonometric part in fact an algebraic function: Chebyshev polynomial.

SLIDE 112

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ Which simple intertwining operators are diagonal?

wα1 = eπγ2, wα2 = eπ(γ0−γ3) Under the Wigner D-matrix basis, exponentials of γ0, γ3 are diagonal, all others are not. Use adjoint action by K to diagonalise Sj,n

m1,m4(z) =

• −j≤m3≤j

(−1)m3Mj,n

m1,m3Nj,n m3,m4Q(z, m3)

◮ We can diagonalize the simple intertwining operators and calculate their inverse

Mellin transform M−1

• π22−2zΓ(2z − 1)

Γ(z + m3)Γ(z − m3)

• (x) =

1

• x(1 − x)

cos(2m3 arcsin(√1−x)) |x|<1

|x|>1 .

The trigonometric part in fact an algebraic function: Chebyshev polynomial.

◮ The diagonalisation uses two change-of-basis matrices Mj

m1,m2, Nj m1,m2. They are

constant Laurent terms of two rational functions Mj

m1,m2 =

• (−1)2j cj

m2

cj

m1

i−m1+m22−m2sm2−j 1 + s 2 j−m1 1 − s 2 j+m1

• Nj

m1,m2 =

• (−1)2j cj

m2

cj

m1

im1−m22−m2tm2−j

• 1 + t

2 j−m1 1 − t 2 j+m1 .

SLIDE 113

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ We can thus write the inverse Mellin transform of Sj,n

m1,m4(z) as a Laurent

coefficient of an analytic function in s and t. However, it is possible to add/drop terms such that it becomes a Laurent coefficient of an analytic function in u:

SLIDE 114

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ We can thus write the inverse Mellin transform of Sj,n

m1,m4(z) as a Laurent

coefficient of an analytic function in s and t. However, it is possible to add/drop terms such that it becomes a Laurent coefficient of an analytic function in u: M−1(Sj,n

m1,m4(z))(x) = cj m1

cj

m4

((−1)2j+m1−m4 + 1)i−m1+m4(1 − θ(|x| − 1)) 2

• (1 − x)x

(u√1 − x + i√x)j+m4(√1 − x + iu√x)j−m4 uj+m1

SLIDE 115

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ We can thus write the inverse Mellin transform of Sj,n

m1,m4(z) as a Laurent

coefficient of an analytic function in s and t. However, it is possible to add/drop terms such that it becomes a Laurent coefficient of an analytic function in u: M−1(Sj,n

m1,m4(z))(x) = cj m1

cj

m4

((−1)2j+m1−m4 + 1)i−m1+m4(1 − θ(|x| − 1)) 2

• (1 − x)x

(u√1 − x + i√x)j+m4(√1 − x + iu√x)j−m4 uj+m1

• Heaviside step function θ(x) =

0 x<0

1 x>0

SLIDE 116

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ We can thus write the inverse Mellin transform of Sj,n

m1,m4(z) as a Laurent

coefficient of an analytic function in s and t. However, it is possible to add/drop terms such that it becomes a Laurent coefficient of an analytic function in u: M−1(Sj,n

m1,m4(z))(x) = cj m1

cj

m4

((−1)2j+m1−m4 + 1)i−m1+m4(1 − θ(|x| − 1)) 2

• (1 − x)x

(u√1 − x + i√x)j+m4(√1 − x + iu√x)j−m4 uj+m1

• Heaviside step function θ(x) =

0 x<0

1 x>0

◮ This is in fact another Jacobi polynomial

M−1(Sj,n

m1,m4(z))(x)

= cj

m1

cj

m4

((−1)2j+m1i2j + (−1)j+m4)(1 − θ(|x| − 1)) 2(1 − x)

m1+m4+1 2

x− 2j+m1+m4−1

2

P(−m1−m4,−2j−1)

j+m1

2 x − 1

• .

SLIDE 117

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ We can thus write the inverse Mellin transform of Sj,n

m1,m4(z) as a Laurent

coefficient of an analytic function in s and t. However, it is possible to add/drop terms such that it becomes a Laurent coefficient of an analytic function in u: M−1(Sj,n

m1,m4(z))(x) = cj m1

cj

m4

((−1)2j+m1−m4 + 1)i−m1+m4(1 − θ(|x| − 1)) 2

• (1 − x)x

(u√1 − x + i√x)j+m4(√1 − x + iu√x)j−m4 uj+m1

• Heaviside step function θ(x) =

0 x<0

1 x>0

◮ This is in fact another Jacobi polynomial

M−1(Sj,n

m1,m4(z))(x)

= cj

m1

cj

m4

((−1)2j+m1i2j + (−1)j+m4)(1 − θ(|x| − 1)) 2(1 − x)

m1+m4+1 2

x− 2j+m1+m4−1

2

P(−m1−m4,−2j−1)

j+m1

2 x − 1

• .

◮ Taking the Mellin transform back, the calculation is finished.

SLIDE 118

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ What about the long intertwining operator?

Choose a correct generating function for 3F2 hypergeometric polynomials such that we don’t sum over the hypergeometric functions, but only the

• Pochhammers. In fact Sj

m1,m2(z) is the constant Laurent term of either of the

following two functions Hj

m1,m2(z; t) = (2j)!(j + m1)!

cj

m1cj m2

√π Γ 1 − m1 − m2 2 (z − 1/2)(− m1−m2

2

)

(z)(j) (1 − t)

−1+m1+m2 2

(−t)j+m1

2F1

• −j+m2,−1−j+z

−2j

; t

• G j

m1,m2(z; t) = (2j)!(j − m2)!

cj

m1cj m2

√π Γ 1 + m1 + m2 2 (z − 1/2)(− m1−m2

2

)

(z)(j) (1 − t)

−1−m1−m2 2

(−t)j−m2

2F1

• −j−m1,−1−j+z

−2j

; t

• .

SLIDE 119

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators

Proof for Sp(4, R)

◮ What about the long intertwining operator?

Choose a correct generating function for 3F2 hypergeometric polynomials such that we don’t sum over the hypergeometric functions, but only the

• Pochhammers. In fact Sj

m1,m2(z) is the constant Laurent term of either of the

following two functions Hj

m1,m2(z; t) = (2j)!(j + m1)!

cj

m1cj m2

√π Γ 1 − m1 − m2 2 (z − 1/2)(− m1−m2

2

)

(z)(j) (1 − t)

−1+m1+m2 2

(−t)j+m1

2F1

• −j+m2,−1−j+z

−2j

; t

• G j

m1,m2(z; t) = (2j)!(j − m2)!

cj

m1cj m2

√π Γ 1 + m1 + m2 2 (z − 1/2)(− m1−m2

2

)

(z)(j) (1 − t)

−1−m1−m2 2

(−t)j−m2

2F1

• −j−m1,−1−j+z

−2j

; t

• .

◮ Then apply special function transformation rules.

SLIDE 120

Clebsch-Gordan Coefficients and Principal Series Representations Thank you

Thank you!