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Im ( U q ( gl m ) End( V n ) ) End H n , 1 ( V n ) S n , 1 . - PowerPoint PPT Presentation

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. Drinfeld type realization of cyclotomic q -Schur algebras . Kentaro Wada Shinshu Univ. 12th March, 2012 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 1 / 25

  1. Introduction r = 1 ( S n , 1 � End H n , 1 ( V ⊗ n ) ) V ⊗ n = ⊕ µ V ⊗ n : weight space decom. as U q ( gl m ) -module. µ µ � Ind H n , 1 � V ⊗ n as H n , 1 -module (permutation module). H ( S µ ) 1 r > 1 M µ : = m µ · H n , r ( m µ ∈ H n , r ) “ permutation module” ( ⊕ µ M µ ) S n , r : = End H n , r ; cyclotomic q -Schur alg. Today Introduce an algebra U ass. to Cartan data of gl m s.t. U ↠ S n , r (Drinfeld type presentation). Representation theory of U and S n , r . highest weight modules. Harish-Chandra Ind and Res ( U L ֒ → U P ֒ → U ) evaluation functor O g → U L -mod ( O g ⊂ U q ( g ) -mod ). . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 3 / 25

  2. Introduction r = 1 ( S n , 1 � End H n , 1 ( V ⊗ n ) ) V ⊗ n = ⊕ µ V ⊗ n : weight space decom. as U q ( gl m ) -module. µ µ � Ind H n , 1 � V ⊗ n as H n , 1 -module (permutation module). H ( S µ ) 1 r > 1 M µ : = m µ · H n , r ( m µ ∈ H n , r ) “ permutation module” ( ⊕ µ M µ ) S n , r : = End H n , r ; cyclotomic q -Schur alg. Today Introduce an algebra U ass. to Cartan data of gl m s.t. U ↠ S n , r (Drinfeld type presentation). Representation theory of U and S n , r . highest weight modules. Harish-Chandra Ind and Res ( U L ֒ → U P ֒ → U ) evaluation functor O g → U L -mod ( O g ⊂ U q ( g ) -mod ). . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 3 / 25

  3. Introduction r = 1 ( S n , 1 � End H n , 1 ( V ⊗ n ) ) V ⊗ n = ⊕ µ V ⊗ n : weight space decom. as U q ( gl m ) -module. µ µ � Ind H n , 1 � V ⊗ n as H n , 1 -module (permutation module). H ( S µ ) 1 r > 1 M µ : = m µ · H n , r ( m µ ∈ H n , r ) “ permutation module” ( ⊕ µ M µ ) S n , r : = End H n , r ; cyclotomic q -Schur alg. Today Introduce an algebra U ass. to Cartan data of gl m s.t. U ↠ S n , r (Drinfeld type presentation). Representation theory of U and S n , r . highest weight modules. Harish-Chandra Ind and Res ( U L ֒ → U P ֒ → U ) evaluation functor O g → U L -mod ( O g ⊂ U q ( g ) -mod ). . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 3 / 25

  4. Cyclotomic q -Schur algebras H n , r : Ariki-Koike alg. over Q ( q ) ass. to S n ⋉ ( Z / r Z ) n . generators: T 0 , T 1 , . . . , T n − 1 . defining relations: ( T 0 − q c 1 )( T 0 − q c 2 ) . . . ( T 0 − q c r ) = 0 ( c 1 , . . . , c r ∈ Z ) , ( T i − q )( T i + q − 1 ) = 0 (1 ≤ i ≤ n − 1) , + braid relations L i : = T i − 1 . . . T 1 T 0 T 1 . . . T i − 1 ( 1 ≤ i ≤ n ) : Jucys-Murphy elements. S n , r : cyclotomic q -Schur algebra ass. to H n , r : ( ⊕ M µ ) S n , r : = End H n , r , µ ∈ Λ n , r ( m ) where M µ = m µ · H n , r ( m µ ∈ H n , r ). . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 4 / 25

  5. Cyclotomic q -Schur algebras H n , r : Ariki-Koike alg. over Q ( q ) ass. to S n ⋉ ( Z / r Z ) n . generators: T 0 , T 1 , . . . , T n − 1 . defining relations: ( T 0 − q c 1 )( T 0 − q c 2 ) . . . ( T 0 − q c r ) = 0 ( c 1 , . . . , c r ∈ Z ) , ( T i − q )( T i + q − 1 ) = 0 (1 ≤ i ≤ n − 1) , + braid relations L i : = T i − 1 . . . T 1 T 0 T 1 . . . T i − 1 ( 1 ≤ i ≤ n ) : Jucys-Murphy elements. S n , r : cyclotomic q -Schur algebra ass. to H n , r : ( ⊕ M µ ) S n , r : = End H n , r , µ ∈ Λ n , r ( m ) where M µ = m µ · H n , r ( m µ ∈ H n , r ). . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 4 / 25

  6. Cyclotomic q -Schur algebras H n , r : Ariki-Koike alg. over Q ( q ) ass. to S n ⋉ ( Z / r Z ) n . generators: T 0 , T 1 , . . . , T n − 1 . defining relations: ( T 0 − q c 1 )( T 0 − q c 2 ) . . . ( T 0 − q c r ) = 0 ( c 1 , . . . , c r ∈ Z ) , ( T i − q )( T i + q − 1 ) = 0 (1 ≤ i ≤ n − 1) , + braid relations L i : = T i − 1 . . . T 1 T 0 T 1 . . . T i − 1 ( 1 ≤ i ≤ n ) : Jucys-Murphy elements. S n , r : cyclotomic q -Schur algebra ass. to H n , r : ( ⊕ M µ ) S n , r : = End H n , r , µ ∈ Λ n , r ( m ) where M µ = m µ · H n , r ( m µ ∈ H n , r ). . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 4 / 25

  7. Borel subalgebras r = 1 e i , f i ( 1 ≤ i ≤ m − 1 ), K ± j ,( 1 ≤ j ≤ m ) : Chevalley gen. of U q ( gl m ) . Recall ρ : U q ( gl m ) ↠ S n , 1 . � U ≥ 0 q : = ⟨ e i , K ± � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j q : = ⟨ f i , K ± � U ≤ 0 � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j ) · ρ ( U ≥ 0 � S n , 1 = ρ ( U ≤ 0 ) q q . Theorem (Du-Rui) . ∃ S ≥ 0 n , r , ∃ S ≤ 0 n , r ⊂ alg. S n , r s.t. S n , r = S ≤ 0 n , r · S ≥ 0 n , r . Moreover, S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 ) , ) . q q . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 5 / 25

  8. Borel subalgebras r = 1 e i , f i ( 1 ≤ i ≤ m − 1 ), K ± j ,( 1 ≤ j ≤ m ) : Chevalley gen. of U q ( gl m ) . Recall ρ : U q ( gl m ) ↠ S n , 1 . � U ≥ 0 q : = ⟨ e i , K ± � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j q : = ⟨ f i , K ± � U ≤ 0 � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j ) · ρ ( U ≥ 0 � S n , 1 = ρ ( U ≤ 0 ) q q . Theorem (Du-Rui) . ∃ S ≥ 0 n , r , ∃ S ≤ 0 n , r ⊂ alg. S n , r s.t. S n , r = S ≤ 0 n , r · S ≥ 0 n , r . Moreover, S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 ) , ) . q q . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 5 / 25

  9. Borel subalgebras r = 1 e i , f i ( 1 ≤ i ≤ m − 1 ), K ± j ,( 1 ≤ j ≤ m ) : Chevalley gen. of U q ( gl m ) . Recall ρ : U q ( gl m ) ↠ S n , 1 . � U ≥ 0 q : = ⟨ e i , K ± � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j q : = ⟨ f i , K ± � U ≤ 0 � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j ) · ρ ( U ≥ 0 � S n , 1 = ρ ( U ≤ 0 ) q q . Theorem (Du-Rui) . ∃ S ≥ 0 n , r , ∃ S ≤ 0 n , r ⊂ alg. S n , r s.t. S n , r = S ≤ 0 n , r · S ≥ 0 n , r . Moreover, S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 ) , ) . q q . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 5 / 25

  10. Borel subalgebras r = 1 e i , f i ( 1 ≤ i ≤ m − 1 ), K ± j ,( 1 ≤ j ≤ m ) : Chevalley gen. of U q ( gl m ) . Recall ρ : U q ( gl m ) ↠ S n , 1 . � U ≥ 0 q : = ⟨ e i , K ± � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j q : = ⟨ f i , K ± � U ≤ 0 � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j ) · ρ ( U ≥ 0 � S n , 1 = ρ ( U ≤ 0 ) q q . Theorem (Du-Rui) . ∃ S ≥ 0 n , r , ∃ S ≤ 0 n , r ⊂ alg. S n , r s.t. S n , r = S ≤ 0 n , r · S ≥ 0 n , r . Moreover, S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 ) , ) . q q . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 5 / 25

  11. Borel subalgebras r = 1 e i , f i ( 1 ≤ i ≤ m − 1 ), K ± j ,( 1 ≤ j ≤ m ) : Chevalley gen. of U q ( gl m ) . Recall ρ : U q ( gl m ) ↠ S n , 1 . � U ≥ 0 q : = ⟨ e i , K ± � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j q : = ⟨ f i , K ± � U ≤ 0 � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j ) · ρ ( U ≥ 0 � S n , 1 = ρ ( U ≤ 0 ) q q . Theorem (Du-Rui) . ∃ S ≥ 0 n , r , ∃ S ≤ 0 n , r ⊂ alg. S n , r s.t. S n , r = S ≤ 0 n , r · S ≥ 0 n , r . Moreover, S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 ) , ) . q q . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 5 / 25

  12. Borel subalgebras r = 1 e i , f i ( 1 ≤ i ≤ m − 1 ), K ± j ,( 1 ≤ j ≤ m ) : Chevalley gen. of U q ( gl m ) . Recall ρ : U q ( gl m ) ↠ S n , 1 . � U ≥ 0 q : = ⟨ e i , K ± � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j q : = ⟨ f i , K ± � U ≤ 0 � 1 ≤ i ≤ m − 1 , 1 ≤ j ≤ m ⟩ alg. ⊂ U q ( gl m ) � j ) · ρ ( U ≥ 0 � S n , 1 = ρ ( U ≤ 0 ) q q . Theorem (Du-Rui) . ∃ S ≥ 0 n , r , ∃ S ≤ 0 n , r ⊂ alg. S n , r s.t. S n , r = S ≤ 0 n , r · S ≥ 0 n , r . Moreover, S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 ) , ) . q q . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 5 / 25

  13. Cartan data m = ( m 1 , . . . , m r ) ∈ Z r s.t. m k ≥ n . Put m = m 1 + · · · + m r . 1:1 � { 1 , 2 , . . . , m } ←→ Γ ( m ) : = { ( i , k ) � 1 ≤ i ≤ m k , 1 ≤ k ≤ r } � ∈ ∈ ∑ k − 1 l = 1 m l + i ←→ ( i , k ) 1:1 ←→ Γ ′ ( m ) : = Γ ( m ) \ { ( m r , r ) } { 1 , 2 , . . . , m − 1 } ⊕ m ⊕ ( i , k ) ∈ Γ ( m ) Z ε ( i , k ) : weight lattice of gl m . P : = i = 1 Z ε i = ⊕ m − 1 ⊕ Q : = i = 1 Z α i = ( i , k ) ∈ Γ ′ ( m ) Z α ( i , k ) : root lattice of gl m . ( α i = ε i − ε i + 1 : simple root) For ( i , k ) , ( j , l ) ∈ Γ ( m ) , put  1 if ( j , l ) = ( i , k )      a ( i , k )( j , l ) =  − 1 if ( j , l ) = ( i + 1 , k ) ( note ( m k + 1 , k ) = (1 , k + 1))     otherwise 0   . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 6 / 25

  14. Cartan data m = ( m 1 , . . . , m r ) ∈ Z r s.t. m k ≥ n . Put m = m 1 + · · · + m r . 1:1 � { 1 , 2 , . . . , m } ←→ Γ ( m ) : = { ( i , k ) � 1 ≤ i ≤ m k , 1 ≤ k ≤ r } � ∈ ∈ ∑ k − 1 l = 1 m l + i ←→ ( i , k ) 1:1 ←→ Γ ′ ( m ) : = Γ ( m ) \ { ( m r , r ) } { 1 , 2 , . . . , m − 1 } ⊕ m ⊕ ( i , k ) ∈ Γ ( m ) Z ε ( i , k ) : weight lattice of gl m . P : = i = 1 Z ε i = ⊕ m − 1 ⊕ Q : = i = 1 Z α i = ( i , k ) ∈ Γ ′ ( m ) Z α ( i , k ) : root lattice of gl m . ( α i = ε i − ε i + 1 : simple root) For ( i , k ) , ( j , l ) ∈ Γ ( m ) , put  1 if ( j , l ) = ( i , k )      a ( i , k )( j , l ) =  − 1 if ( j , l ) = ( i + 1 , k ) ( note ( m k + 1 , k ) = (1 , k + 1))     otherwise 0   . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 6 / 25

  15. Cartan data m = ( m 1 , . . . , m r ) ∈ Z r s.t. m k ≥ n . Put m = m 1 + · · · + m r . 1:1 � { 1 , 2 , . . . , m } ←→ Γ ( m ) : = { ( i , k ) � 1 ≤ i ≤ m k , 1 ≤ k ≤ r } � ∈ ∈ ∑ k − 1 l = 1 m l + i ←→ ( i , k ) 1:1 ←→ Γ ′ ( m ) : = Γ ( m ) \ { ( m r , r ) } { 1 , 2 , . . . , m − 1 } ⊕ m ⊕ ( i , k ) ∈ Γ ( m ) Z ε ( i , k ) : weight lattice of gl m . P : = i = 1 Z ε i = ⊕ m − 1 ⊕ Q : = i = 1 Z α i = ( i , k ) ∈ Γ ′ ( m ) Z α ( i , k ) : root lattice of gl m . ( α i = ε i − ε i + 1 : simple root) For ( i , k ) , ( j , l ) ∈ Γ ( m ) , put  1 if ( j , l ) = ( i , k )      a ( i , k )( j , l ) =  − 1 if ( j , l ) = ( i + 1 , k ) ( note ( m k + 1 , k ) = (1 , k + 1))     otherwise 0   . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 6 / 25

  16. Cartan data m = ( m 1 , . . . , m r ) ∈ Z r s.t. m k ≥ n . Put m = m 1 + · · · + m r . 1:1 � { 1 , 2 , . . . , m } ←→ Γ ( m ) : = { ( i , k ) � 1 ≤ i ≤ m k , 1 ≤ k ≤ r } � ∈ ∈ ∑ k − 1 l = 1 m l + i ←→ ( i , k ) 1:1 ←→ Γ ′ ( m ) : = Γ ( m ) \ { ( m r , r ) } { 1 , 2 , . . . , m − 1 } ⊕ m ⊕ ( i , k ) ∈ Γ ( m ) Z ε ( i , k ) : weight lattice of gl m . P : = i = 1 Z ε i = ⊕ m − 1 ⊕ Q : = i = 1 Z α i = ( i , k ) ∈ Γ ′ ( m ) Z α ( i , k ) : root lattice of gl m . ( α i = ε i − ε i + 1 : simple root) For ( i , k ) , ( j , l ) ∈ Γ ( m ) , put  1 if ( j , l ) = ( i , k )      a ( i , k )( j , l ) =  − 1 if ( j , l ) = ( i + 1 , k ) ( note ( m k + 1 , k ) = (1 , k + 1))     otherwise 0   . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 6 / 25

  17. Cartan data m = ( m 1 , . . . , m r ) ∈ Z r s.t. m k ≥ n . Put m = m 1 + · · · + m r . 1:1 � { 1 , 2 , . . . , m } ←→ Γ ( m ) : = { ( i , k ) � 1 ≤ i ≤ m k , 1 ≤ k ≤ r } � ∈ ∈ ∑ k − 1 l = 1 m l + i ←→ ( i , k ) 1:1 ←→ Γ ′ ( m ) : = Γ ( m ) \ { ( m r , r ) } { 1 , 2 , . . . , m − 1 } ⊕ m ⊕ ( i , k ) ∈ Γ ( m ) Z ε ( i , k ) : weight lattice of gl m . P : = i = 1 Z ε i = ⊕ m − 1 ⊕ Q : = i = 1 Z α i = ( i , k ) ∈ Γ ′ ( m ) Z α ( i , k ) : root lattice of gl m . ( α i = ε i − ε i + 1 : simple root) For ( i , k ) , ( j , l ) ∈ Γ ( m ) , put  1 if ( j , l ) = ( i , k )      a ( i , k )( j , l ) =  − 1 if ( j , l ) = ( i + 1 , k ) ( note ( m k + 1 , k ) = (1 , k + 1))     otherwise 0   . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 6 / 25

  18. Cartan data m = ( m 1 , . . . , m r ) ∈ Z r s.t. m k ≥ n . Put m = m 1 + · · · + m r . 1:1 � { 1 , 2 , . . . , m } ←→ Γ ( m ) : = { ( i , k ) � 1 ≤ i ≤ m k , 1 ≤ k ≤ r } � ∈ ∈ ∑ k − 1 l = 1 m l + i ←→ ( i , k ) 1:1 ←→ Γ ′ ( m ) : = Γ ( m ) \ { ( m r , r ) } { 1 , 2 , . . . , m − 1 } ⊕ m ⊕ ( i , k ) ∈ Γ ( m ) Z ε ( i , k ) : weight lattice of gl m . P : = i = 1 Z ε i = ⊕ m − 1 ⊕ Q : = i = 1 Z α i = ( i , k ) ∈ Γ ′ ( m ) Z α ( i , k ) : root lattice of gl m . ( α i = ε i − ε i + 1 : simple root) For ( i , k ) , ( j , l ) ∈ Γ ( m ) , put  1 if ( j , l ) = ( i , k )      a ( i , k )( j , l ) =  − 1 if ( j , l ) = ( i + 1 , k ) ( note ( m k + 1 , k ) = (1 , k + 1))     otherwise 0   . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 6 / 25

  19. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  20. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  21. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  22. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  23. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  24. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  25. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  26. generators of S n , r ⊕ Z ≥ 0 ε ( i , k ) ⊂ P P ≥ 0 : = ( i , k ) ∈ Γ ( m )    �  ∑ ∑   Λ n , r ( m ) : =  λ = λ ( i , k ) ε ( i , k ) ∈ P ≥ 0 � λ ( i , k ) = n   �   �    ( i , k ) ∈ Γ ( m ) ( i , k ) ∈ Γ ( m ) Denote λ ∈ Λ n , r ( m ) by λ = ( λ [1] , . . . , λ [ r ] ) , where λ [ k ] = ( λ (1 , k ) , λ (2 , k ) , . . . , λ ( m k , k ) ) . Recall S n , r : = S ≤ 0 n , r · S ≥ 0 and S ≤ 0 n , r � ρ ( U ≤ 0 S ≥ 0 n , r � ρ ( U ≥ 0 q ) , q ) n , r ρ : U q ( gl m ) ↠ S n , 1 = ρ ( U ≤ 0 q ) · ρ ( U ≥ 0 q ) . ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) by Define X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( i , k ) , 0 : image of e ( i , k ) in S ≥ 0 n , r � ρ ( U ≥ 0 X + q ) . X − ( i , k ) , 0 : image of f ( i , k ) in S ≤ 0 n , r � ρ ( U ≤ 0 q ) . K ± ( j , l ) : image of K ± ( j , l ) in S ≥ 0 n , r � ρ ( U ≥ 0 q ) or S ≤ 0 n , r � ρ ( U ≤ 0 q ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 7 / 25

  27. Symmetric polynomials Φ ± t Define Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] ( t ≥ 1 ) by Φ ± 1 ( X 1 , . . . , X k ) : = X 1 + X 2 + · · · + X k . Φ ± t + 1 ( X 1 , . . . , X k ) k ∑ ( ) Φ ± t ( X 1 , . . . , X s ) X s − q ∓ 2 Φ ± : = X t + 1 + t ( X 1 , . . . , X s − 1 ) X s 1 s = 2 ( ) Z [ q , q − 1 ][ X 1 , . . . , X s ] ֒ → Z [ q , q − 1 ][ X 1 , . . . , X k ] , X i �→ X i . Lemma . Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] S k . . . Remark . Φ ± t ( X 1 , . . . , X k − 1 , 0) = Φ ± t ( X 1 , . . . , X k − 1 ) � ∃ Φ ± t ( X 1 , X 2 , . . . ) : symmetric function s.t. Φ ± t ( X 1 , . . . , X k , 0 , . . . ) = Φ ± t ( X 1 , . . . , X k ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 8 / 25

  28. Symmetric polynomials Φ ± t Define Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] ( t ≥ 1 ) by Φ ± 1 ( X 1 , . . . , X k ) : = X 1 + X 2 + · · · + X k . Φ ± t + 1 ( X 1 , . . . , X k ) k ∑ ( ) Φ ± t ( X 1 , . . . , X s ) X s − q ∓ 2 Φ ± : = X t + 1 + t ( X 1 , . . . , X s − 1 ) X s 1 s = 2 ( ) Z [ q , q − 1 ][ X 1 , . . . , X s ] ֒ → Z [ q , q − 1 ][ X 1 , . . . , X k ] , X i �→ X i . Lemma . Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] S k . . . Remark . Φ ± t ( X 1 , . . . , X k − 1 , 0) = Φ ± t ( X 1 , . . . , X k − 1 ) � ∃ Φ ± t ( X 1 , X 2 , . . . ) : symmetric function s.t. Φ ± t ( X 1 , . . . , X k , 0 , . . . ) = Φ ± t ( X 1 , . . . , X k ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 8 / 25

  29. Symmetric polynomials Φ ± t Define Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] ( t ≥ 1 ) by Φ ± 1 ( X 1 , . . . , X k ) : = X 1 + X 2 + · · · + X k . Φ ± t + 1 ( X 1 , . . . , X k ) k ∑ ( ) Φ ± t ( X 1 , . . . , X s ) X s − q ∓ 2 Φ ± : = X t + 1 + t ( X 1 , . . . , X s − 1 ) X s 1 s = 2 ( ) Z [ q , q − 1 ][ X 1 , . . . , X s ] ֒ → Z [ q , q − 1 ][ X 1 , . . . , X k ] , X i �→ X i . Lemma . Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] S k . . . Remark . Φ ± t ( X 1 , . . . , X k − 1 , 0) = Φ ± t ( X 1 , . . . , X k − 1 ) � ∃ Φ ± t ( X 1 , X 2 , . . . ) : symmetric function s.t. Φ ± t ( X 1 , . . . , X k , 0 , . . . ) = Φ ± t ( X 1 , . . . , X k ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 8 / 25

  30. Symmetric polynomials Φ ± t Define Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] ( t ≥ 1 ) by Φ ± 1 ( X 1 , . . . , X k ) : = X 1 + X 2 + · · · + X k . Φ ± t + 1 ( X 1 , . . . , X k ) k ∑ ( ) Φ ± t ( X 1 , . . . , X s ) X s − q ∓ 2 Φ ± : = X t + 1 + t ( X 1 , . . . , X s − 1 ) X s 1 s = 2 ( ) Z [ q , q − 1 ][ X 1 , . . . , X s ] ֒ → Z [ q , q − 1 ][ X 1 , . . . , X k ] , X i �→ X i . Lemma . Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] S k . . . Remark . Φ ± t ( X 1 , . . . , X k − 1 , 0) = Φ ± t ( X 1 , . . . , X k − 1 ) � ∃ Φ ± t ( X 1 , X 2 , . . . ) : symmetric function s.t. Φ ± t ( X 1 , . . . , X k , 0 , . . . ) = Φ ± t ( X 1 , . . . , X k ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 8 / 25

  31. Symmetric polynomials Φ ± t Define Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] ( t ≥ 1 ) by Φ ± 1 ( X 1 , . . . , X k ) : = X 1 + X 2 + · · · + X k . Φ ± t + 1 ( X 1 , . . . , X k ) k ∑ ( ) Φ ± t ( X 1 , . . . , X s ) X s − q ∓ 2 Φ ± : = X t + 1 + t ( X 1 , . . . , X s − 1 ) X s 1 s = 2 ( ) Z [ q , q − 1 ][ X 1 , . . . , X s ] ֒ → Z [ q , q − 1 ][ X 1 , . . . , X k ] , X i �→ X i . Lemma . Φ ± t ( X 1 , . . . , X k ) ∈ Z [ q , q − 1 ][ X 1 , . . . , X k ] S k . . . Remark . Φ ± t ( X 1 , . . . , X k − 1 , 0) = Φ ± t ( X 1 , . . . , X k − 1 ) � ∃ Φ ± t ( X 1 , X 2 , . . . ) : symmetric function s.t. Φ ± t ( X 1 , . . . , X k , 0 , . . . ) = Φ ± t ( X 1 , . . . , X k ) . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 8 / 25

  32. generators of S n , r ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) . We already defined X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( ⊕ ) Recall S n , r = End H n , r µ ∈ Λ n , r ( m ) m µ · H n , r . ( ( j , l ) ∈ Γ ( m ) , t ≥ 1 ) by Define H ± ( j , l ) , t ∈ S n , r 1 H + q − t + 1 ( q − q − 1 ) t − 1 m µ Φ + ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , − 1 H − q t − 1 ( q − q − 1 ) t − 1 m µ Φ − ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , where N = ∑ l − 1 s = 1 | µ [ s ] | + ∑ j i = 1 µ ( i , l ) . ( ( i , k ) ∈ Γ ′ ( m ) , t ≥ 1 ) by Define X ± ( i , k ) , t ∈ S n , r 1 ( ) X ± ( i , k ) , 1 X ± ( i , k ) , t − X ± H + ( i , k ) , t H + ( i , k ) , t + 1 : = . ( i , k ) , 1 q − q − 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 9 / 25

  33. generators of S n , r ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) . We already defined X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( ⊕ ) Recall S n , r = End H n , r µ ∈ Λ n , r ( m ) m µ · H n , r . ( ( j , l ) ∈ Γ ( m ) , t ≥ 1 ) by Define H ± ( j , l ) , t ∈ S n , r 1 H + q − t + 1 ( q − q − 1 ) t − 1 m µ Φ + ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , − 1 H − q t − 1 ( q − q − 1 ) t − 1 m µ Φ − ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , where N = ∑ l − 1 s = 1 | µ [ s ] | + ∑ j i = 1 µ ( i , l ) . ( ( i , k ) ∈ Γ ′ ( m ) , t ≥ 1 ) by Define X ± ( i , k ) , t ∈ S n , r 1 ( ) X ± ( i , k ) , 1 X ± ( i , k ) , t − X ± H + ( i , k ) , t H + ( i , k ) , t + 1 : = . ( i , k ) , 1 q − q − 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 9 / 25

  34. generators of S n , r ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) . We already defined X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( ⊕ ) Recall S n , r = End H n , r µ ∈ Λ n , r ( m ) m µ · H n , r . ( ( j , l ) ∈ Γ ( m ) , t ≥ 1 ) by Define H ± ( j , l ) , t ∈ S n , r 1 H + q − t + 1 ( q − q − 1 ) t − 1 m µ Φ + ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , − 1 H − q t − 1 ( q − q − 1 ) t − 1 m µ Φ − ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , where N = ∑ l − 1 s = 1 | µ [ s ] | + ∑ j i = 1 µ ( i , l ) . ( ( i , k ) ∈ Γ ′ ( m ) , t ≥ 1 ) by Define X ± ( i , k ) , t ∈ S n , r 1 ( ) X ± ( i , k ) , 1 X ± ( i , k ) , t − X ± H + ( i , k ) , t H + ( i , k ) , t + 1 : = . ( i , k ) , 1 q − q − 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 9 / 25

  35. generators of S n , r ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) . We already defined X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( ⊕ ) Recall S n , r = End H n , r µ ∈ Λ n , r ( m ) m µ · H n , r . ( ( j , l ) ∈ Γ ( m ) , t ≥ 1 ) by Define H ± ( j , l ) , t ∈ S n , r 1 H + q − t + 1 ( q − q − 1 ) t − 1 m µ Φ + ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , − 1 H − q t − 1 ( q − q − 1 ) t − 1 m µ Φ − ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , where N = ∑ l − 1 s = 1 | µ [ s ] | + ∑ j i = 1 µ ( i , l ) . ( ( i , k ) ∈ Γ ′ ( m ) , t ≥ 1 ) by Define X ± ( i , k ) , t ∈ S n , r 1 ( ) X ± ( i , k ) , 1 X ± ( i , k ) , t − X ± H + ( i , k ) , t H + ( i , k ) , t + 1 : = . ( i , k ) , 1 q − q − 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 9 / 25

  36. generators of S n , r ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) ) . We already defined X ± ( i , k ) , 0 , K ± ( j , l ) ∈ S n , r ( ⊕ ) Recall S n , r = End H n , r µ ∈ Λ n , r ( m ) m µ · H n , r . ( ( j , l ) ∈ Γ ( m ) , t ≥ 1 ) by Define H ± ( j , l ) , t ∈ S n , r 1 H + q − t + 1 ( q − q − 1 ) t − 1 m µ Φ + ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , − 1 H − q t − 1 ( q − q − 1 ) t − 1 m µ Φ − ( j , l ) , t ( m µ ) : = t ( L N , L N − 1 , . . . , L N − µ ( l ) j + 1 ) , where N = ∑ l − 1 s = 1 | µ [ s ] | + ∑ j i = 1 µ ( i , l ) . ( ( i , k ) ∈ Γ ′ ( m ) , t ≥ 1 ) by Define X ± ( i , k ) , t ∈ S n , r 1 ( ) X ± ( i , k ) , 1 X ± ( i , k ) , t − X ± H + ( i , k ) , t H + ( i , k ) , t + 1 : = . ( i , k ) , 1 q − q − 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 9 / 25

  37. Algebra U . Definition . m = ( m 1 , . . . , m r ) ∈ Z r > 0 . U : associative algebra over Q ( q ) defined by generators: X ± ( i , k ) , t , K ± ( j , l ) , H ± ( j , l ) , t , C k ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ) defining relations: C k : central elements , ( j , l ) K − ( j , l ) = K − H ± K + ( j , l ) K + ( j , l ) = 1 , ( i , k ) , 0 = 1 , ( i , k ) , K ε ′ ( i , k ) , H ε ′ ( i , k ) , t , H ε ′ [ K ε ( j , l ) ] = [ K ε ( j , l ) , s ] = [ H ε ( j , l ) , s ] = 0 ( ε, ε ′ ∈ { + , −} ) K ( j , l ) X ± ( i , k ) , t K − ( j , l ) = q ± a ( i , k )( j , l ) X ± ( i , k ) , t , ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ± a ( i , k )( j , l ) H + ( j , l ) , s X ± ( i , k ) , t + 1 − q ∓ a ( i , k )( j , l ) X ± [ H + ( i , k ) , t + 1 H + ( j , l ) , s [ H − ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ∓ a ( i , k )( j , l ) H − ( j , l ) , s X ± ( i , k ) , t + 1 − q ± a ( i , k )( j , l ) X ± ( i , k ) , t + 1 H − . ( j , l ) , s . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 10 / 25

  38. Algebra U . Definition . m = ( m 1 , . . . , m r ) ∈ Z r > 0 . U : associative algebra over Q ( q ) defined by generators: X ± ( i , k ) , t , K ± ( j , l ) , H ± ( j , l ) , t , C k ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ) defining relations: C k : central elements , ( j , l ) K − ( j , l ) = K − H ± K + ( j , l ) K + ( j , l ) = 1 , ( i , k ) , 0 = 1 , ( i , k ) , K ε ′ ( i , k ) , H ε ′ ( i , k ) , t , H ε ′ [ K ε ( j , l ) ] = [ K ε ( j , l ) , s ] = [ H ε ( j , l ) , s ] = 0 ( ε, ε ′ ∈ { + , −} ) K ( j , l ) X ± ( i , k ) , t K − ( j , l ) = q ± a ( i , k )( j , l ) X ± ( i , k ) , t , ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ± a ( i , k )( j , l ) H + ( j , l ) , s X ± ( i , k ) , t + 1 − q ∓ a ( i , k )( j , l ) X ± [ H + ( i , k ) , t + 1 H + ( j , l ) , s [ H − ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ∓ a ( i , k )( j , l ) H − ( j , l ) , s X ± ( i , k ) , t + 1 − q ± a ( i , k )( j , l ) X ± ( i , k ) , t + 1 H − . ( j , l ) , s . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 10 / 25

  39. Algebra U . Definition . m = ( m 1 , . . . , m r ) ∈ Z r > 0 . U : associative algebra over Q ( q ) defined by generators: X ± ( i , k ) , t , K ± ( j , l ) , H ± ( j , l ) , t , C k ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ) defining relations: C k : central elements , ( j , l ) K − ( j , l ) = K − H ± K + ( j , l ) K + ( j , l ) = 1 , ( i , k ) , 0 = 1 , ( i , k ) , K ε ′ ( i , k ) , H ε ′ ( i , k ) , t , H ε ′ [ K ε ( j , l ) ] = [ K ε ( j , l ) , s ] = [ H ε ( j , l ) , s ] = 0 ( ε, ε ′ ∈ { + , −} ) K ( j , l ) X ± ( i , k ) , t K − ( j , l ) = q ± a ( i , k )( j , l ) X ± ( i , k ) , t , ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ± a ( i , k )( j , l ) H + ( j , l ) , s X ± ( i , k ) , t + 1 − q ∓ a ( i , k )( j , l ) X ± [ H + ( i , k ) , t + 1 H + ( j , l ) , s [ H − ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ∓ a ( i , k )( j , l ) H − ( j , l ) , s X ± ( i , k ) , t + 1 − q ± a ( i , k )( j , l ) X ± ( i , k ) , t + 1 H − . ( j , l ) , s . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 10 / 25

  40. Algebra U . Definition . m = ( m 1 , . . . , m r ) ∈ Z r > 0 . U : associative algebra over Q ( q ) defined by generators: X ± ( i , k ) , t , K ± ( j , l ) , H ± ( j , l ) , t , C k ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ) defining relations: C k : central elements , ( j , l ) K − ( j , l ) = K − H ± K + ( j , l ) K + ( j , l ) = 1 , ( i , k ) , 0 = 1 , ( i , k ) , K ε ′ ( i , k ) , H ε ′ ( i , k ) , t , H ε ′ [ K ε ( j , l ) ] = [ K ε ( j , l ) , s ] = [ H ε ( j , l ) , s ] = 0 ( ε, ε ′ ∈ { + , −} ) K ( j , l ) X ± ( i , k ) , t K − ( j , l ) = q ± a ( i , k )( j , l ) X ± ( i , k ) , t , ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ± a ( i , k )( j , l ) H + ( j , l ) , s X ± ( i , k ) , t + 1 − q ∓ a ( i , k )( j , l ) X ± [ H + ( i , k ) , t + 1 H + ( j , l ) , s [ H − ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ∓ a ( i , k )( j , l ) H − ( j , l ) , s X ± ( i , k ) , t + 1 − q ± a ( i , k )( j , l ) X ± ( i , k ) , t + 1 H − . ( j , l ) , s . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 10 / 25

  41. Algebra U . Definition . m = ( m 1 , . . . , m r ) ∈ Z r > 0 . U : associative algebra over Q ( q ) defined by generators: X ± ( i , k ) , t , K ± ( j , l ) , H ± ( j , l ) , t , C k ( ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ) defining relations: C k : central elements , ( j , l ) K − ( j , l ) = K − H ± K + ( j , l ) K + ( j , l ) = 1 , ( i , k ) , 0 = 1 , ( i , k ) , K ε ′ ( i , k ) , H ε ′ ( i , k ) , t , H ε ′ [ K ε ( j , l ) ] = [ K ε ( j , l ) , s ] = [ H ε ( j , l ) , s ] = 0 ( ε, ε ′ ∈ { + , −} ) K ( j , l ) X ± ( i , k ) , t K − ( j , l ) = q ± a ( i , k )( j , l ) X ± ( i , k ) , t , ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ± a ( i , k )( j , l ) H + ( j , l ) , s X ± ( i , k ) , t + 1 − q ∓ a ( i , k )( j , l ) X ± [ H + ( i , k ) , t + 1 H + ( j , l ) , s [ H − ( j , l ) , s + 1 , X ± ( i , k ) , t ] = q ∓ a ( i , k )( j , l ) H − ( j , l ) , s X ± ( i , k ) , t + 1 − q ± a ( i , k )( j , l ) X ± ( i , k ) , t + 1 H − . ( j , l ) , s . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 10 / 25

  42. . defining relations (cont.): . X ε ( i , k ) , t X ε ( j , l ) , s − X ε ( j , l ) , s X ε if ( j , l ) � ( i , k ) , ( i ± 1 , k ) , ( i , k ) , t = 0 ( i , k ) , 0 ) 2 − ( q + q − 1 ) X ε X ε ( i ± 1 , k ) , 0 ( X ε ( i , k ) , 0 X ε ( i ± 1 , 0) , 0 X ε ( i , k ) , 0 + ( X ε ( i , k ) , 0 ) 2 X ε ( i ± 1 , k ) , 0 = 0 [ X + ( i , k ) , t , X − ( j , l ) , s ] J + ( i , k ) , s + t − J −  ( i , k ) , s + t   if i � m k ,   q − q − 1        = δ ( i , k )( j , l ) ( m k , k ) , s + t − J −  J + ( m k , k ) , s + t  + ( J + J −  − C k + 1 ( m k , k ) , s + t + 1 − )   ( m k , k ) , s + t + 1 q − q − 1       if i = m k ,   ( i , k ) K − ( i + 1 , k ) , J − ( i , k ) , 0 : = K − where J + ( i , k ) , 0 : = K + ( i , k ) K + ( i + 1 , k ) . t − 1 q − 1 ∑ ( ) J + ( i , k ) , t : = K + ( i , k ) K − q − t H + q t − 2 h H + ( i , k ) , h H − ( i , k ) , t − ( i + 1 , k ) ( i + 1 , k ) , t − h q − q − 1 h = 1 t − 1 q ∑ ( ) J − ( i , k ) K − − q t H − q t − 2 h H + ( i , k ) , h H − ( i , k ) , t : = K + ( i + 1 , k ) , t − ( i + 1 , k ) ( i + 1 , k ) , t − h q − q − 1 . h = 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 11 / 25

  43. . defining relations (cont.): . X ε ( i , k ) , t X ε ( j , l ) , s − X ε ( j , l ) , s X ε if ( j , l ) � ( i , k ) , ( i ± 1 , k ) , ( i , k ) , t = 0 ( i , k ) , 0 ) 2 − ( q + q − 1 ) X ε X ε ( i ± 1 , k ) , 0 ( X ε ( i , k ) , 0 X ε ( i ± 1 , 0) , 0 X ε ( i , k ) , 0 + ( X ε ( i , k ) , 0 ) 2 X ε ( i ± 1 , k ) , 0 = 0 [ X + ( i , k ) , t , X − ( j , l ) , s ] J + ( i , k ) , s + t − J −  ( i , k ) , s + t   if i � m k ,   q − q − 1        = δ ( i , k )( j , l ) ( m k , k ) , s + t − J −  J + ( m k , k ) , s + t  + ( J + J −  − C k + 1 ( m k , k ) , s + t + 1 − )   ( m k , k ) , s + t + 1 q − q − 1       if i = m k ,   ( i , k ) K − ( i + 1 , k ) , J − ( i , k ) , 0 : = K − where J + ( i , k ) , 0 : = K + ( i , k ) K + ( i + 1 , k ) . t − 1 q − 1 ∑ ( ) J + ( i , k ) , t : = K + ( i , k ) K − q − t H + q t − 2 h H + ( i , k ) , h H − ( i , k ) , t − ( i + 1 , k ) ( i + 1 , k ) , t − h q − q − 1 h = 1 t − 1 q ∑ ( ) J − ( i , k ) K − − q t H − q t − 2 h H + ( i , k ) , h H − ( i , k ) , t : = K + ( i + 1 , k ) , t − ( i + 1 , k ) ( i + 1 , k ) , t − h q − q − 1 . h = 1 . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 11 / 25

  44. . Theorem (W) . Assume that m k ≥ n for all k = 1 , . . . , r − 1 . There exists a surjective homomorphism U ↠ S n , r s.t. C k �→ q c k , X ± ( i , k ) , t �→ X ± ( i , k ) , t , K ± ( j , l ) �→ K ± ( j , l ) , H ± ( j , l ) , t �→ H ± ( j , l ) , t . . . Proposition . . There exists a surjective homomorphism 1 U ↠ U q ( gl m ) s.t. C k �→ − 1 , X + ( i , k ) , 0 �→ e ( i , k ) , X − ( i , k ) , 0 �→ f ( i , k ) , K ± ( j , l ) �→ K ± ( j , l ) , X ± ( i , k ) , t , H ± ( j , l ) , t �→ 0 ( t ≥ 1) . . . There exists a injective homomorphism 2 U q ( g ) ֒ → U s.t. e ( i , k ) �→ X + ( i , k ) , 0 , f ( i , k ) �→ X − ( i , k ) , 0 , K ± ( j , l ) �→ K ± ( j , l ) . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 12 / 25

  45. . Theorem (W) . Assume that m k ≥ n for all k = 1 , . . . , r − 1 . There exists a surjective homomorphism U ↠ S n , r s.t. C k �→ q c k , X ± ( i , k ) , t �→ X ± ( i , k ) , t , K ± ( j , l ) �→ K ± ( j , l ) , H ± ( j , l ) , t �→ H ± ( j , l ) , t . . . Proposition . . There exists a surjective homomorphism 1 U ↠ U q ( gl m ) s.t. C k �→ − 1 , X + ( i , k ) , 0 �→ e ( i , k ) , X − ( i , k ) , 0 �→ f ( i , k ) , K ± ( j , l ) �→ K ± ( j , l ) , X ± ( i , k ) , t , H ± ( j , l ) , t �→ 0 ( t ≥ 1) . . . There exists a injective homomorphism 2 U q ( g ) ֒ → U s.t. e ( i , k ) �→ X + ( i , k ) , 0 , f ( i , k ) �→ X − ( i , k ) , 0 , K ± ( j , l ) �→ K ± ( j , l ) . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 12 / 25

  46. . Theorem (W) . Assume that m k ≥ n for all k = 1 , . . . , r − 1 . There exists a surjective homomorphism U ↠ S n , r s.t. C k �→ q c k , X ± ( i , k ) , t �→ X ± ( i , k ) , t , K ± ( j , l ) �→ K ± ( j , l ) , H ± ( j , l ) , t �→ H ± ( j , l ) , t . . . Proposition . . There exists a surjective homomorphism 1 U ↠ U q ( gl m ) s.t. C k �→ − 1 , X + ( i , k ) , 0 �→ e ( i , k ) , X − ( i , k ) , 0 �→ f ( i , k ) , K ± ( j , l ) �→ K ± ( j , l ) , X ± ( i , k ) , t , H ± ( j , l ) , t �→ 0 ( t ≥ 1) . . . There exists a injective homomorphism 2 U q ( g ) ֒ → U s.t. e ( i , k ) �→ X + ( i , k ) , 0 , f ( i , k ) �→ X − ( i , k ) , 0 , K ± ( j , l ) �→ K ± ( j , l ) . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 12 / 25

  47. . Theorem (W) . Assume that m k ≥ n for all k = 1 , . . . , r − 1 . There exists a surjective homomorphism U ↠ S n , r s.t. C k �→ q c k , X ± ( i , k ) , t �→ X ± ( i , k ) , t , K ± ( j , l ) �→ K ± ( j , l ) , H ± ( j , l ) , t �→ H ± ( j , l ) , t . . . Proposition . . There exists a surjective homomorphism 1 U ↠ U q ( gl m ) s.t. C k �→ − 1 , X + ( i , k ) , 0 �→ e ( i , k ) , X − ( i , k ) , 0 �→ f ( i , k ) , K ± ( j , l ) �→ K ± ( j , l ) , X ± ( i , k ) , t , H ± ( j , l ) , t �→ 0 ( t ≥ 1) . . . There exists a injective homomorphism 2 U q ( g ) ֒ → U s.t. e ( i , k ) �→ X + ( i , k ) , 0 , f ( i , k ) �→ X − ( i , k ) , 0 , K ± ( j , l ) �→ K ± ( j , l ) . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 12 / 25

  48. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  49. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  50. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  51. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  52. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  53. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  54. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  55. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  56. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  57. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  58. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  59. Highest weight modules of U Define subalgebras U 0 and U ± of U by U 0 : = ⟨ K ± � ( j , l ) , H ± � ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r ⟩ ( j , l ) , t , C k alg. . � U + : = ⟨ X + � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t U − : = ⟨ X − � � ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 ⟩ alg. . � ( i , k ) , t . Lemma . ( U � U − ⊗ U 0 ⊗ U + ??? ) U = U − U 0 U + . For λ ∈ P ≥ 0 , φ = ( φ ± ( i , k ) , t ) ( i , k ) ∈ Γ ( m ) , t ≥ 1 ( φ ± ( i , k ) , t ∈ Q ( q )) and c = ( c 1 , . . . , c r ) ∈ Z r , U -module V is a highest weight module (of type 1) with h.w. ( λ, φ, c ) if ∃ v 0 ∈ V s.t. V = U · v 0 . X + ( i , k ) , t · v o = 0 for all ( i , k ) ∈ Γ ′ ( m ) , t ≥ 0 . ( i , k ) · v 0 = q λ ( i , k ) v 0 , H ± ( i , k ) , t · v 0 = φ ± K + ( i , k ) , t v 0 , C k · v 0 = q c k v 0 . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 13 / 25

  60. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  61. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  62. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  63. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  64. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  65. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  66. Highest weight modules of U For ( λ, φ, c ) , define the Verma module V ( λ, φ, c ) in usual way. h.w. module with h.w. ( λ, φ, c ) is a quotient of V ( λ, φ, c ) . V ( λ, φ, c ) has the unique simple top L ( λ, φ, c ) . . Problem . When is L ( λ, φ, c ) finite dimensional?. . . Lemma . L ( λ, φ, c ) : finite dim. � λ : r -partition. i.e. λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) ≥ 0 for k = 1 , . . . , r . . ∵ ) Recall U q ( g ) ֒ → U . v 0 : h.w. vector of L ( λ, φ, c ) � v 0 : h.w. vector with weight λ as U q ( g ) -module . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 14 / 25

  67. Category O c For c = ( c 1 , . . . , c r ) ∈ Z r , let O c be the category of finite dim. U -modules s.t. C k ( 1 ≤ k ≤ r ) acts on M ∈ O c as the multiplication by q c k . M ∈ O c has the weight space decom.: ⊕ � � K ( j , l ) · m = q λ ( j , l ) m } . M = M λ , where M λ = { m ∈ M � λ ∈ P ≥ 0 All eigenvalues of H ± ( i , k ) , t ( ( i , k ) ∈ Γ ( m ) , t ≥ 0 ) are elements of Z [ q , q − 1 , ( q − q − 1 ) − 1 ] . . Proposition . . simple object of O c is a h.w. module. 1 . . If m k ≥ n for all k = 1 , . . . , r − 1 , 2 S n , r -mod ⊂ O c . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 15 / 25

  68. Category O c For c = ( c 1 , . . . , c r ) ∈ Z r , let O c be the category of finite dim. U -modules s.t. C k ( 1 ≤ k ≤ r ) acts on M ∈ O c as the multiplication by q c k . M ∈ O c has the weight space decom.: ⊕ � � K ( j , l ) · m = q λ ( j , l ) m } . M = M λ , where M λ = { m ∈ M � λ ∈ P ≥ 0 All eigenvalues of H ± ( i , k ) , t ( ( i , k ) ∈ Γ ( m ) , t ≥ 0 ) are elements of Z [ q , q − 1 , ( q − q − 1 ) − 1 ] . . Proposition . . simple object of O c is a h.w. module. 1 . . If m k ≥ n for all k = 1 , . . . , r − 1 , 2 S n , r -mod ⊂ O c . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 15 / 25

  69. Category O c For c = ( c 1 , . . . , c r ) ∈ Z r , let O c be the category of finite dim. U -modules s.t. C k ( 1 ≤ k ≤ r ) acts on M ∈ O c as the multiplication by q c k . M ∈ O c has the weight space decom.: ⊕ � � K ( j , l ) · m = q λ ( j , l ) m } . M = M λ , where M λ = { m ∈ M � λ ∈ P ≥ 0 All eigenvalues of H ± ( i , k ) , t ( ( i , k ) ∈ Γ ( m ) , t ≥ 0 ) are elements of Z [ q , q − 1 , ( q − q − 1 ) − 1 ] . . Proposition . . simple object of O c is a h.w. module. 1 . . If m k ≥ n for all k = 1 , . . . , r − 1 , 2 S n , r -mod ⊂ O c . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 15 / 25

  70. Category O c For c = ( c 1 , . . . , c r ) ∈ Z r , let O c be the category of finite dim. U -modules s.t. C k ( 1 ≤ k ≤ r ) acts on M ∈ O c as the multiplication by q c k . M ∈ O c has the weight space decom.: ⊕ � � K ( j , l ) · m = q λ ( j , l ) m } . M = M λ , where M λ = { m ∈ M � λ ∈ P ≥ 0 All eigenvalues of H ± ( i , k ) , t ( ( i , k ) ∈ Γ ( m ) , t ≥ 0 ) are elements of Z [ q , q − 1 , ( q − q − 1 ) − 1 ] . . Proposition . . simple object of O c is a h.w. module. 1 . . If m k ≥ n for all k = 1 , . . . , r − 1 , 2 S n , r -mod ⊂ O c . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 15 / 25

  71. Category O c For c = ( c 1 , . . . , c r ) ∈ Z r , let O c be the category of finite dim. U -modules s.t. C k ( 1 ≤ k ≤ r ) acts on M ∈ O c as the multiplication by q c k . M ∈ O c has the weight space decom.: ⊕ � � K ( j , l ) · m = q λ ( j , l ) m } . M = M λ , where M λ = { m ∈ M � λ ∈ P ≥ 0 All eigenvalues of H ± ( i , k ) , t ( ( i , k ) ∈ Γ ( m ) , t ≥ 0 ) are elements of Z [ q , q − 1 , ( q − q − 1 ) − 1 ] . . Proposition . . simple object of O c is a h.w. module. 1 . . If m k ≥ n for all k = 1 , . . . , r − 1 , 2 S n , r -mod ⊂ O c . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 15 / 25

  72. Category O c For c = ( c 1 , . . . , c r ) ∈ Z r , let O c be the category of finite dim. U -modules s.t. C k ( 1 ≤ k ≤ r ) acts on M ∈ O c as the multiplication by q c k . M ∈ O c has the weight space decom.: ⊕ � � K ( j , l ) · m = q λ ( j , l ) m } . M = M λ , where M λ = { m ∈ M � λ ∈ P ≥ 0 All eigenvalues of H ± ( i , k ) , t ( ( i , k ) ∈ Γ ( m ) , t ≥ 0 ) are elements of Z [ q , q − 1 , ( q − q − 1 ) − 1 ] . . Proposition . . simple object of O c is a h.w. module. 1 . . If m k ≥ n for all k = 1 , . . . , r − 1 , 2 S n , r -mod ⊂ O c . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 15 / 25

  73. Example ⊕ V : = ( i , k ) ∈ Γ ( m ) Q ( q ) v ( i , k ) with the following action:  qv ( i , k ) if ( j , l ) = ( i , k ) ,   K + ( j , l ) · v ( i , k ) =   othewise. 0    q tc 1 if ( j , l ) = ( i , k ) ,  H ±  ( j , l ) . t · v ( i , k ) =   0 otherwise.   q tc 1   ( q − q − 1 ) t v ( i + 1 , k ) if ( j , l ) = ( i , k )    X − ( j , l ) , t · v ( i , k ) =     0 otherwise   X + ( j , l ) , t · v ( i , k ) q tc 1  ( q − q − 1 ) t v ( i − 1 , k ) if i � 1 and ( j , l ) = ( i − 1 , k )        q tc 1  =  ( q c 1 − q c k ) v ( m k − 1 , k − 1) if i = 1 and ( j , l ) = ( m k − 1 , k − 1)   ( q − q − 1 ) t       0 otherwise  . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 16 / 25

  74. Example ⊕ V : = ( i , k ) ∈ Γ ( m ) Q ( q ) v ( i , k ) with the following action:  qv ( i , k ) if ( j , l ) = ( i , k ) ,   K + ( j , l ) · v ( i , k ) =   othewise. 0    q tc 1 if ( j , l ) = ( i , k ) ,  H ±  ( j , l ) . t · v ( i , k ) =   0 otherwise.   q tc 1   ( q − q − 1 ) t v ( i + 1 , k ) if ( j , l ) = ( i , k )    X − ( j , l ) , t · v ( i , k ) =     0 otherwise   X + ( j , l ) , t · v ( i , k ) q tc 1  ( q − q − 1 ) t v ( i − 1 , k ) if i � 1 and ( j , l ) = ( i − 1 , k )        q tc 1  =  ( q c 1 − q c k ) v ( m k − 1 , k − 1) if i = 1 and ( j , l ) = ( m k − 1 , k − 1)   ( q − q − 1 ) t       0 otherwise  . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 16 / 25

  75. Example ⊕ V : = ( i , k ) ∈ Γ ( m ) Q ( q ) v ( i , k ) with the following action:  qv ( i , k ) if ( j , l ) = ( i , k ) ,   K + ( j , l ) · v ( i , k ) =   othewise. 0    q tc 1 if ( j , l ) = ( i , k ) ,  H ±  ( j , l ) . t · v ( i , k ) =   0 otherwise.   q tc 1   ( q − q − 1 ) t v ( i + 1 , k ) if ( j , l ) = ( i , k )    X − ( j , l ) , t · v ( i , k ) =     0 otherwise   X + ( j , l ) , t · v ( i , k ) q tc 1  ( q − q − 1 ) t v ( i − 1 , k ) if i � 1 and ( j , l ) = ( i − 1 , k )        q tc 1  =  ( q c 1 − q c k ) v ( m k − 1 , k − 1) if i = 1 and ( j , l ) = ( m k − 1 , k − 1)   ( q − q − 1 ) t       0 otherwise  . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 16 / 25

  76. Example ⊕ V : = ( i , k ) ∈ Γ ( m ) Q ( q ) v ( i , k ) with the following action:  qv ( i , k ) if ( j , l ) = ( i , k ) ,   K + ( j , l ) · v ( i , k ) =   othewise. 0    q tc 1 if ( j , l ) = ( i , k ) ,  H ±  ( j , l ) . t · v ( i , k ) =   0 otherwise.   q tc 1   ( q − q − 1 ) t v ( i + 1 , k ) if ( j , l ) = ( i , k )    X − ( j , l ) , t · v ( i , k ) =     0 otherwise   X + ( j , l ) , t · v ( i , k ) q tc 1  ( q − q − 1 ) t v ( i − 1 , k ) if i � 1 and ( j , l ) = ( i − 1 , k )        q tc 1  =  ( q c 1 − q c k ) v ( m k − 1 , k − 1) if i = 1 and ( j , l ) = ( m k − 1 , k − 1)   ( q − q − 1 ) t       0 otherwise  . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 16 / 25

  77. Example ⊕ V : = ( i , k ) ∈ Γ ( m ) Q ( q ) v ( i , k ) with the following action:  qv ( i , k ) if ( j , l ) = ( i , k ) ,   K + ( j , l ) · v ( i , k ) =   othewise. 0    q tc 1 if ( j , l ) = ( i , k ) ,  H ±  ( j , l ) . t · v ( i , k ) =   0 otherwise.   q tc 1   ( q − q − 1 ) t v ( i + 1 , k ) if ( j , l ) = ( i , k )    X − ( j , l ) , t · v ( i , k ) =     0 otherwise   X + ( j , l ) , t · v ( i , k ) q tc 1  ( q − q − 1 ) t v ( i − 1 , k ) if i � 1 and ( j , l ) = ( i − 1 , k )        q tc 1  =  ( q c 1 − q c k ) v ( m k − 1 , k − 1) if i = 1 and ( j , l ) = ( m k − 1 , k − 1)   ( q − q − 1 ) t       0 otherwise  . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 16 / 25

  78. U -module V : Q ( q ) v (1 , 1) Q ( q ) v (1 , 2) Q ( q ) v (1 , k ) Q ( q ) v (1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + (1 , 1) , t (1 , 1) , t (1 , 2) , t (1 , 2) , t (1 , k ) , t (1 , k ) , t (1 , r ) , t (1 , r ) , t Q ( q ) v (2 , 1) Q ( q ) v (2 , 2) Q ( q ) v (2 , k ) Q ( q ) v (2 , r ) . . . . . . . . . . . . Q ( q ) v ( i − 1 , 1) Q ( q ) v ( i − 1 , 2) Q ( q ) v ( i − 1 , k ) Q ( q ) v ( i − 1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + ( i − 1 , 1) , t ( i − 1 , 1) , t ( i − 1 , 2) , t ( i − 1 , 2) , t ( i − 1 , k ) , t ( i − 1 , k ) , t ( i − 1 , r ) , t ( i − 1 , r ) , t Q ( q ) v ( i , 1) Q ( q ) v ( i , 2) Q ( q ) v ( i , k ) Q ( q ) v ( i , r ) · · · · · · ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X − X − X − X + X + X + X + ( i , 1) , t ( i , 1) , t ( i , 2) , t ( i , 2) , t ( i , k ) , t ( i , k ) , t ( i , r ) , t ( i , r ) , t Q ( q ) v ( i + 1 , 1) Q ( q ) v ( i + 1 , 2) Q ( q ) v ( i + 1 , k ) Q ( q ) v ( i + 1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X − X − X − X + X + X + X + ( i + 1 , 1) , t . ( i + 1 , 1) , t ( i + 1 , 2) , t . ( i + 1 , 2) , t ( i + 1 , k ) , t . ( i + 1 , k ) , t ( i + 1 , r ) , t . ( i + 1 , r ) , t . . . . . . . . Q ( q ) v ( m 1 − 1 , 1) Q ( q ) v ( m 2 − 1 , 2) Q ( q ) v ( m k − 1 , k ) Q ( q ) v ( m r − 1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + ( m 1 − 1 , 1) , t ( m 1 − 1 , 1) , t ( m 2 − 1 , 2) , t ( m 2 − 1 , 2) , t ( m k − 1 , k ) , t ( m k − 1 , k ) , t ( m r − 1 , r ) , t ( m r − 1 , r ) , Q ( q ) v ( m 1 , 1) Q ( q ) v ( m 2 , 2) Q ( q ) v ( m k , k ) Q ( q ) v ( m r , r ) ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + ( m 1 , 1) , t ( m 1 , 1) , t ( m 2 , 2) , t ( m 2 , 2) , t ( m k , k ) , t ( m k , k ) , t . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 17 / 25

  79. Res U U q ( g ) V (red : omit, only t = 0 ) : Q ( q ) v (1 , 1) Q ( q ) v (1 , 2) Q ( q ) v (1 , k ) Q ( q ) v (1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + (1 , 1) , t (1 , 1) , t (1 , 2) , t (1 , 2) , t (1 , k ) , t (1 , k ) , t (1 , r ) , t (1 , r ) , t Q ( q ) v (2 , 1) Q ( q ) v (2 , 2) Q ( q ) v (2 , k ) Q ( q ) v (2 , r ) . . . . . . . . . . . . Q ( q ) v ( i − 1 , 1) Q ( q ) v ( i − 1 , 2) Q ( q ) v ( i − 1 , k ) Q ( q ) v ( i − 1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + ( i − 1 , 1) , t ( i − 1 , 1) , t ( i − 1 , 2) , t ( i − 1 , 2) , t ( i − 1 , k ) , t ( i − 1 , k ) , t ( i − 1 , r ) , t ( i − 1 , r ) , t Q ( q ) v ( i , 1) Q ( q ) v ( i , 2) Q ( q ) v ( i , k ) Q ( q ) v ( i , r ) · · · · · · ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + ( i , 1) , t ( i , 1) , t ( i , 2) , t ( i , 2) , t ( i , r ) , t ( i , r ) , t ( i , k ) , t ( i , k ) , t Q ( q ) v ( i + 1 , 1) Q ( q ) v ( i + 1 , 2) Q ( q ) v ( i + 1 , k ) Q ( q ) v ( i + 1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X + X − X + X − X + X − X + ( i + 1 , 1) , t . ( i + 1 , 1) , t ( i + 1 , 2) , t . ( i + 1 , 2) , t ( i + 1 , k ) , t . ( i + 1 , k ) , t ( i + 1 , r ) , t . ( i + 1 , r ) , t . . . . . . . . Q ( q ) v ( m 1 − 1 , 1) Q ( q ) v ( m 2 − 1 , 2) Q ( q ) v ( m k − 1 , k ) Q ( q ) v ( m r − 1 , r ) ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ X − X − X − X − X + X + X + X + ( m 1 − 1 , 1) , t ( m 1 − 1 , 1) , t ( m 2 − 1 , 2) , t ( m 2 − 1 , 2) , t ( m k − 1 , k ) , t ( m k − 1 , k ) , t ( m r − 1 , r ) , t ( m r − 1 , r ) , Q ( q ) v ( m 1 , 1) Q ( q ) v ( m 2 , 2) Q ( q ) v ( m k , k ) Q ( q ) v ( m r , r ) ↓ ↑ ↓ ↑ ↓ ↑ X − X − X − X + X + X + ( m 1 , 1) , t ( m 1 , 1) , t ( m 2 , 2) , t ( m 2 , 2) , t ( m k , k ) , t ( m k , k ) , t . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 17 / 25

  80. Weyl modules and simple modules of S n , r Assume that m k ≥ n for all k = 1 , . . . , r − 1 . Then we have U ↠ S n , r . S n , r : quasi-hereditary algebra. � Λ + n , r ( m ) : = { λ ∈ Λ n , r ( m ) � λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) for all k = 1 , . . . , r } � n , r ( m ) } : a set of standard (Weyl) modules of S n , r . � � λ ∈ Λ + { W ( λ ) � n , r ( m ) } = { simple S n , r -modules } / iso. . � � λ ∈ Λ + { L ( λ ) : = Top W ( λ ) � . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 18 / 25

  81. Weyl modules and simple modules of S n , r Assume that m k ≥ n for all k = 1 , . . . , r − 1 . Then we have U ↠ S n , r . S n , r : quasi-hereditary algebra. � Λ + n , r ( m ) : = { λ ∈ Λ n , r ( m ) � λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) for all k = 1 , . . . , r } � n , r ( m ) } : a set of standard (Weyl) modules of S n , r . � � λ ∈ Λ + { W ( λ ) � n , r ( m ) } = { simple S n , r -modules } / iso. . � � λ ∈ Λ + { L ( λ ) : = Top W ( λ ) � . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 18 / 25

  82. Weyl modules and simple modules of S n , r Assume that m k ≥ n for all k = 1 , . . . , r − 1 . Then we have U ↠ S n , r . S n , r : quasi-hereditary algebra. � Λ + n , r ( m ) : = { λ ∈ Λ n , r ( m ) � λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) for all k = 1 , . . . , r } � n , r ( m ) } : a set of standard (Weyl) modules of S n , r . � � λ ∈ Λ + { W ( λ ) � n , r ( m ) } = { simple S n , r -modules } / iso. . � � λ ∈ Λ + { L ( λ ) : = Top W ( λ ) � . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 18 / 25

  83. Weyl modules and simple modules of S n , r Assume that m k ≥ n for all k = 1 , . . . , r − 1 . Then we have U ↠ S n , r . S n , r : quasi-hereditary algebra. � Λ + n , r ( m ) : = { λ ∈ Λ n , r ( m ) � λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) for all k = 1 , . . . , r } � n , r ( m ) } : a set of standard (Weyl) modules of S n , r . � � λ ∈ Λ + { W ( λ ) � n , r ( m ) } = { simple S n , r -modules } / iso. . � � λ ∈ Λ + { L ( λ ) : = Top W ( λ ) � . Theorem (W) . Under U q ( g ) ֒ → U , ) ⊕ β λµ as U q ( g ) -modules , ⊕ ( W ( µ [1] ) ⊗ W ( µ [2] ) ⊗· · ·⊗ W ( µ [ r ] ) W ( λ ) � µ ∈ Λ + n , r ( m ) β λµ is computed by a generalization of Littlewood-Richardson rule. . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 18 / 25

  84. Weyl modules and simple modules of S n , r Assume that m k ≥ n for all k = 1 , . . . , r − 1 . Then we have U ↠ S n , r . S n , r : quasi-hereditary algebra. � Λ + n , r ( m ) : = { λ ∈ Λ n , r ( m ) � λ (1 , k ) ≥ λ (2 , k ) ≥ · · · ≥ λ ( m k , k ) for all k = 1 , . . . , r } � n , r ( m ) } : a set of standard (Weyl) modules of S n , r . � � λ ∈ Λ + { W ( λ ) � n , r ( m ) } = { simple S n , r -modules } / iso. . � � λ ∈ Λ + { L ( λ ) : = Top W ( λ ) � . Proposition . As a U -module, W ( λ ) (resp. L ( λ ) ) is a h.w. module with h.w. ( λ, φ, c ) , where ± 1 φ ± q ± (1 − t ) ( q − q − 1 ) Φ ± t ( q c k + 2(1 − i ) , q c k + 2(2 − i ) , . . . , q c k + 2( λ ( i , k ) − i ) ) ( i , k ) , t = . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 18 / 25

  85. Harish-Chandra Ind and Res Recall the generators of U : { } X ± ( i , k ) , t , K ± ( j , l ) , H ± � � ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r X = ( j , l ) , t , C k . � Put U P : subalg. of U gen. by X \ { X − � � 1 ≤ k ≤ r − 1 , t ≥ 0 } � ( m k , k ) , t U L : subalg. of U gen. by X \ { X ± � � 1 ≤ k ≤ r − 1 , t ≥ 0 } � ( m k , k ) , t . Lemma . U L ֒ → U P ֒ → U − ↠ → g id U L � U [1] ⊗ U [2] ⊗ · · · ⊗ U [ r ] where U [ k ] is an ass. algebra generated by � { } X ± ( i , k ) , t , K ± ( j , k ) , H ± ( j , k ) , t , C k � � 1 ≤ i ≤ m k − 1 , 1 ≤ j ≤ m k , t ≥ 0 � with the same defining relations of U . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 19 / 25

  86. Harish-Chandra Ind and Res Recall the generators of U : { } X ± ( i , k ) , t , K ± ( j , l ) , H ± � � ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r X = ( j , l ) , t , C k . � Put U P : subalg. of U gen. by X \ { X − � � 1 ≤ k ≤ r − 1 , t ≥ 0 } � ( m k , k ) , t U L : subalg. of U gen. by X \ { X ± � � 1 ≤ k ≤ r − 1 , t ≥ 0 } � ( m k , k ) , t . Lemma . U L ֒ → U P ֒ → U − ↠ → g id U L � U [1] ⊗ U [2] ⊗ · · · ⊗ U [ r ] where U [ k ] is an ass. algebra generated by � { } X ± ( i , k ) , t , K ± ( j , k ) , H ± ( j , k ) , t , C k � � 1 ≤ i ≤ m k − 1 , 1 ≤ j ≤ m k , t ≥ 0 � with the same defining relations of U . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 19 / 25

  87. Harish-Chandra Ind and Res Recall the generators of U : { } X ± ( i , k ) , t , K ± ( j , l ) , H ± � � ( i , k ) ∈ Γ ′ ( m ) , ( j , l ) ∈ Γ ( m ) , t ≥ 0 , 1 ≤ k ≤ r X = ( j , l ) , t , C k . � Put U P : subalg. of U gen. by X \ { X − � � 1 ≤ k ≤ r − 1 , t ≥ 0 } � ( m k , k ) , t U L : subalg. of U gen. by X \ { X ± � � 1 ≤ k ≤ r − 1 , t ≥ 0 } � ( m k , k ) , t . Lemma . U L ֒ → U P ֒ → U − ↠ → g id U L � U [1] ⊗ U [2] ⊗ · · · ⊗ U [ r ] where U [ k ] is an ass. algebra generated by � { } X ± ( i , k ) , t , K ± ( j , k ) , H ± ( j , k ) , t , C k � � 1 ≤ i ≤ m k − 1 , 1 ≤ j ≤ m k , t ≥ 0 � with the same defining relations of U . . . . . . . . Kentaro Wada ( Shinshu University) Drinfeld type realization of cyclotomic q -Schur algebras 12th March, 2012 19 / 25

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