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

� � � � � � � � � � � 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 ( χ δ,λ ) ։ W | λ | U + U + � U + � U + U + � e − 2 i k θ e − 2 i θ e 2 i θ e 2 i k θ � . . . 1 � . . . U − U − U − U − U − ◮ λ < 0: W | λ | ֒ → I ( χ δ,λ ) ։ D + | λ | ⊕ D − | λ | U + U + � U + U + U + � � 1 e − 2 i k θ e − 2 i θ e 2 i θ e 2 i k θ � . . . . . . U − U − U − U − U −

� � � � � � � � � � � 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 ( χ δ,λ ) ։ W | λ | U + U + � U + � U + U + � e − 2 i k θ e − 2 i θ e 2 i θ e 2 i k θ � . . . 1 � . . . U − U − U − U − U − ◮ λ < 0: W | λ | ֒ → I ( χ δ,λ ) ։ D + | λ | ⊕ D − | λ | U + U + � U + U + U + � � 1 e − 2 i k θ e − 2 i θ e 2 i θ e 2 i k θ � . . . . . . U − U − U − U − U − 0 ⊕ D − ◮ I ( χ − 1 , 0 ) = D + (limit of discrete series) 0

� � � � � � � � � � � � � � � 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 ( χ δ,λ ) ։ W | λ | U + U + � U + � U + U + � e − 2 i k θ e − 2 i θ e 2 i θ e 2 i k θ � . . . 1 � . . . U − U − U − U − U − ◮ λ < 0: W | λ | ֒ → I ( χ δ,λ ) ։ D + | λ | ⊕ D − | λ | U + U + � U + U + U + � � 1 e − 2 i k θ e − 2 i θ e 2 i θ e 2 i k θ � . . . . . . U − U − U − U − U − 0 ⊕ D − ◮ I ( χ − 1 , 0 ) = D + (limit of discrete series) 0 U + U + � U + U + � e − i k θ e − i θ e i θ e i k θ � . . . � . . . U − U − U − U −

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 γ 0 = i � 1 0 , γ 1 = i � 0 1 , γ 2 = i � 0 − i , γ 3 = i � 1 0 � � � � 0 1 1 0 0 − 1 i 0 2 2 2 2 The matrices γ 1 , γ 2 , γ 3 can be interpreted as rotations in R 3 around x , y , z axis respectively.

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 γ 0 = i � 1 0 , γ 1 = i � 0 1 , γ 2 = i � 0 − i , γ 3 = i � 1 0 � � � � 0 1 1 0 0 − 1 i 0 2 2 2 2 The matrices γ 1 , γ 2 , γ 3 can be interpreted as rotations in R 3 around x , y , z axis respectively. ◮ Euler angles on U (2): a general element in U (2) can be expressed as e ζ γ 0 e φ γ 3 e θ γ 2 e ψ γ 3

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Representations of U (2): π ( j , n ) = Pol 2 j ( C 2 ) ⊗ det j + n . The action on f ∈ Pol 2 j ( C 2 ) is given by f ( z ) �→ f ( g − 1 z ).

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Representations of U (2): π ( j , n ) = Pol 2 j ( C 2 ) ⊗ det j + n . The action on f ∈ Pol 2 j ( C 2 ) is given by f ( z ) �→ f ( g − 1 z ). ◮ Highest weight ( j , n ), j , n half integers, j ≥ 0; weight vectors z j − m z j + m v ( j , n ) √ = ( j − m )!( j + m )! with − j ≤ m ≤ j . 1 2 m

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Representations of U (2): π ( j , n ) = Pol 2 j ( C 2 ) ⊗ det j + n . The action on f ∈ Pol 2 j ( C 2 ) is given by f ( z ) �→ f ( g − 1 z ). ◮ Highest weight ( j , n ), j , n half integers, j ≥ 0; weight vectors z j − m z j + m v ( j , n ) √ = ( j − m )!( j + m )! with − j ≤ m ≤ j . 1 2 m ◮ Action of Pauli matrices: weight vectors are normalized such that γ 0 v ( j , n ) = i nv ( j , n ) , γ 3 v ( j , n ) = i mv ( j , n ) m m m m ( j ∓ m )( j ± m + 1) v j ( γ 1 ± i γ 2 ) v j � m = − i m ± 1

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Representations of U (2): π ( j , n ) = Pol 2 j ( C 2 ) ⊗ det j + n . The action on f ∈ Pol 2 j ( C 2 ) is given by f ( z ) �→ f ( g − 1 z ). ◮ Highest weight ( j , n ), j , n half integers, j ≥ 0; weight vectors z j − m z j + m v ( j , n ) √ = ( j − m )!( j + m )! with − j ≤ m ≤ j . 1 2 m ◮ Action of Pauli matrices: weight vectors are normalized such that γ 0 v ( j , n ) = i nv ( j , n ) , γ 3 v ( j , n ) = i mv ( j , n ) m m m m ( j ∓ m )( j ± m + 1) v j ( γ 1 ± i γ 2 ) v j � m = − i m ± 1 ◮ The raising and lowering operators

� � � � � � Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Representations of U (2): π ( j , n ) = Pol 2 j ( C 2 ) ⊗ det j + n . The action on f ∈ Pol 2 j ( C 2 ) is given by f ( z ) �→ f ( g − 1 z ). ◮ Highest weight ( j , n ), j , n half integers, j ≥ 0; weight vectors z j − m z j + m v ( j , n ) √ = ( j − m )!( j + m )! with − j ≤ m ≤ j . 1 2 m ◮ Action of Pauli matrices: weight vectors are normalized such that γ 0 v ( j , n ) = i nv ( j , n ) , γ 3 v ( j , n ) = i mv ( j , n ) m m m m ( j ∓ m )( j ± m + 1) v j ( γ 1 ± i γ 2 ) v j � m = − i m ± 1 ◮ The raising and lowering operators ◮ 2 j ≡ 0 mod 2: γ 1+ i γ 2 � . . . γ 1+ i γ 2 � γ 1+ i γ 2 � γ 1 − i γ 2 � γ 1+ i γ 2 � . . . γ 1+ i γ 2 � − j − 1 0 1 j γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2

� � � � � � � � � � � Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Representations of U (2): π ( j , n ) = Pol 2 j ( C 2 ) ⊗ det j + n . The action on f ∈ Pol 2 j ( C 2 ) is given by f ( z ) �→ f ( g − 1 z ). ◮ Highest weight ( j , n ), j , n half integers, j ≥ 0; weight vectors z j − m z j + m v ( j , n ) √ = ( j − m )!( j + m )! with − j ≤ m ≤ j . 1 2 m ◮ Action of Pauli matrices: weight vectors are normalized such that γ 0 v ( j , n ) = i nv ( j , n ) , γ 3 v ( j , n ) = i mv ( j , n ) m m m m ( j ∓ m )( j ± m + 1) v j ( γ 1 ± i γ 2 ) v j � m = − i m ± 1 ◮ The raising and lowering operators ◮ 2 j ≡ 0 mod 2: γ 1+ i γ 2 � . . . γ 1+ i γ 2 � γ 1+ i γ 2 � γ 1 − i γ 2 � γ 1+ i γ 2 � . . . γ 1+ i γ 2 � − j − 1 0 1 j γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 ◮ 2 j ≡ 1 mod 2: γ 1+ i γ 2 � . . . γ 1+ i γ 2 � γ 1+ i γ 2 � γ 1+ i γ 2 � . . . γ 1+ i γ 2 � − 1 1 − j j 2 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2 γ 1 − i γ 2

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Matrix coefficients: Wigner D-functions D ( j , n ) m 2 , e ζ γ 0 e φ γ 3 e θ γ 2 e ψ γ 3 v j m 1 , m 2 ( ζ, φ, θ, ψ ) = � v j m 1 � K m 2 e i n ζ e i ( m 1 ψ + m 2 φ ) d ( j , n ) = c j m 1 c j m 1 , m 2 ( θ ) . in which

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Matrix coefficients: Wigner D-functions D ( j , n ) m 2 , e ζ γ 0 e φ γ 3 e θ γ 2 e ψ γ 3 v j m 1 , m 2 ( ζ, φ, θ, ψ ) = � v j m 1 � K m 2 e i n ζ e i ( m 1 ψ + m 2 φ ) d ( j , n ) = c j m 1 c j m 1 , m 2 ( θ ) . in which ◮ c j � m = ( j + m )!( j − m )! is a normalization factor

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) ◮ Matrix coefficients: Wigner D-functions D ( j , n ) m 2 , e ζ γ 0 e φ γ 3 e θ γ 2 e ψ γ 3 v j m 1 , m 2 ( ζ, φ, θ, ψ ) = � v j m 1 � K m 2 e i n ζ e i ( m 1 ψ + m 2 φ ) d ( j , n ) = c j m 1 c j m 1 , m 2 ( θ ) . in which ◮ c j � m = ( j + m )!( j − m )! is a normalization factor ◮ d ( j , n ) m 1 , m 2 ( θ ) is given by the hypergeometric sum: min( j − m 2 , j + m 1) ( − 1) m 2 − m 1+ p d ( j , n ) � m 1 , m 2 ( θ ) = ( j + m 1 − p )! p !( m 2 − m 1 + p )!( j − m 2 − p )! p =max(0 , m 1 − m 2) � θ � θ � � sin m 2 − m 1+2 p cos 2 j + m 1 − m 2 − 2 p . 2 2

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. � � x + 1 � n � n + α � − n , − n − β, α + 1; x − 1 � P α,β ( x ) = . 2 F 1 n 2 x + 1 n where n ≥ 0.

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. � � x + 1 � n � n + α � − n , − n − β, α + 1; x − 1 � P α,β ( x ) = . 2 F 1 n 2 x + 1 n where n ≥ 0. ◮ d ( j , n ) m 1 , m 2 ( θ ) can be expressed in terms of Jacobi polynomials � m 1 − m 2 � � m 1 + m 2 � sin θ cos θ 2 2 d ( j , n ) P ( m 1 − m 2 , m 1 + m 2 ) m 1 , m 2 ( θ ) = (cos θ ) j − m 1 ( j + m 2 )!( j − m 2 )!

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) Why we do this? ◮ Generating function for Jacobi polynomials ∞ � α � � β � 1 + 1 1 + 1 ( x ) t n = � P ( α − n ,β − n ) 2 ( x + 1) t 2 ( x − 1) t n n =0

Clebsch-Gordan Coefficients and Principal Series Representations Preparation: SU(2) The Compact Group U (2) Why we do this? ◮ Generating function for Jacobi polynomials ∞ � α � � β � 1 + 1 1 + 1 ( x ) t n = � P ( α − n ,β − n ) 2 ( x + 1) t 2 ( x − 1) t n n =0 ◮ Product of Wigner D -functions: Clebsch-Gordan coefficients � � � � D ( j 1 , n 1 ) m 11 , m 12 D ( j 2 , n 2 ) � J , M 1 J , M 2 D ( J , n 1 + n 2 ) m 21 , m 22 = j 1 , m 11 , j 2 , m 21 j 1 , m 12 , j 2 , m 22 M 1 , M 2 | j 1 − j 2 |≤ J ≤ j 1 + j 2 J −| j 1 − j 2 |∈ Z The Clebsch-Gordan coefficients can be computed following a recursive procedure.

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.

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 T e ( g / k ) ⊗ τ ( j , n ) ∼ = p C ⊗ τ ( j , n ) − → C ∞ ( K ) which sends X ⊗ v to X · v . (compare SL (3 , R ) Buttcane-Miller ’17)

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 T e ( g / k ) ⊗ τ ( j , n ) ∼ = p C ⊗ 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 g α is a subset of k or p .

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)).

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)). � 1 2 � 0 ◮ SU (2 , 1) = { g ∈ SL (3 , C ) | ¯ g t Jg = J } , J = ; 0 − 1

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)). � 1 2 � 0 ◮ SU (2 , 1) = { g ∈ SL (3 , C ) | ¯ g t Jg = J } , J = ; 0 − 1 g t ) − 1 J − 1 an antiholomorphic involution, SU (2 , 1) = SL (3 , C ) σ ; ◮ σ ( g ) = J (¯

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)). � 1 2 � 0 ◮ SU (2 , 1) = { g ∈ SL (3 , C ) | ¯ g t Jg = J } , J = ; 0 − 1 g t ) − 1 J − 1 an antiholomorphic involution, SU (2 , 1) = SL (3 , C ) σ ; ◮ σ ( g ) = J (¯ � U (2) 0 � g t ) − 1 a Cartan involution, K = ◮ θ ( g ) = (¯ = G θ ; (det) − 1 0

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)). � 1 2 � 0 ◮ SU (2 , 1) = { g ∈ SL (3 , C ) | ¯ g t Jg = J } , J = ; 0 − 1 g t ) − 1 J − 1 an antiholomorphic involution, SU (2 , 1) = SL (3 , C ) σ ; ◮ σ ( g ) = J (¯ � U (2) 0 � g t ) − 1 a Cartan involution, K = ◮ θ ( g ) = (¯ = G θ ; (det) − 1 0 ◮ For SU (2 , 1), ∆ + c = { α 1 } , ∆ + nc = { α 2 , α 1 + α 2 } α 2 α 1 + α 2 ̟ 2 ̟ 1 α 1 Green: ∆ nc ; Blue: ∆ c

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)). � 1 2 � 0 ◮ SU (2 , 1) = { g ∈ SL (3 , C ) | ¯ g t Jg = J } , J = ; 0 − 1 g t ) − 1 J − 1 an antiholomorphic involution, SU (2 , 1) = SL (3 , C ) σ ; ◮ σ ( g ) = J (¯ � U (2) 0 � g t ) − 1 a Cartan involution, K = ◮ θ ( g ) = (¯ = G θ ; (det) − 1 0 ◮ For SU (2 , 1), ∆ + c = { α 1 } , ∆ + nc = { α 2 , α 1 + α 2 } α 2 α 1 + α 2 ̟ 2 ̟ 1 α 1 Green: ∆ nc ; Blue: ∆ c ◮ The two irreducible pieces of p C are p ± C = span { u (1 / 2 , ± 3 / 2) } m ∈{− 1 / 2 , 1 / 2 } m

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU ( n , 1) Example: SU (2 , 1) Induction Data ◮ Minimal parabolic P = MAN , � i w z − i w � � cosh t 0 sinh t � e i s � 0 0 � exp 0 e − 2 i s − ¯ z 0 z ¯ 0 0 0 0 sinh t 0 cosh t i w z − i w e i s 0 0

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU ( n , 1) Example: SU (2 , 1) Induction Data ◮ Minimal parabolic P = MAN , � i w z − i w � � cosh t 0 sinh t � e i s � 0 0 � exp 0 e − 2 i s − ¯ z 0 z ¯ 0 0 0 0 sinh t 0 cosh t i w z − i w e i s 0 0 ◮ M ∼ = S 1 , character on M given by δ ∈ Z

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU ( n , 1) Example: SU (2 , 1) Induction Data ◮ Minimal parabolic P = MAN , � i w z − i w � � cosh t 0 sinh t � e i s � 0 0 � exp 0 e − 2 i s − ¯ z 0 z ¯ 0 0 0 0 sinh t 0 cosh t i w z − i w e i s 0 0 ◮ M ∼ = S 1 , character on M given by δ ∈ Z ◮ A ∼ = R > 0 , character on A given by a λ ∈ C

Clebsch-Gordan Coefficients and Principal Series Representations Rank 1 Groups SU ( n , 1) Example: SU (2 , 1) Induction Data ◮ Minimal parabolic P = MAN , � i w z − i w � � cosh t 0 sinh t � e i s � 0 0 � exp 0 e − 2 i s − ¯ z 0 ¯ z 0 0 0 0 sinh t 0 cosh t i w z − i w e i s 0 0 ◮ M ∼ = S 1 , character on M given by δ ∈ Z ◮ A ∼ = R > 0 , character on A given by a λ ∈ C ◮ They define a character χ δ,λ sending the above element to e i δ t + λ t .

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 I P ( χ δ,λ ) decompose into K -isotypic spaces: � τ ( j , n ) I P ( χ δ,λ ) = − 3 j + δ ≤ n ≤ 3 j + δ

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 I P ( χ δ,λ ) decompose into K -isotypic spaces: � τ ( j , n ) I P ( χ δ,λ ) = − 3 j + δ ≤ n ≤ 3 j + δ ◮ Each τ ( j , n ) has decomposition: τ ( j , n ) = � C D ( j , n ) m 1 , m 2 m 2 ∈{− j , − j +1 ,..., j } m 1 =( n − δ ) / 3

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 I P ( χ δ,λ ) decompose into K -isotypic spaces: � τ ( j , n ) I P ( χ δ,λ ) = − 3 j + δ ≤ n ≤ 3 j + δ ◮ Each τ ( j , n ) has decomposition: τ ( j , n ) = � C D ( j , n ) m 1 , m 2 m 2 ∈{− j , − j +1 ,..., j } m 1 =( n − δ ) / 3 ◮ 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 } .

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 d l : p ⊗ τ ( j , n ) − → C ∞ ( K ) is(compare Buttcane-Miller ’17): ( 1 2 , ± 3 2 ) ) D ( j , n ) d l ( u m 1 , m 2 = m � j + j 0 , m 2 − m 1 ( j + j 0 , n ∓ 3 � 2 ) � 2 √ 2 j + 1 q j 0 , ∓ κ j 0 , ∓ ( j , n , m 1 ; λ ) D J , m 2 , 1 2 , − m m 1 ∓ 1 2 , m 2 − m j 0 ∈{± 1 2 }

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 d l : p ⊗ τ ( j , n ) − → C ∞ ( K ) is(compare Buttcane-Miller ’17): ( 1 2 , ± 3 2 ) ) D ( j , n ) d l ( u m 1 , m 2 = m � j + j 0 , m 2 − m 1 ( j + j 0 , n ∓ 3 � 2 ) � 2 √ 2 j + 1 q j 0 , ∓ κ j 0 , ∓ ( j , n , m 1 ; λ ) D J , m 2 , 1 2 , − m m 1 ∓ 1 2 , m 2 − m j 0 ∈{± 1 2 } ◮ The coefficients q j 0 , ± are in the following tables: q j 0 , ∓ - + √ j + m 1 √ j − m 1 j 0 = − 1 2 √ j − m 1 + 1 √ j + m 1 + 1 j 0 = 1 2

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 d l : p ⊗ τ ( j , n ) − → C ∞ ( K ) is(compare Buttcane-Miller ’17): ( 1 2 , ± 3 2 ) ) D ( j , n ) d l ( u m 1 , m 2 = m � j + j 0 , m 2 − m 1 ( j + j 0 , n ∓ 3 � 2 ) � 2 √ 2 j + 1 q j 0 , ∓ κ j 0 , ∓ ( j , n , m 1 ; λ ) D J , m 2 , 1 2 , − m m 1 ∓ 1 2 , m 2 − m j 0 ∈{± 1 2 } ◮ The coefficients q j 0 , ± are in the following tables: q j 0 , ∓ - + √ j + m 1 √ j − m 1 j 0 = − 1 2 √ j − m 1 + 1 √ j + m 1 + 1 j 0 = 1 2 ◮ The κ j 0 , ∓ ’s are affine linear functions in λ : κ j 0 , ∓ - + j 0 = − 1 2 j − m 1 + n − ( λ + ρ 0 ) + 2 2 j + m 1 − n − ( λ + ρ 0 ) + 2 2 j 0 = 1 − 2 j − m 1 + n − ( λ + ρ 0 ) 2 j − m 1 + n + ( λ + ρ 0 ) 2

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) V W Legend: means there exists a submodule V ⊂ I P ( χ δ,λ ) such that I P ( χ δ,λ ) / V ∼ = W . λ − δ ( λ − δ , 3 λ + δ ) 4 4 l I 1 ( λ 2 , − δ (0 , δ ) k 2 ) λ + δ , − 3 λ + δ ( λ + δ ) 4 4 The composition series when χ δ,λ ∈ Weyl Chamber I 1 : V H V fin W 1 W 2 ⊕

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) V W Legend: means there exists a submodule V ⊂ I P ( χ δ,λ ) such that I P ( χ δ,λ ) / V ∼ = W . λ − δ II 1 ( λ − δ , 3 λ + δ ) 4 4 λ + δ ( − λ − δ , − 3 λ + δ ) 4 4 (0 , δ ) The composition series when χ δ,λ ∈ Weyl Chamber II 1 : V 2 V H W 2 ⊕

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) V W Legend: means there exists a submodule V ⊂ I P ( χ δ,λ ) such that I P ( χ δ,λ ) / V ∼ = W . λ − δ II 2 ( − λ − δ , − 3 λ + δ ) 4 4 λ + δ ( λ − δ , 3 λ + δ ) 4 4 (0 , δ ) The composition series when χ δ,λ ∈ Weyl Chamber II 2 : V 2 V H W 2 ⊕

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) V W Legend: means there exists a submodule V ⊂ I P ( χ δ,λ ) such that I P ( χ δ,λ ) / V ∼ = W . λ − δ ( λ + δ , − 3 λ + δ ) 4 4 ( λ 2 , − δ (0 , δ ) 2 ) λ + δ I 2 ( λ − δ , 3 λ + δ ) 4 4 The composition series when χ δ,λ ∈ Weyl Chamber I 2 : V fin V H W 1 W 2 ⊕

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) V W Legend: means there exists a submodule V ⊂ I P ( χ δ,λ ) such that I P ( χ δ,λ ) / V ∼ = W . λ − δ (0 , δ ) ( − λ + δ , 3 λ + δ ) 4 4 ( λ + δ , − 3 λ + δ ) 4 4 λ + δ III 1 The composition series when χ δ,λ ∈ Weyl Chamber III 1 : V 1 V H W 1 ⊕

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) V W Legend: means there exists a submodule V ⊂ I P ( χ δ,λ ) such that I P ( χ δ,λ ) / V ∼ = W . λ − δ (0 , δ ) ( λ + δ , − 3 λ + δ ) 4 4 ( − λ + δ , 3 λ + δ ) 4 4 λ + δ III 2 The composition series when χ δ,λ ∈ Weyl Chamber III 2 : V 1 V H W 1 ⊕

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).

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). � 0 2 � 1 2 ◮ Sp (4 , R ) = { g ∈ SL (4 , C ) | g t Jg = J } , J = − 1 2 0 2

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). � 0 2 � 1 2 ◮ Sp (4 , R ) = { g ∈ SL (4 , C ) | g t Jg = J } , J = − 1 2 0 2 g an antiholomorphic involution, Sp (4 , R ) = Sp (4 , C ) σ ; ◮ σ ( g ) = ¯

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). � 0 2 � 1 2 ◮ Sp (4 , R ) = { g ∈ SL (4 , C ) | g t Jg = J } , J = − 1 2 0 2 g an antiholomorphic involution, Sp (4 , R ) = Sp (4 , C ) σ ; ◮ σ ( g ) = ¯ � � A B ◮ θ ( g ) = ( g t ) − 1 a Cartan involution, k = = g θ ; − B A

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). � 0 2 � 1 2 ◮ Sp (4 , R ) = { g ∈ SL (4 , C ) | g t Jg = J } , J = − 1 2 0 2 g an antiholomorphic involution, Sp (4 , R ) = Sp (4 , C ) σ ; ◮ σ ( g ) = ¯ � � A B ◮ θ ( g ) = ( g t ) − 1 a Cartan involution, k = = g θ ; − B A � � A B ◮ k ∼ = u (2) via �→ A + i B . − B A

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). � 0 2 � 1 2 ◮ Sp (4 , R ) = { g ∈ SL (4 , C ) | g t Jg = J } , J = − 1 2 0 2 g an antiholomorphic involution, Sp (4 , R ) = Sp (4 , C ) σ ; ◮ σ ( g ) = ¯ � � A B ◮ θ ( g ) = ( g t ) − 1 a Cartan involution, k = = g θ ; − B A � � A B ◮ k ∼ = u (2) via �→ A + i B . − B A ◮ ∆ + c = { α 1 } , ∆ + nc = { α 2 , α 1 + α 2 , 2 α 1 + α 2 } α 2 ̟ 2 α 1 + α 2 ̟ 1 2 α 1 + α 2 α 1 Green: ∆ nc ; Blue: ∆ c

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). � 0 2 � 1 2 ◮ Sp (4 , R ) = { g ∈ SL (4 , C ) | g t Jg = J } , J = − 1 2 0 2 g an antiholomorphic involution, Sp (4 , R ) = Sp (4 , C ) σ ; ◮ σ ( g ) = ¯ � � A B ◮ θ ( g ) = ( g t ) − 1 a Cartan involution, k = = g θ ; − B A � � A B ◮ k ∼ = u (2) via �→ A + i B . − B A ◮ ∆ + c = { α 1 } , ∆ + nc = { α 2 , α 1 + α 2 , 2 α 1 + α 2 } α 2 ̟ 2 α 1 + α 2 ̟ 1 2 α 1 + α 2 α 1 Green: ∆ nc ; Blue: ∆ c C = span { u (1 , ± 1) ◮ The two irreducible pieces of p C are p ± } m ∈{− 1 , 0 , 1 } m

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Example: Sp (4 , R ) Induction Data ◮ Minimal parabolic: P = MAN � ǫ 2 � t 1 0 � ǫ 1 � 0 0 � � 1 0 0 0 − 1 0 0 0 0 t 2 0 0 0 − 1 0 0 0 1 0 0 N 0 0 1 0 0 0 − 1 0 0 0 1 / t 1 0 0 0 0 − 1 0 0 0 1 0 0 0 1 / t 2

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Example: Sp (4 , R ) Induction Data ◮ Minimal parabolic: P = MAN � ǫ 2 � t 1 0 � ǫ 1 � 0 0 � � 1 0 0 0 − 1 0 0 0 0 t 2 0 0 0 − 1 0 0 0 1 0 0 N 0 0 1 0 0 0 − 1 0 0 0 1 / t 1 0 0 0 0 − 1 0 0 0 1 0 0 0 1 / t 2 ◮ M ∼ = Z 2 × Z 2 , character on M given by ( δ 1 , δ 2 ), δ i ∈ { 0 , 1 }

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Example: Sp (4 , R ) Induction Data ◮ Minimal parabolic: P = MAN � ǫ 2 � t 1 0 � ǫ 1 � 0 0 � � 1 0 0 0 − 1 0 0 0 0 t 2 0 0 0 − 1 0 0 0 1 0 0 N 0 0 1 0 0 0 − 1 0 0 0 1 / t 1 0 0 0 0 − 1 0 0 0 1 0 0 0 1 / t 2 ◮ M ∼ = Z 2 × Z 2 , character on M given by ( δ 1 , δ 2 ), δ i ∈ { 0 , 1 } ◮ A ∼ = ( R > 0 ) 2 , character on A given by ( λ 1 , λ 2 ) ∈ C 2

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Example: Sp (4 , R ) Induction Data ◮ Minimal parabolic: P = MAN � ǫ 2 � t 1 0 � ǫ 1 � 0 0 � � 1 0 0 0 − 1 0 0 0 0 t 2 0 0 0 − 1 0 0 0 1 0 0 N 0 0 1 0 0 0 − 1 0 0 0 1 / t 1 0 0 0 0 − 1 0 0 0 1 0 0 0 1 / t 2 ◮ M ∼ = Z 2 × Z 2 , character on M given by ( δ 1 , δ 2 ), δ i ∈ { 0 , 1 } ◮ A ∼ = ( R > 0 ) 2 , character on A given by ( λ 1 , λ 2 ) ∈ C 2 ◮ They define a character χ δ,λ sending the above element to ( − 1) δ 1 ǫ 1 + δ 2 ǫ 2 t λ 1 1 t λ 2 2

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Sp (4 , R ) Principal Series ◮ The restriction of the principal series I P ( χ δ,λ ) to K can be decomposed as a direct sum of K -isotypic spaces τ ( j , n ) : � τ ( j , n ) . I P ( χ δ,λ ) = ( j , n ) ∈ KTypes ( δ 1 ,δ 2 )

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Sp (4 , R ) Principal Series ◮ The restriction of the principal series I P ( χ δ,λ ) to K can be decomposed as a direct sum of K -isotypic spaces τ ( j , n ) : � τ ( j , n ) . I P ( χ δ,λ ) = ( j , n ) ∈ KTypes ( δ 1 ,δ 2 ) ◮ Each space τ ( j , n ) is τ ( j , n ) = C D ( j , n ) � m 1 , m 2 m 2 ∈{− j , − j +1 ,..., j } m 1 ∈ M ( j , n ; δ 1 ,δ 2 )

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Sp (4 , R ) Principal Series ◮ The restriction of the principal series I P ( χ δ,λ ) to K can be decomposed as a direct sum of K -isotypic spaces τ ( j , n ) : � τ ( j , n ) . I P ( χ δ,λ ) = ( j , n ) ∈ KTypes ( δ 1 ,δ 2 ) ◮ Each space τ ( j , n ) is τ ( j , n ) = C D ( j , n ) � m 1 , m 2 m 2 ∈{− j , − j +1 ,..., j } m 1 ∈ M ( j , n ; δ 1 ,δ 2 ) ◮ The two sets of admissible j , n , m 1 , m 2 ’s are defined as: KTypes ( δ 1 , δ 2 ) = { ( j , n ) ∈ 1 2 Z + × 1 2 Z | 2 j ≡ δ 2 − δ 1 and 2 n ≡ δ 2 + δ 1 mod 2 } M ( j , n ; δ 1 , δ 2 ) = { m 1 ∈ {− j , − j + 1 , . . . , j − 1 , j }| n − m 1 ≡ δ 1 and n + m 1 ≡ δ 2 mod 2 } .

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Sp (4 , R ) Principal Series The homomorphism d l : p ⊗ τ ( j , n ) − → C ∞ ( K ) is m 1 , m 2 = ( − 1) m 1 i d l ( u (1 , ± 1) ) D ( j , n ) m 2 � � � � j + j 0 , m 2 − m C j + j 0 q j 0 ,ε κ ± , j 0 ,ε ( j , n , m 1 ; λ ) D ( j + j 0 , n ∓ 1) m 1 + ε, m 2 − m j , m 2 , 1 , − m j 0 ∈{− 1 , 0 , 1 } ε = ± 1 with the coefficients given the tables in the next frame.

Clebsch-Gordan Coefficients and Principal Series Representations Sp (4 , R ) Example Sp (4 , R ) Principal Series C j + j 0 j − 1 2 (2 j + 1) − 1 j 0 = − 1 2 j − 1 2 ( j + 1) − 1 j 0 = 0 2 ( j + 1) − 1 2 (2 j + 1) − 1 j 0 = 1 2 q j 0 ,ε ε = − 1 ε = 1 � � j 0 = − 1 ( j + m 1 − 1)( j + m 1 ) ( j − m 1 − 1)( j − m 1 ) � � j 0 = 0 ( j + m 1 )( j − m 1 + 1) ( j − m 1 )( j + m 1 + 1) � � j 0 = 1 ( j − m 1 + 1)( j − m 1 + 2) ( j + m 1 + 1)( j + m 1 + 2) κ + , j 0 ,ε ε = − 1 ε = 1 j 0 = − 1 2 j − m 1 + n − � ˇ α 1 , λ + ρ 0 � + 2 n − m 1 − � ˇ α 2 , λ + ρ 0 � j 0 = 0 n − m 1 − � ˇ α 1 , λ + ρ 0 � + 2 m 1 − n + � ˇ α 2 , λ + ρ 0 � j 0 = 1 − 2 j − m 1 + n − � ˇ α 1 , λ + ρ 0 � n − m 1 − � ˇ α 2 , λ + ρ 0 � κ − , j 0 ,ε ε = − 1 ε = 1 j 0 = − 1 − n + m 1 − � ˇ α 2 , λ + ρ 0 � 2 j + m 1 − n − � ˇ α 1 , λ + ρ 0 � + 2 j 0 = 0 − n + m 1 − � ˇ α 2 , λ + ρ 0 � − m 1 + n + � ˇ α 1 , λ + ρ 0 � − 2 j 0 = 1 − n + m 1 − � ˇ α 2 , λ + ρ 0 � 2 j + m 1 − n − � ˇ α 1 , λ + ρ 0 � + 2

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 ( χ δ,λ ) − → I ( χ δ, − λ )

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 ( χ δ,λ ) − → I ( χ δ, − λ ) ◮ It is defines as the integral operator: � ∞ � 0 − 1 � � 1 � x ( A ( λ ) f )( g ) = f ( g ) 1 0 0 1 −∞

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 ( χ δ,λ ) − → I ( χ δ, − λ ) ◮ It is defines as the integral operator: � ∞ � 0 − 1 � � 1 � x ( A ( λ ) f )( g ) = f ( g ) 1 0 0 1 −∞ ◮ By computing the Iwasawa decompositions and use the vector space structure of the principal series, denote the function e i k θ ∈ I P ( χ δ,λ ) by φ λ, k Γ( λ +1 2 )Γ( λ 2 ) A ( λ ) φ λ, k = ( − i ) k √ π φ − λ, k Γ( λ +1+ k )Γ( λ − k +1 ) 2 2 When δ = 0 , λ ∈ R , then all k ∈ Z are even. The intertwining operator defines a Hermitian pairing on I ( χ δ,λ ).

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators What are Intertwining Operators

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators What are Intertwining Operators ◮ For w ∈ W , f ∈ I P ( χ δ,λ ), define the intertwining operator � A ( w , χ δ,λ ) f ( g ) = N ∩ N w f ( gw ¯ n ) d ¯ n ¯ such that it intertwines I P ( χ δ,λ ) and I P ( w χ δ,λ ) A ( w , χ δ,λ ) π P ( χ δ,λ )( g ) = π P ( w χ δ,λ )( g ) A ( w , χ δ,λ )

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators What are Intertwining Operators ◮ For w ∈ W , f ∈ I P ( χ δ,λ ), define the intertwining operator � A ( w , χ δ,λ ) f ( g ) = N ∩ N w f ( gw ¯ n ) d ¯ n ¯ such that it intertwines I P ( χ δ,λ ) and I P ( w χ δ,λ ) A ( w , χ δ,λ ) π P ( χ δ,λ )( g ) = π P ( w χ δ,λ )( g ) A ( w , χ δ,λ ) ◮ It can be meromorphically continuated to all λ

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators What are Intertwining Operators ◮ For w ∈ W , f ∈ I P ( χ δ,λ ), define the intertwining operator � A ( w , χ δ,λ ) f ( g ) = N ∩ N w f ( gw ¯ n ) d ¯ n ¯ such that it intertwines I P ( χ δ,λ ) and I P ( w χ δ,λ ) A ( w , χ δ,λ ) π P ( χ δ,λ )( g ) = π P ( w χ δ,λ )( g ) A ( w , χ δ,λ ) ◮ It can be meromorphically continuated to all λ ◮ For dominant λ , A ( w 0 ) defines the Langlands quotient of the principal series by quotienting out its kernel.

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators What are Intertwining Operators ◮ For w ∈ W , f ∈ I P ( χ δ,λ ), define the intertwining operator � A ( w , χ δ,λ ) f ( g ) = N ∩ N w f ( gw ¯ n ) d ¯ n ¯ such that it intertwines I P ( χ δ,λ ) and I P ( w χ δ,λ ) A ( w , χ δ,λ ) π P ( χ δ,λ )( g ) = π P ( w χ δ,λ )( g ) A ( w , χ δ,λ ) ◮ It can be meromorphically continuated to all λ ◮ For dominant λ , A ( w 0 ) defines the Langlands quotient of the principal series by quotienting out its kernel. ◮ (Knapp, Wallach, Barbasch)If w 0 δ ∼ = δ, w 0 λ = − ¯ λ Choose an isomorphism τ : w 0 δ ∼ = δ , an hermitian form on the Langlands quotient is given by � v 1 , v 2 � = � v 1 , τ ◦ A ′ ( w 0 , χ δ,λ ) v 2 �

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 ) m 1 , m 2 ,

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 ) m 1 , m 2 , ◮ (Kostant ’62 Johnson-Wallach ’75, Fabec ’91) The long intertwining operator A ( w , χ δ,λ ) acts on each D ( j , n ) � � m 1 , m 2 as a scalar A ( w , χ δ,λ ) m 1 , with a closed form formula: � � A ( w , χ δ,λ ) m 1 = � � � � j − m 1 − λ + δ j + m 1 − λ − δ Γ + 1 Γ + 1 − π 2 2 − λ − 2 ( − 1) λ + δ Γ( λ ) 2 2 � � � � � � � � 1 − λ − δ 1 − λ + δ j − m 1 + λ − δ j + m 1 + λ + δ Γ Γ Γ + 1 Γ + 1 2 2 2 2

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 ) m 1 , m 2 , ◮ (Kostant ’62 Johnson-Wallach ’75, Fabec ’91) The long intertwining operator A ( w , χ δ,λ ) acts on each D ( j , n ) � � m 1 , m 2 as a scalar A ( w , χ δ,λ ) m 1 , with a closed form formula: � � A ( w , χ δ,λ ) m 1 = � � � � j − m 1 − λ + δ j + m 1 − λ − δ Γ + 1 Γ + 1 − π 2 2 − λ − 2 ( − 1) λ + δ Γ( λ ) 2 2 � � � � � � � � 1 − λ − δ 1 − λ + δ j − m 1 + λ − δ j + m 1 + λ + δ Γ Γ Γ + 1 Γ + 1 2 2 2 2 ◮ Normalized intertwining operator: divide by the Harish-Chandra c-function , the normalized intertwining operator becomes a rational function � ( j − m 1 ) � � ( j + m 1 ) � 2 − λ − δ 2 − λ + δ 2 2 � � A ( w , χ δ,λ ) m 1 = � ( j − m 1 ) � � ( j + m 1 ) � 2+ λ − δ 2+ λ + δ 2 2

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ◮ A ( w 0 , λ ) can be factorized into 4 intertwining operators corresponding to simple reflections: A ( w 0 , λ ) = A ( w α 2 , w α 1 w α 2 w α 1 λ ) A ( w α 1 , w α 2 w α 1 λ ) A ( w α 2 , w α 1 λ ) A ( w α 1 , λ ) .

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ◮ A ( w 0 , λ ) can be factorized into 4 intertwining operators corresponding to simple reflections: A ( w 0 , λ ) = A ( w α 2 , w α 1 w α 2 w α 1 λ ) A ( w α 1 , w α 2 w α 1 λ ) A ( w α 2 , w α 1 λ ) A ( w α 1 , λ ) . � λ 1 − λ 2+1 w α 1 � ◮ ( λ 1 , λ 2 ) → ( λ 2 , λ 1 ): [ A ( w α 1 , λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ◮ A ( w 0 , λ ) can be factorized into 4 intertwining operators corresponding to simple reflections: A ( w 0 , λ ) = A ( w α 2 , w α 1 w α 2 w α 1 λ ) A ( w α 1 , w α 2 w α 1 λ ) A ( w α 2 , w α 1 λ ) A ( w α 1 , λ ) . � λ 1 − λ 2+1 w α 1 � ◮ ( λ 1 , λ 2 ) → ( λ 2 , λ 1 ): [ A ( w α 1 , λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2 � λ 1+1 w α 2 � ◮ ( λ 2 , λ 1 ) → ( λ 2 , − λ 1 ): [ A ( w α 2 , w α 1 λ )] j , n m 1 , m 2 = T n − − − δ m 1 , m 2 m 1 2

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ◮ A ( w 0 , λ ) can be factorized into 4 intertwining operators corresponding to simple reflections: A ( w 0 , λ ) = A ( w α 2 , w α 1 w α 2 w α 1 λ ) A ( w α 1 , w α 2 w α 1 λ ) A ( w α 2 , w α 1 λ ) A ( w α 1 , λ ) . � λ 1 − λ 2+1 w α 1 � ◮ ( λ 1 , λ 2 ) → ( λ 2 , λ 1 ): [ A ( w α 1 , λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2 � λ 1+1 w α 2 � ◮ ( λ 2 , λ 1 ) → ( λ 2 , − λ 1 ): [ A ( w α 2 , w α 1 λ )] j , n m 1 , m 2 = T n − − − δ m 1 , m 2 m 1 2 � λ 1+ λ 2+1 w α 1 � ◮ ( λ 2 , − λ 1 ) → ( − λ 1 , λ 2 ): [ A ( w α 1 , w α 2 w α 1 λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ◮ A ( w 0 , λ ) can be factorized into 4 intertwining operators corresponding to simple reflections: A ( w 0 , λ ) = A ( w α 2 , w α 1 w α 2 w α 1 λ ) A ( w α 1 , w α 2 w α 1 λ ) A ( w α 2 , w α 1 λ ) A ( w α 1 , λ ) . � λ 1 − λ 2+1 w α 1 � ◮ ( λ 1 , λ 2 ) → ( λ 2 , λ 1 ): [ A ( w α 1 , λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2 � λ 1+1 w α 2 � ◮ ( λ 2 , λ 1 ) → ( λ 2 , − λ 1 ): [ A ( w α 2 , w α 1 λ )] j , n m 1 , m 2 = T n − − − δ m 1 , m 2 m 1 2 � λ 1+ λ 2+1 w α 1 � ◮ ( λ 2 , − λ 1 ) → ( − λ 1 , λ 2 ): [ A ( w α 1 , w α 2 w α 1 λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2 � λ 2+1 w α 2 � ◮ ( − λ 1 , λ 2 ) → ( − λ 1 , − λ 2 ): [ A ( w α 2 , w α 1 w α 2 w α 1 λ )] j , n m 1 , m 2 = T n − − − δ m 1 , m 2 m 1 2 ◮ Normalization: divide each simple intertwining operator S j , n m 1 , m 2 ( z ) , T n m 1 ( z ) by √ π Γ( z − 1 / 2) , get rational functions S j , n m 1 , m 2 ( z ) , T n m 1 ( z ) such that the corresponding Γ( z ) normalized simple intertwining operator satisfies A ( − λ ) A ( λ ) = 1.

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ◮ A ( w 0 , λ ) can be factorized into 4 intertwining operators corresponding to simple reflections: A ( w 0 , λ ) = A ( w α 2 , w α 1 w α 2 w α 1 λ ) A ( w α 1 , w α 2 w α 1 λ ) A ( w α 2 , w α 1 λ ) A ( w α 1 , λ ) . � λ 1 − λ 2+1 w α 1 � ◮ ( λ 1 , λ 2 ) → ( λ 2 , λ 1 ): [ A ( w α 1 , λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2 � λ 1+1 w α 2 � ◮ ( λ 2 , λ 1 ) → ( λ 2 , − λ 1 ): [ A ( w α 2 , w α 1 λ )] j , n m 1 , m 2 = T n − − − δ m 1 , m 2 m 1 2 � λ 1+ λ 2+1 w α 1 � ◮ ( λ 2 , − λ 1 ) → ( − λ 1 , λ 2 ): [ A ( w α 1 , w α 2 w α 1 λ )] j , n m 1 , m 2 = S j , n − − − m 1 , m 2 2 � λ 2+1 w α 2 � ◮ ( − λ 1 , λ 2 ) → ( − λ 1 , − λ 2 ): [ A ( w α 2 , w α 1 w α 2 w α 1 λ )] j , n m 1 , m 2 = T n − − − δ m 1 , m 2 m 1 2 ◮ Normalization: divide each simple intertwining operator S j , n m 1 , m 2 ( z ) , T n m 1 ( z ) by √ π Γ( z − 1 / 2) , get rational functions S j , n m 1 , m 2 ( z ) , T n m 1 ( z ) such that the corresponding Γ( z ) normalized simple intertwining operator satisfies A ( − λ ) A ( λ ) = 1. ◮ For the convenience of calculation, our principal series is induced from the character � ǫ 2 � t 1 0 � ǫ 1 � 0 0 � � 1 0 0 0 − 1 0 0 0 0 t 2 0 0 N �→ ( − 1) ( ǫ 1 + ǫ 2 ) δ t λ 1 1 t λ 2 0 − 1 0 0 0 1 0 0 0 0 1 0 0 0 − 1 0 0 0 1 / t 1 0 2 0 0 0 − 1 0 0 0 1 0 0 0 1 / t 2

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) ( j , n ) A ( λ ) (0,0) ( 1 ) � � ( λ 1 − 1 )( λ 2 − 1 ) (0,2) ( λ 1+1 )( λ 2+1 ) � � ( λ 1 − 1 )( λ 1 − λ 2 − 1 )( λ 2 − 1 )( λ 1+ λ 2 − 1 ) (1,-2) ( λ 1+1 )( λ 1 − λ 2+1 )( λ 2+1 )( λ 1+ λ 2+1 ) λ 3 1 − λ 2 1 − λ 2 2 λ 1 − λ 1+ λ 2  2 − 1 2 λ 1 ( λ 2 − 1 )  − − ( λ 1+1 )( λ 1 − λ 2+1 )( λ 1+ λ 2+1 ) ( λ 1+1 )( λ 1 − λ 2+1 )( λ 1+ λ 2+1 ) (1,-1)  ( λ 1+1 )( λ 1 − λ 2+1 )( λ 1+ λ 2+1 ) − ( λ 2 − 1 ) ( λ 3 1+ λ 2 1 − λ 2 2 λ 1 − λ 1 − λ 2  2+1 ) 2 λ 1 ( λ 2 − 1 ) − ( λ 1+1 )( λ 1 − λ 2+1 )( λ 2+1 )( λ 1+ λ 2+1 ) − ( λ 2 − 1 ) ( λ 3 1+ λ 2 1 − λ 2 2 λ 1 − λ 1 − λ 2 2+1 )   2 λ 1 ( λ 2 − 1 ) ( λ 1+1 )( λ 1 − λ 2+1 )( λ 2+1 )( λ 1+ λ 2+1 ) − ( λ 1+1 )( λ 1 − λ 2+1 )( λ 1+ λ 2+1 ) (1,1)  λ 3 1 − λ 2 1 − λ 2 2 λ 1 − λ 1+ λ 2  2 λ 1 ( λ 2 − 1 ) 2 − 1 − − ( λ 1+1 )( λ 1 − λ 2+1 )( λ 1+ λ 2+1 ) ( λ 1+1 )( λ 1 − λ 2+1 )( λ 1+ λ 2+1 )

Clebsch-Gordan Coefficients and Principal Series Representations Calculating Intertwining Operators Intertwining Operator for Sp (4 , R ) S j , n ( j , n ) m 1 , m 2 ( z ) (0,0) ( 1 ) ( − z − 1 (1,0) ) z � 3   2 (2 z − 1) (2 z − 1)(2 z +1) 3 4 z ( z +1) 2 z ( z +1) 4 z ( z +1) � 3 � 3    2 (2 z − 1) 2 z 2 − 4 z +3 2 (2 z − 1)  (2,0)   2 z ( z +1) 2 z ( z +1) 2 z ( z +1)   � 3   2 (2 z − 1) (2 z − 1)(2 z +1) 3 4 z ( z +1) 2 z ( z +1) 4 z ( z +1) � 15   2 ( z − 1)(2 z − 1) 15( z − 1) − ( z − 1)(2 z − 1)(2 z +1) − − 4 z ( z +1)( z +2) 2 z ( z +1)( z +2) 4 z ( z +1)( z +2) � 15 � 15   ( z − 1) ( 2 z 2 − 4 z +9 )  2 ( z − 1)(2 z − 1) 2 ( z − 1)(2 z − 1)  (3,0)  − − −  2 z ( z +1)( z +2) 2 z ( z +1)( z +2) 2 z ( z +1)( z +2)   � 15   2 ( z − 1)(2 z − 1) − ( z − 1)(2 z − 1)(2 z +1) 15( z − 1) − − 4 z ( z +1)( z +2) 2 z ( z +1)( z +2) 4 z ( z +1)( z +2)

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: i − m 1 − m 2 (2 j )! √ π � ( z − 1 / 2) ( − m 1 − m 2 ) � � 2 − j + z − 1 , − j − m 1 , − j + m 2 S j , n m 1 , m 2 ( z ) = 3 F 2 ; 1 − 2 j , − j − m 1 − m 2 + 1 � ( z ) ( j ) c j m 1 c j 1 − 2 j − m 1 + m 2 m 2 Γ 2 2 2 i n − m 1 T n m 1 ( z ) = ) . ( z ) ( m 1 − n ) ( z ) ( − m 1+ n 2 2

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 m 1 , m 2 ( t 1 , t 2 ) = ( − 1) j + n − ǫ j , n δ ((2 j )!) 2 � j + n − ǫ j , n j + n − ǫ j , n δ δ � − � ( j ) � � ( j ) � � � λ 1 +1 λ 1 +1 λ 1 − λ 2 +1 λ 1 + λ 2 +1 2 2 2 2 2 2 j + m 2 − ǫ j , n − j − m 1+ ǫ j , n � 1 − ǫ j , n � δ δ δ + j − m 2 � ( ) � � ( ) � λ 1 + λ 2 2 λ 1 − λ 2 2 Γ i − m 1 − m 2 2 2 2 � m 2 − n � − m 2 − n c j m 1 c j � 1 − ǫ j , n � δ + j − m 1 � � m 2 λ 2 +1 λ 2 +1 2 2 Γ 2 2 2 − 1 − ǫ j , n − 1+ ǫ j , n + j − m 1 − j + m 2 ǫ j , n − ǫ j , n δ − 2 j δ δ (1 − t 1 ) (1 − t 2 ) δ t t 2 2 1 2 � � � � − j − m 1 , λ 1 − λ 2 − 2 j − 1 − j + m 2 , λ 1+ λ 2 − 2 j − 1 2 F 1 ; t 1 2 F 1 ; t 2 2 2 − 2 j − 2 j  − j + m 1+1+ ǫ j , n − j − n − λ 1+1+ ǫ j , n − j − m 1+ λ 1 − λ 2+ ǫ j , n  ; t 2 1 (1 − t 2 ) δ δ δ  1 , , , 2 2 2 4 F 3  − j + m 2+1+ ǫ j , n − j − n + λ 1+1+ ǫ j , n − j − m 2 − λ 1 − λ 2+ ǫ j , n (1 − t 1 ) t 2 δ δ δ , , +1 2 2 2 2 � where ǫ j , n 0 j − n ≡ δ i mod 2 = 1 j − n �≡ δ i mod 2 . δ