Parabolic Positive Representations of Uq

(1)

Parabolic Positive Representations of U _q (g _R )

Ivan Ip

Hong Kong University of Science and Technology

October 8, 2020

Recent advances in combinatorial representation theory

RIMS, Kyoto University

(2)

Positive Representations ofU_q(g_R)

Positive Representations of U _q (g _R )

(3)

Definition of U _q (sl(2, R ))

Definition

U _q (sl ₂ )= Hopf-algebra hE, F, K ^±1 i over C (q) such that

KE = q ² EK, KF = q ⁻² F K, [E, F ] = K − K ⁻¹ q − q ⁻¹

Coproduct:

∆(E) = 1 ⊗ E + E ⊗ K, ∆(F) = F ⊗ 1 + K ⁻¹ ⊗ F

∆(K) = K ⊗ K

(Also counit , antipode S)

(4)

Definition of U _q (sl(2, R ))

Definition

U _q (sl ₂ )= Hopf-algebra hE, F, K ^±1 i over C (q) such that

KE = q ² EK, KF = q ⁻² F K, [E, F ] = K − K ⁻¹ q − q ⁻¹

Coproduct:

∆(E) = 1 ⊗ E + E ⊗ K, ∆(F) = F ⊗ 1 + K ⁻¹ ⊗ F

∆(K) = K ⊗ K

(Also counit , antipode S)

(5)

Definition of U _q (g _R )

Definition

U _q (g)= Hopf-algebra hE _i , F _i , K _i ^±1 i _i∈I over C (q) such that K i E j = q ^a

^ij

E j K i , K i F j = q ^−a

^ij

F j K i , [E i , F j ] = δ ij

K _i − K _i ⁻¹ q − q ⁻¹ + Serre relations.

Coproduct:

∆(E i ) = 1 ⊗ E i + E i ⊗ K i , ∆(F i ) = F i ⊗ 1 + K _i ⁻¹ ⊗ F i

∆(K _i ) = K _i ⊗ K _i

(Also counit , antipode S)

(6)

Definition of U _q (g _R )

Definition

D _q (g)= Drinfeld’s Double: hE _i , F i , K _i ^±1 , K _i ^0±1 i _i∈I

K _i E _j = q ^a

^ij

E _j K _i , K _i F _j = q ^−a

^ij

F _j K _i , [E _i , F _j ] = δ _ij K _i − K _i ⁰ q − q ⁻¹ + Serre relations + Similar for K _i ⁰

Coproduct:

∆(E _i ) = 1 ⊗ E _i + E _i ⊗ K _i , ∆(F _i ) = F _i ⊗ 1 + K _i ⁰ ⊗ F _i

∆(K i ) = K i ⊗ K i , ∆(K _i ⁰ ) = K i ⊗ K _i ⁰

(Also counit , antipode S)

(7)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(8)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(9)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(10)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(11)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(12)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(13)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(14)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(15)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(16)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(17)

Positive Representations of U _q (g _R )

Research program started in [Frenkel-I. (2012)]

Representations by positive operators on Hilbert space.

Generalization of Teschner’s representations of U _q (sl(2, R )) Closure under taking tensor product A

n

: [Schrader-Shapiro 2018]

Braiding structure [I. 2012]

Peter-Weyl Theorem A

n

: [I.-Schrader-Shapiro 2020]

=“Quantization of principal series representations”

Constructed for all semisimple Lie types.

Construction:

Lusztig’s total positive space L ² ((G/B) >0 ) ' L ² ( R ^N _>0 ^=`(w

⁰

⁾ )

Mellin transformation: L ² (R ^N _>0 ) ' L ² (R ^N )

(18)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(19)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(20)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(21)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(22)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(23)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(24)

Positive Representations of U _q (g _R )

Rescale generators by (q = e ^πib

²

, b ∈ (0, 1))

e _k = −i(q − q ⁻¹ )E _k , f _k = −i(q − q ⁻¹ )F _k

Theorem (I. (2012))

There exists a family of irreducible representations P _λ of U _q (g _R ):

Parametrized by λ ∈ R ≥0 P ⁺ ' R ^n=rankg _≥0

Positivity: {e _i , f _i , K _i } are represented by positive, essentially self-adjoint (unbounded) operators on L ² (R ^N )

e _i , f _i , K _i are expressed in terms of Laurent polynomials of {e ^πbx

^k

, e ^2πbp

^k

} ^N _k=1

Characterized by modular double structure (Langland’s duality)

(25)

Example: U _q (sl ₃ )

Coordinates on (G/B) >0 :

1 a 0

0 1 0

0 0 1

1 0 0

0 1 b

0 0 1

1 c 0

0 1 0

0 0 1

·

1 0 0

0 1 0

0 t 1

a, b, c > 0

=

1 0 0

0 1

0 _1+bt^t 1

1 0 0

0 1 +bt 0

0 0 (1 +bt)⁻¹

1 a+abt 0

0 1 0

0 0 1

1 0 0

0 1 _1+bt^b

0 0 1

1 c 0

0 1 0

0 0 1

e ^tF

²

· f (a, b, c) = (1 + bt) ^2λ f (a + abt, b

1 + bt , c), λ ∈ R ≥0

F ₂ := d dt e ^tF

²

t=0

= ab ∂

∂a − b ² ∂

∂b + bλ

(26)

Example: U _q (sl ₃ )

Coordinates on (G/B) >0 :

1 a 0

0 1 0

0 0 1

1 0 0

0 1 b

0 0 1

1 c 0

0 1 0

0 0 1

·

1 0 0

0 1 0

0 t 1

a, b, c > 0

=

1 0 0

0 1

0 _1+bt^t 1

1 0 0

0 1 +bt 0

0 0 (1 +bt)⁻¹

1 a+abt 0

0 1 0

0 0 1

1 0 0

0 1 _1+bt^b

0 0 1

1 c 0

0 1 0

0 0 1

e ^tF

²

· f (a, b, c) = (1 + bt) ^2λ f (a + abt, b

1 + bt , c), λ ∈ R ≥0

F ₂ := d dt e ^tF

²

t=0

= ab ∂

∂a − b ² ∂

∂b + bλ

(27)

Example: U _q (sl ₃ )

Coordinates on (G/B) >0 :

1 a 0

0 1 0

0 0 1

1 0 0

0 1 b

0 0 1

1 c 0

0 1 0

0 0 1

·

1 0 0

0 1 0

0 t 1

a, b, c > 0

=

1 0 0

0 1

0 _1+bt^t 1

1 0 0

0 1 +bt 0

0 0 (1 +bt)⁻¹

1 a+abt 0

0 1 0

0 0 1

1 0 0

0 1 _1+bt^b

0 0 1

1 c 0

0 1 0

0 0 1

e ^tF

²

· f (a, b, c) = (1 + bt) ^2λ f (a + abt, b

1 + bt , c), λ ∈ R ≥0

F ₂ := d dt e ^tF

²

t=0

= ab ∂

∂a − b ² ∂

∂b + bλ

(28)

Example: U _q (sl ₃ )

Coordinates on (G/B) >0 :

1 a 0

0 1 0

0 0 1

1 0 0

0 1 b

0 0 1

1 c 0

0 1 0

0 0 1

·

1 0 0

0 1 0

0 t 1

a, b, c > 0

=

1 0 0

0 1

0 _1+bt^t 1

1 0 0

0 1 +bt 0

0 0 (1 +bt)⁻¹

1 a+abt 0

0 1 0

0 0 1

1 0 0

0 1 _1+bt^b

0 0 1

1 c 0

0 1 0

0 0 1

e ^tF

²

· f (a, b, c) = (1 + bt) ^2λ f (a + abt, b

1 + bt , c), λ ∈ R ≥0

F ₂ := d dt e ^tF

²

t=0

= ab ∂

∂a − b ² ∂

∂b + bλ

(29)

Example: U _q (sl ₃ )

F 2 = ab ∂

∂a − b ² ∂

∂b + bλ (Formal) Mellin transform: F(u, v, w) :=

Z

f (a, b, c)a ^u b ^v c ^w dadbdc F 2 : F(u, v, w) 7→ (2λ + u − v + 1)F(u, v − 1, w) Quantum Twist (n 7→ [n] _q + “Wick’s rotation”)

F 2 :=

i q − q ⁻¹

e πb(2λ+u−v+2p

_v

) + e πb(−2λ−u+v+2p

_v

)

(30)

Example: U _q (sl ₃ )

F 2 = ab ∂

∂a − b ² ∂

∂b + bλ (Formal) Mellin transform: F(u, v, w) :=

Z

f (a, b, c)a ^u b ^v c ^w dadbdc F 2 : F(u, v, w) 7→ (2λ + u − v + 1)F(u, v − 1, w) Quantum Twist (n 7→ [n] _q + “Wick’s rotation”)

F 2 :=

i q − q ⁻¹

e πb(2λ+u−v+2p

_v

) + e πb(−2λ−u+v+2p

_v

)

(31)

Example: U _q (sl ₃ )

F 2 = ab ∂

∂a − b ² ∂

∂b + bλ (Formal) Mellin transform: F(u, v, w) :=

Z

f (a, b, c)a ^u b ^v c ^w dadbdc F 2 : F(u, v, w) 7→ (2λ + u − v + 1)F(u, v − 1, w) Quantum Twist (n 7→ [n] _q + “Wick’s rotation”)

F 2 :=

i q − q ⁻¹

e πb(2λ+u−v+2p

_v

) + e πb(−2λ−u+v+2p

_v

)

(32)

The goal of this talk

Definition

Parabolic positive representations is a new family of positive

representations of U _q (g _R ) based on quantizing the parabolic induction representations on L ² ((G/P ) _>0 ), where P ⊂ G is a parabolic subgroup.

It answers some combinatorial mysteries of quantum group embedding (cluster realization)

Gives a new realization of the evaluation module of U _q ( sl b _n ).

(33)

The goal of this talk

Definition

Parabolic positive representations is a new family of positive

representations of U _q (g _R ) based on quantizing the parabolic induction representations on L ² ((G/P ) _>0 ), where P ⊂ G is a parabolic subgroup.

It answers some combinatorial mysteries of quantum group embedding (cluster realization)

Gives a new realization of the evaluation module of U _q ( sl b _n ).

(34)

Quantum Cluster Variety

(35)

Quantum Torus Algebra

“Quantization of cluster X variety” [Fock-Goncharov]

Definition

Seed Q = (Q, Q 0 , B):

Q = nodes (finite set) Q 0 ⊂ Q = frozen nodes

B = (b _ij ) exchange matrix (|Q| × |Q|, skew-symmetric, ¹ ₂ Z -valued) Quantum torus algebra X _q ^Q = algebra generated by {X _i } _i∈Q over C [q]

such that

X i X j = q ^−2b

^ij

X j X i

X i = quantum cluster variables

Exchange Matrix B ; Quiver.

(36)

Quantum Cluster Variety Quantum Torus Algebra

Quantum Torus Algebra

“Quantization of cluster X variety” [Fock-Goncharov]

Definition

Seed Q = (Q, Q 0 , B):

Q = nodes (finite set) Q 0 ⊂ Q = frozen nodes

B = (b _ij ) exchange matrix (|Q| × |Q|, skew-symmetric, ¹ ₂ Z -valued) Quantum torus algebra X _q ^Q = algebra generated by {X _i } _i∈Q over C [q]

such that

X i X j = q ^−2b

^ij

X j X i

X i = quantum cluster variables

Exchange Matrix B ; Quiver.

(37)

Quantum Torus Algebra

“Quantization of cluster X variety” [Fock-Goncharov]

Definition

Seed Q = (Q, Q 0 , B):

Q = nodes (finite set) Q 0 ⊂ Q = frozen nodes

B = (b _ij ) exchange matrix (|Q| × |Q|, skew-symmetric, ¹ ₂ Z -valued) Quantum torus algebra X _q ^Q = algebra generated by {X _i } _i∈Q over C [q]

such that

X i X j = q ^−2b

^ij

X j X i

X i = quantum cluster variables

Exchange Matrix B ; Quiver.

(38)

Quantum Torus Algebra

“Quantization of cluster X variety” [Fock-Goncharov]

Definition

Seed Q = (Q, Q 0 , B):

Q = nodes (finite set) Q 0 ⊂ Q = frozen nodes

B = (b _ij ) exchange matrix (|Q| × |Q|, skew-symmetric, ¹ ₂ Z -valued) Quantum torus algebra X _q ^Q = algebra generated by {X _i } _i∈Q over C [q]

such that

X i X j = q ^−2b

^ij

X j X i

X i = quantum cluster variables

Exchange Matrix B ; Quiver.

(39)

Quantum Torus Algebra

“Quantization of cluster X variety” [Fock-Goncharov]

Definition

Seed Q = (Q, Q 0 , B):

Λ Q = Z-Lattice with basis {e _i } i∈Q

(−, −) skew-symmetric form, (e _i , e _j ) := b _ij .

Quantum torus algebra X _q ^Q =algebra generated by {X _λ } _λ∈Λ

_Q

over C [q

¹²

] such that

X _λ+µ = q ^(λ,µ) X _λ X µ

X i := X e

i

, X i

1

,i

2

,...,i

_k

:= X e

i1

+e

i2

+···+e

_ik

Exchange Matrix B ; Quiver.

(40)

Quantum Cluster Mutations

T ^Q q := (non-commutative) field of fractions of X _q ^Q . Cluster mutation µ _k induces µ ^q _k : T ^Q q

⁰

−→ T ^Q q :

µ ^q _k ( X b i ) :=



 

 

X _k ⁻¹ i = k

X i Q |b

_ki

|

r=1 (1 + q ^2r−1 _i X _k ) i 6= k, b _ki < 0 X _i Q b

_ki

r=1 (1 + q ^2r−1 _i X _k ⁻¹ ) ⁻¹ i 6= k, b _ki > 0 Can be rewritten as

µ ^q _k = µ ^# _k ◦ µ ⁰ _k

µ ⁰ _k ( X b i ) :=







X _k ⁻¹ i = k

X _i i 6= k, b _ki < 0

q ^b _i

^ik

^b

^ki

X _i X _k ^b

^ik

i 6= k, b _ki > 0

(41)

Quantum Cluster Mutations

T ^Q q := (non-commutative) field of fractions of X _q ^Q . Cluster mutation µ _k induces µ ^q _k : T ^Q q

⁰

−→ T ^Q q :

µ ^q _k ( X b i ) :=



 

 

X _k ⁻¹ i = k

X i Q |b

_ki

|

r=1 (1 + q ^2r−1 _i X _k ) i 6= k, b _ki < 0 X _i Q b

_ki

r=1 (1 + q ^2r−1 _i X _k ⁻¹ ) ⁻¹ i 6= k, b _ki > 0 Can be rewritten as

µ ^q _k = µ ^# _k ◦ µ ⁰ _k

µ ⁰ _k ( X b i ) :=







X _k ⁻¹ i = k

X _i i 6= k, b _ki < 0

q ^b _i

^ik

^b

^ki

X _i X _k ^b

^ik

i 6= k, b _ki > 0

(42)

Polarization of X _q ^Q

Recall q = e ^πib

²

such that |q| = 1.

Definition

A polarization of X _q ^Q is a choice of representation of the cluster variables X _k ∈ X _q ^Q of the form X _k = e ^2πbx

^k

such that

x j is self-adjoint

x k satisfies the Heisenberg algebra relations [x _j , x _k ] = 1

2πi b _jk , acting on some Hilbert space H _Q ' L ² (R ^N ).

Remark

(43)

Polarization of X _q ^Q

Recall q = e ^πib

²

such that |q| = 1.

Definition

A polarization of X _q ^Q is a choice of representation of the cluster variables X _k ∈ X _q ^Q of the form X _k = e ^2πbx

^k

such that

x j is self-adjoint

x k satisfies the Heisenberg algebra relations [x _j , x _k ] = 1

2πi b _jk , acting on some Hilbert space H _Q ' L ² (R ^N ).

Remark

(44)

Polarization of X _q ^Q

Example

For X ₁ X ₂ = q ² X ₂ X ₁ , we have

X 1 = e ^2πbx X 2 = e ^2πbp acting on L ² (R), where p = _2πi ¹ _dx ^d . Proposition

Different polarizations (with the same central characters) are

unitary equivalent (via Sp(2N )-action)

(45)

Polarization of X _q ^Q

Example

For X ₁ X ₂ = q ² X ₂ X ₁ , we have

X 1 = e ^2πbx X 2 = e ^2πbp acting on L ² (R), where p = _2πi ¹ _dx ^d . Proposition

Different polarizations (with the same central characters) are

unitary equivalent (via Sp(2N )-action)

(46)

Polarization of X _q ^Q

Example

For X ₁ X ₂ = q ² X ₂ X ₁ , we have

X 1 = e ^2πbx X 2 = e ^2πbp acting on L ² (R), where p = _2πi ¹ _dx ^d . Proposition

Different polarizations (with the same central characters) are

unitary equivalent (via Sp(2N )-action)

(47)

Quantum cluster variety

S=Riemann surface with marked points on ∂S and punctures.

Fock-Goncharov’s X _G,S -space= “(framed) local G-system”

X _G,S has Poisson cluster X variety structure ; quantization X _G,S ^q To each triangle of ideal triangulation of S, assign a basic quiver.

G = P GL n+1 : “n-triangulation”

Q _sl

₄

(48)

Quantum Cluster Variety Fock-GoncharovX_G,S-Space

Quantum cluster variety

S=Riemann surface with marked points on ∂S and punctures.

Fock-Goncharov’s X _G,S -space= “(framed) local G-system”

X _G,S has Poisson cluster X variety structure ; quantization X _G,S ^q To each triangle of ideal triangulation of S, assign a basic quiver.

G = P GL n+1 : “n-triangulation”

Q _sl

₄

(49)

Quantum cluster variety

S=Riemann surface with marked points on ∂S and punctures.

Fock-Goncharov’s X _G,S -space= “(framed) local G-system”

X _G,S has Poisson cluster X variety structure ; quantization X _G,S ^q To each triangle of ideal triangulation of S, assign a basic quiver.

G = P GL n+1 : “n-triangulation”

Q _sl

₄

(50)

Quantum cluster variety

S=Riemann surface with marked points on ∂S and punctures.

Fock-Goncharov’s X _G,S -space= “(framed) local G-system”

X _G,S has Poisson cluster X variety structure ; quantization X _G,S ^q To each triangle of ideal triangulation of S, assign a basic quiver.

G = P GL n+1 : “n-triangulation”

Q _sl

₄

(51)

Quantum cluster variety

S=Riemann surface with marked points on ∂S and punctures.

Fock-Goncharov’s X _G,S -space= “(framed) local G-system”

X _G,S has Poisson cluster X variety structure ; quantization X _G,S ^q To each triangle of ideal triangulation of S, assign a basic quiver.

G = P GL n+1 : “n-triangulation”

Q _sl

₄

(52)

Basic Quiver

[I. (2016), Goncharov-Shen (2019)]

Definition

Elementary quiver J _k (i), i, k ∈ I

Q = Q 0 = (I \ {i}) ∪ {i _l } ∪ {i _r } ∪ {k _e } c _i

_l

_,j = c _j,i

_r

= a ij

2 , c _i,i

_r

= c _i

_r

_,k

_e

= c _k

_e

_,i

_l

= 1

J(i): without {k _e }.

(53)

Basic Quiver

[I. (2016), Goncharov-Shen (2019)]

Definition

Elementary quiver

H(i), i = (i ₁ , ..., i _m ) reduced words Q = I

c ij :=

sgn(r − s) ^a ₂

^ij

β s = α i and β r = α j

0 otherwise

β j := s i

m

s i

m−1

· · · s i

j+1

(α i

j

), α i ∈ ∆ +

(If i = i 0 , orientation of Dynkin diagram)

(54)

Basic Quiver

[I. (2016), Goncharov-Shen (2019)]

Definition Basic Quiver

Q(i), i = (i 1 , ..., i m ) reduced words Q = J ^# _i (i ₁ ) ∗ J ^# _i (i ₂ ) ∗ · · · ∗ J ^# _i (i _m ) ∗ H(i) J ^# _i (i _j ) =

J _k (i _j ) if β _j = α _k

J(i _j ) otherwise

(55)

Basic Quiver

Example

g = sl ₄ , i = (3, 2, 1).

3 l 3 r

2 J(3)

←→

3 2 _l 2 _r

1 J(2)

←→ 2

1 l 1 r

1 _e

J ₁ (1)

= ⇒

f₃⁰ f₃¹ f₂⁰ f₂¹ f₁⁰ f₁¹ e⁰₁

Q(i)

(56)

Basic Quiver

Example

g = sl ₄ , i ₀ = (3, 2, 1, 3, 2, 3).

Q _sl

₄

(57)

Example: Type A _n Case

Q

Q ^op

(58)

Quantum Cluster Variety Embedding ofU_q(sl_n)

Example: Type A _n Case

13 15

8 12

9 11

1 7

2 6

3 5

4

10

14 16

17

18

D _sl

_n+1

-quiver ; X := X _sl

_n+1

[Schrader-Shapiro]

ι : D _q (sl _n+1 ) , → X

(59)

Example: Type A _n Case

f 3

13 ¹⁵

e 1

f ₂

8 ¹²

e ₁

9 11

f ₁

1 ⁷

e 1

2 6

3 5

4

10

14 16

17

18

Embedding of F i ∈ D _sl

₄

, → X

f₁=X₁+X_1,2+X_1,2,3+X_1,2,3,4+X_1,2,3,4,5+X1,2,3,4,5,6 f₂=X₈+X_8,9+X_8,9,10+X_8,9,10,11

(60)

Example: Type A _n Case

f 1

13 ¹⁵

e 3

f ₁

8 ¹²

e ₂

9 11

f ₁

1 ⁷

e 1

2 6

3 5

4

10

14 16

17

18

Embedding of E i ∈ D _sl

₄

, → X

e₁=X₇+X_7,16

e₂=X₁₂+X_12,6+X_12,6,17+X_12,6,17,2

(61)

Positive Representations of U _q (g _R )

Theorem (Schrader-Shapiro, I. (2016)) There exists an embedding

D _q (g) , → X corresponding to the quiver D _g associated to

We recover the positive representations P _λ ' H ^J through a polarization of X .

Theorem (I. (2016))

The generators e _i , f _i , K _i are represented by positive polynomials

(i.e. over N [q, q ⁻¹ ]) in the cluster variables X _i ∈ X .

(62)

Positive Representations of U _q (g _R )

Theorem (Schrader-Shapiro, I. (2016)) There exists an embedding

D _q (g) , → X corresponding to the quiver D _g associated to

We recover the positive representations P _λ ' H ^J through a polarization of X .

Theorem (I. (2016))

The generators e _i , f _i , K _i are represented by positive polynomials

(i.e. over N [q, q ⁻¹ ]) in the cluster variables X _i ∈ X .

(63)

Positive Representations of U _q (g _R )

Theorem (Schrader-Shapiro, I. (2016)) There exists an embedding

D _q (g) , → X corresponding to the quiver D _g associated to

We recover the positive representations P _λ ' H ^J through a polarization of X .

Theorem (I. (2016))

The generators e _i , f _i , K _i are represented by positive polynomials

(i.e. over N [q, q ⁻¹ ]) in the cluster variables X _i ∈ X .

(64)

Positive Representations of U _q (g _R )

Theorem (Schrader-Shapiro, I. (2016)) There exists an embedding

D _q (g) , → X corresponding to the quiver D _g associated to

We recover the positive representations P _λ ' H ^J through a polarization of X .

Theorem (I. (2016))

The generators e _i , f _i , K _i are represented by positive polynomials

(i.e. over N [q, q ⁻¹ ]) in the cluster variables X _i ∈ X .

(65)

Positive Representations of U _q (g _R )

Theorem (Schrader-Shapiro, I. (2016)) There exists an embedding

D _q (g) , → X corresponding to the quiver D _g associated to

We recover the positive representations P _λ ' H ^J through a polarization of X .

Theorem (I. (2016))

The generators e _i , f _i , K _i are represented by positive polynomials

(i.e. over N [q, q ⁻¹ ]) in the cluster variables X _i ∈ X .

(66)

E ₆ embedding

i 0 = (3 43 034 230432 12340321 5432103243054321)

f1 e1

f₂ e2

f₃ e₃

f₄ e₄

f₀ e₀

e⁰₁

e⁰₂ e⁰₃ e⁰₄ e⁰₅

(67)

E ₆ embedding

i 0 = (3 43 034 230432 12340321 5432103243054321)

f1 e1

f₂ e2

f₃ e₃

f₄ e₄

f₀ e₀

e⁰₁

e⁰₂ e⁰₃ e⁰₄ e⁰₅

(68)

Minimal Positive Representation

Minimal Positive Representation for

U _q (sl(n + 1, R ))

(69)

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(70)

Minimal Positive Representation Construction

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(71)

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(72)

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(73)

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(74)

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(75)

Minimal Positive Representation

Parabolic subgroups ←→ J ⊂ I

P _J := B − L _J , Levi subgroup L _J = hT, U _j ⁺ , U _j ⁻ i _j∈J P _∅ := B − .

Example

For G = SL 4 , J = {1, 2} ⊂ I = {1, 2, 3}

P_J=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗







(G/P_J)>0=







∗ ∗ ∗ 0

∗ ∗ ∗ ∗













1 a 0 0

0 1 b 0

0 0 1 c

0 0 0 1







, a, b, c >0

(76)

Minimal Positive Representation

Previous recipe produces a representation P _λ ^J for U _q (sl(4, R )), (λ ∈ R ) π _λ ^J (e 1 ) = e ^πb(u−2p

^u

⁾ + e ^{πb(−u−2p}

^u

⁾

π _λ ^J (e 2 ) = e ^{πb(−u+v−2p}

^v

⁾ + e ^{πb(u−v−2p}

^v

⁾ π _λ ^J (e ₃ ) = e ^{πb(−v+w−2p}

^w

⁾ + e ^{πb(v−w−2p}

^w

⁾

π ^J _λ (f ₁ ) = e ^{πb(−u+v+2p}

^u

⁾ + e ^πb(u−v+2p

^u

⁾ π ^J _λ (f ₂ ) = e ^{πb(−v+w+2p}

^v

⁾ + e ^πb(v−w+2p

^v

⁾ π ^J _λ (f 3 ) = e ^{πb(2λ−w+2p}

^w

⁾ + e πb(−2λ+w+2p

_w

)

π ^J _λ (K 1 ) = e ^πb(−2u+v) π ^J _λ (K ₂ ) = e ^{πb(u−2v+w)}

J πb(v−2w+2λ)

(77)

Minimal Positive Representation

1 0

3

4 2

6

7 5

9 8

D(i) := Q(i ^op ) ∗ Q(i), i = (3, 2, 1)

e₁=X₃+X_3,0 K₁=X_3,0,1

e2=X6+X6,2 K2=X6,2,4

e₃=X₉+X_9,5 K₃=X_9,5,7

f₁=X₁+X_1,2 K₁⁰=X_1,2,3

f₂=X₄+X_4,5 K₂⁰=X_4,5,6

(78)

Minimal Positive Representation

Theorem (I. (2020))

The polarization of the quiver D(i) for i = (n, ..., 3, 2, 1) gives a representation P _λ ^J of U _q (sl(n + 1, R )) acting on L ² ( R ⁿ ) as positive self-adjoint operators.

.. . .. . .. .

(79)

Minimal Positive Representation

Theorem (I. (2020))

The non-simple generators

e _α := T _i

₁

· · · T _i

_k−1

(e _k ) f _α := T _i

₁

· · · T _i

_k−1

(f _k )

is non-zero, where T _i = Lusztig’s braid group action.

The universal R operator is well-defined R = K Y

α∈Φ

+

g _b (e _α ⊗ f _α )

(80)

Minimal Positive Representation

Theorem (I. (2020))

The non-simple generators

e _α := T _i

₁

· · · T _i

_k−1

(e _k ) f _α := T _i

₁

· · · T _i

_k−1

(f _k )

is non-zero, where T _i = Lusztig’s braid group action.

The universal R operator is well-defined R = K Y

α∈Φ

+

g _b (e _α ⊗ f _α )

(81)

Minimal Positive Representation

Theorem (I. (2020))

The non-simple generators

e _α := T _i

₁

· · · T _i

_k−1

(e _k ) f _α := T _i

₁

· · · T _i

_k−1

(f _k )

is non-zero, where T _i = Lusztig’s braid group action.

The universal R operator is well-defined R = K Y

α∈Φ

+

g _b (e _α ⊗ f _α )

(82)

Casimirs

Example

U _q (sl(3, R )), the possible action of (C ₁ , C ₂ ) (by scalars) on P _λ and P _λ ^J :

(83)

Evaluation Module of U _q ( sl b n+1 )

(84)

Evaluation Module of U _q ( sl b n+1 )

.. . .. . .. .

= ⇒

.. . .. . .. .

(85)

Evaluation Module of U _q ( sl b n+1 )

.. . .. . .. .

= ⇒

.. . .. . .. .

(86)

Evaluation Module of U _q ( sl b n+1 )

Theorem (I. (2020))

The positive representation of U _q ( sl b _n+1 ) defined by the polarization of the previous quiver is unitarily equivalent to Jimbo’s evaluation module P _λ ^µ , µ ∈ R

U _q ( sl b _n+1 ) −→ U _q (sl _n+1 )

of the minimal positive representations P _λ ^J of U _q (sl _n+1 ), where e ^πbµ := π(D

1 n+1

0 D ₁ )

(D ₀ =product of all middle vertices, D ₁ = product of all right vertices.)

(87)

Positive representation of U _q ( sl b 2 )

Example

1 2 3

4 5 6

f ₀ = X ₁ + X _1,2 e ₀ = X ₃ + X _3,5 f 1 = X 4 + X 4,5 e 1 = X 6 + X 6,2

Serre relation (a 01 = a 10 = −2):

(88)

General Construction

(89)

Main Theorem

Parabolic induction ←→ truncating i _J ⊂ i ₀ where i _J , i ₀ are the longest word of the Weyl groups W _J ⊂ W .

w ₀ = w _J w w ←→ i Example

W _sl

₄

⊂ W _sl

₅

i 0 = (1, 2, 1, 3, 2, 1, 4, 3, 2, 1) Observe that

Q(i) = Q(i J ) ∗ Q(i)

(90)

Main Theorem

Parabolic induction ←→ truncating i _J ⊂ i ₀ where i _J , i ₀ are the longest word of the Weyl groups W _J ⊂ W .

w ₀ = w _J w w ←→ i Example

W _sl

₄

⊂ W _sl

₅

i 0 = (1, 2, 1, 3, 2, 1, 4, 3, 2, 1) Observe that

Q(i) = Q(i J ) ∗ Q(i)

(91)

Main Theorem

Parabolic induction ←→ truncating i _J ⊂ i ₀ where i _J , i ₀ are the longest word of the Weyl groups W _J ⊂ W .

w ₀ = w _J w w ←→ i Example

W _sl

₄

⊂ W _sl

₅

i 0 = (1, 2, 1, 3, 2, 1, 4, 3, 2, 1) Observe that

Q(i) = Q(i J ) ∗ Q(i)

(92)

Main Theorem

Theorem (I. (2020))

There is a homomorphism

D _q (g) −→ X _q ^D(i)

such that the image of universally Laurent polynomials.

A polarization of X _q ^D(i) induces a family of irreducible

representations P _λ ^J of U _q (g _R ) parametrized by λ ∈ R ^|I\J| as positive self-adjoint operators on L ² ( R ^l(w) ).

Corollary

The parabolic positive representations P _λ ^J is obtained as a quantum twist of the parabolic induction, by ignoring the variables u _i

corresponding to the Levi subgroups L of P in the quotient G/P .

(93)

Main Theorem

Theorem (I. (2020))

There is a homomorphism

D _q (g) −→ X _q ^D(i)

such that the image of universally Laurent polynomials.

A polarization of X _q ^D(i) induces a family of irreducible

representations P _λ ^J of U _q (g _R ) parametrized by λ ∈ R ^|I\J| as positive self-adjoint operators on L ² ( R ^l(w) ).

Corollary

The parabolic positive representations P _λ ^J is obtained as a quantum twist of the parabolic induction, by ignoring the variables u _i

corresponding to the Levi subgroups L of P in the quotient G/P .

(94)

Main Theorem

Theorem (I. (2020))

There is a homomorphism

D _q (g) −→ X _q ^D(i)

such that the image of universally Laurent polynomials.

A polarization of X _q ^D(i) induces a family of irreducible

representations P _λ ^J of U _q (g _R ) parametrized by λ ∈ R ^|I\J| as positive self-adjoint operators on L ² ( R ^l(w) ).

Corollary

The parabolic positive representations P _λ ^J is obtained as a quantum twist of the parabolic induction, by ignoring the variables u _i

corresponding to the Levi subgroups L of P in the quotient G/P .

(95)

Main Theorem

Theorem (I. (2020))

There is a homomorphism

D _q (g) −→ X _q ^D(i)

such that the image of universally Laurent polynomials.

A polarization of X _q ^D(i) induces a family of irreducible

representations P _λ ^J of U _q (g _R ) parametrized by λ ∈ R ^|I\J| as positive self-adjoint operators on L ² ( R ^l(w) ).

Corollary

The parabolic positive representations P _λ ^J is obtained as a quantum twist of the parabolic induction, by ignoring the variables u _i

corresponding to the Levi subgroups L of P in the quotient G/P .

(96)

Idea of Proof

Definition

The Heisenberg double H ^± _q (g) := he ^± _i , f _i ^± , K ^± _i , K ⁰ _i ^± i satisfying [e ⁺ _i , f _j ⁺ ]

q − q ⁻¹ = δ ij K ⁰ _i ⁺ , [e ⁻ _i , f _j ⁻ ]

q − q ⁻¹ = δ ij K i −

and other standard quantum group relations.

Proposition

The embedding D _q (g) , → X _q ^D(i

⁰

⁾ ⊂ X ^Q(i

op 0

Parabolic Positive Representations of Uq