KnizhnikZamolodchikov functor for degenerate double affine Hecke - PowerPoint PPT Presentation

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : definition (cont.) O ( H rat ) : category of coherent H rat -modules on which D ξ acts locally nilpotently for ξ ∈ h ∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O ( H rat ) → H W -mod is defined by the assignement O ( H rat ) ∋ M �→ M ∇ ∈ H W -mod . Recall the Iwahori–Hecke algebra H W is generated by T α for simple roots α ∈ Π modulo the braid relations T α T β T α · · · = T β T α T β · · ·

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : definition (cont.) O ( H rat ) : category of coherent H rat -modules on which D ξ acts locally nilpotently for ξ ∈ h ∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O ( H rat ) → H W -mod is defined by the assignement O ( H rat ) ∋ M �→ M ∇ ∈ H W -mod . Recall the Iwahori–Hecke algebra H W is generated by T α for simple roots α ∈ Π modulo the braid relations T α T β T α · · · = T β T α T β · · · and the quadratic relations ( T α − v α )( T α + v − 1 α ) = 0

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : definition (cont.) O ( H rat ) : category of coherent H rat -modules on which D ξ acts locally nilpotently for ξ ∈ h ∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O ( H rat ) → H W -mod is defined by the assignement O ( H rat ) ∋ M �→ M ∇ ∈ H W -mod . Recall the Iwahori–Hecke algebra H W is generated by T α for simple roots α ∈ Π modulo the braid relations T α T β T α · · · = T β T α T β · · · and the quadratic relations ( T α − v α )( T α + v − 1 α ) = 0 Parameters are given by v α = exp( π √− 1 h α )

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : properties KZ functor V : O ( H rat ) → H W -mod V ( M ) = M ∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03] 1 O ( H rat ) is a highest weight category with index set Irrep( W )

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : properties KZ functor V : O ( H rat ) → H W -mod V ( M ) = M ∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03] 1 O ( H rat ) is a highest weight category with index set Irrep( W ) 2 V is a quotient functor of abelian categories, inducing equivalence O ( H rat ) / ker V ∼ = H W -mod

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : properties KZ functor V : O ( H rat ) → H W -mod V ( M ) = M ∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03] 1 O ( H rat ) is a highest weight category with index set Irrep( W ) 2 V is a quotient functor of abelian categories, inducing equivalence O ( H rat ) / ker V ∼ = H W -mod 3 V satisfies the double centraliser property, i.e. V is fully faithful on projective objects of O ( H rat )

RDAHAs dDAHAs Algebraic KZ KZ functor for RDAHAs : properties KZ functor V : O ( H rat ) → H W -mod V ( M ) = M ∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03] 1 O ( H rat ) is a highest weight category with index set Irrep( W ) 2 V is a quotient functor of abelian categories, inducing equivalence O ( H rat ) / ker V ∼ = H W -mod 3 V satisfies the double centraliser property, i.e. V is fully faithful on projective objects of O ( H rat ) 4 L ∈ O ( H rat ) : simple module, P L ∈ O ( H rat ) : projective cover of L . Then L ∈ ker V ⇔ P L is not injective

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 ,

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 , KZ : dz f − h 1 − s d f = 0 z

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 , KZ : dz f − h 1 − s d f = 0 z Dunkl : D = d dz − hz − 1 (1 − s )

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 , KZ : dz f − h 1 − s d f = 0 z Dunkl : D = d dz − hz − 1 (1 − s ) RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) .

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 , KZ : dz f − h 1 − s d f = 0 z Dunkl : D = d dz − hz − 1 (1 − s ) RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C .

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module L triv = C · u 0 .

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module L triv = C · u 0 . zu 0 = Du 0 = 0 , su 0 = u 0 .

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module L triv = C · u 0 . zu 0 = Du 0 = 0 , su 0 = u 0 . In this case,

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module L triv = C · u 0 . zu 0 = Du 0 = 0 , su 0 = u 0 . In this case, v = exp( π √− 1 h ) = √− 1

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module L triv = C · u 0 . zu 0 = Du 0 = 0 , su 0 = u 0 . In this case, v = exp( π √− 1 h ) = √− 1 H W = C [ T ] / ( T − √− 1) 2

RDAHAs dDAHAs Algebraic KZ Example : SL2 RDAHA : H rat = C � z, s, D � / ( sz = − zs, sD = − Ds, [ D, z ] = 1 − 2 hs ) . Iwahori–Hecke H W = C [ T ] / ( T − v )( T + v − 1 ) , v = exp( π √− 1 h ) . T = monodromy of half-turn around 0 ∈ C . Consider h = 1 / 2 . Then H rat has a one-dimensional module L triv = C · u 0 . zu 0 = Du 0 = 0 , su 0 = u 0 . In this case, v = exp( π √− 1 h ) = √− 1 H W = C [ T ] / ( T − √− 1) 2 ker V = � L triv � .

Recall (trigonometrically) degenerate DAHAs !

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n generated by subalgebras C [ x 1 , · · · , x n ] and C S n

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n generated by subalgebras C [ x 1 , · · · , x n ] and C S n modulo relations : [ s i , x j ] = 0 for j / ∈ { i, i + 1 } , s i x i − x i +1 s i = h and s i x i +1 − x i s i = − h .

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n M f.d. H gr − aff -module n

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n M f.d. H gr − aff -module n Affine Knizhnik–Zamolodchikov (AKZ) equations for M are the following PDEs for f with values in M : i − 1 � ∂ 1 − s i,k z i f + x i · f − h f ( ∗ ) ∂z i 1 − z k /z i k =1 n � 1 − s i,k f − h � ρ, ω ∨ − h i � = 0 , i ∈ [1 , n ] 1 − z i /z k k = i +1 h ∈ C : parameter

RDAHAs dDAHAs Algebraic KZ AKZ equations for H gr − aff n H gr − aff : graded affine Hecke algebra for GL n , n M f.d. H gr − aff -module n Affine Knizhnik–Zamolodchikov (AKZ) equations for M are the following PDEs for f with values in M : i − 1 � ∂ 1 − s i,k z i f + x i · f − h f ( ∗ ) ∂z i 1 − z k /z i k =1 n � 1 − s i,k f − h � ρ, ω ∨ − h i � = 0 , i ∈ [1 , n ] 1 − z i /z k k = i +1 h ∈ C : parameter Schur–Weyl ⇒ KZ equations for GL m on P 1 (WZW ( ∗ ) = = = = = = = conformal blocks).

RDAHAs dDAHAs Algebraic KZ AKZ equations for root systems ∆ ⊂ h ∗ P ⊂ h ∗ R reduced root system as above, R weight Q ∨ = Z ∆ ∨ dual root lattice, lattice, h = h R ⊗ R C

RDAHAs dDAHAs Algebraic KZ AKZ equations for root systems ∆ ⊂ h ∗ P ⊂ h ∗ R reduced root system as above, R weight Q ∨ = Z ∆ ∨ dual root lattice, lattice, h = h R ⊗ R C T ∨ = P ⊗ C × dual torus, C [ T ∨ ] = C [ z µ ; µ ∈ Q ∨ ]

RDAHAs dDAHAs Algebraic KZ AKZ equations for root systems ∆ ⊂ h ∗ P ⊂ h ∗ R reduced root system as above, R weight Q ∨ = Z ∆ ∨ dual root lattice, lattice, h = h R ⊗ R C T ∨ = P ⊗ C × dual torus, C [ T ∨ ] = C [ z µ ; µ ∈ Q ∨ ] H gr − aff : graded affine Hecke W : Weyl group of ∆ , M : f.d. H gr − aff -module algebra for ∆ ,

RDAHAs dDAHAs Algebraic KZ AKZ equations for root systems ∆ ⊂ h ∗ P ⊂ h ∗ R reduced root system as above, R weight Q ∨ = Z ∆ ∨ dual root lattice, lattice, h = h R ⊗ R C T ∨ = P ⊗ C × dual torus, C [ T ∨ ] = C [ z µ ; µ ∈ Q ∨ ] H gr − aff : graded affine Hecke W : Weyl group of ∆ , M : f.d. H gr − aff -module algebra for ∆ , AKZ equations Affine Knizhnik–Zamolodchikov (AKZ) equations are the following PDEs for analytic functions f : T ∨ → M � h α � ξ, α ∨ � 1 − s α 1 − z − α ∨ f − � ξ, ρ ∨ ∂ ξ ( f ) + ξ · f − h � f = 0 α ∈ ∆ + for ξ ∈ h ∗ . h = (1 / 2) � ρ ∨ α ∈ ∆ + h α α ∨ h α ∈ C : parameters,

RDAHAs dDAHAs Algebraic KZ dDAHAs : definition AKZ equations : � h α � ξ, α ∨ � 1 − s α 1 − z − α ∨ f − � ξ, ρ ∨ ∂ ξ ( f ) + ξ · f − h � f = 0 α ∈ ∆ +

RDAHAs dDAHAs Algebraic KZ dDAHAs : definition AKZ equations : � h α � ξ, α ∨ � 1 − s α 1 − z − α ∨ f − � ξ, ρ ∨ ∂ ξ ( f ) + ξ · f − h � f = 0 α ∈ ∆ + � � z ∈ T ∨ ; z α ∨ � = 1 , ∀ α ∈ ∆ Regular part : T ∨ ◦ =

RDAHAs dDAHAs Algebraic KZ dDAHAs : definition AKZ equations : � h α � ξ, α ∨ � 1 − s α 1 − z − α ∨ f − � ξ, ρ ∨ ∂ ξ ( f ) + ξ · f − h � f = 0 α ∈ ∆ + � � z ∈ T ∨ ; z α ∨ � = 1 , ∀ α ∈ ∆ Regular part : T ∨ ◦ = D ( T ∨ ◦ ) : ring of algebraic differential operators on the affine variety T ∨ ◦ , acted on by W

RDAHAs dDAHAs Algebraic KZ dDAHAs : definition AKZ equations : � h α � ξ, α ∨ � 1 − s α 1 − z − α ∨ f − � ξ, ρ ∨ ∂ ξ ( f ) + ξ · f − h � f = 0 α ∈ ∆ + � � z ∈ T ∨ ; z α ∨ � = 1 , ∀ α ∈ ∆ Regular part : T ∨ ◦ = D ( T ∨ ◦ ) : ring of algebraic differential operators on the affine variety T ∨ ◦ , acted on by W ◦ ) ⋊ C W for ξ ∈ h ∗ : Dunkl operator D ξ ∈ D ( T ∨ � h α � ξ, α ∨ � (1 − z − α ∨ ) − 1 (1 − s α ) + � ξ, ρ ∨ D ξ := ∂ ξ − h � α ∈ ∆ +

RDAHAs dDAHAs Algebraic KZ dDAHAs : definition AKZ equations : � h α � ξ, α ∨ � 1 − s α 1 − z − α ∨ f − � ξ, ρ ∨ ∂ ξ ( f ) + ξ · f − h � f = 0 α ∈ ∆ + � � z ∈ T ∨ ; z α ∨ � = 1 , ∀ α ∈ ∆ Regular part : T ∨ ◦ = D ( T ∨ ◦ ) : ring of algebraic differential operators on the affine variety T ∨ ◦ , acted on by W ◦ ) ⋊ C W for ξ ∈ h ∗ : Dunkl operator D ξ ∈ D ( T ∨ � h α � ξ, α ∨ � (1 − z − α ∨ ) − 1 (1 − s α ) + � ξ, ρ ∨ D ξ := ∂ ξ − h � α ∈ ∆ + The degenerate double affine Hecke algebra (dDAHA), is the subalgebra H trig ⊂ D ( T ∨ ◦ ) ⋊ C W generated by C [ T ∨ ] , C W and D ξ for ξ ∈ h ∗ .

RDAHAs dDAHAs Algebraic KZ dDAHAs : structure Dunkl operator : � h α � ξ, α ∨ � (1 − z − α ∨ ) − 1 (1 − s α )+ � ξ, ρ ∨ h � ∈ D ( T ∨ D ξ := ∂ ξ − ◦ ) ⋊ C W α ∈ ∆ +

RDAHAs dDAHAs Algebraic KZ dDAHAs : structure Dunkl operator : � h α � ξ, α ∨ � (1 − z − α ∨ ) − 1 (1 − s α )+ � ξ, ρ ∨ h � ∈ D ( T ∨ D ξ := ∂ ξ − ◦ ) ⋊ C W α ∈ ∆ + [ D ξ , D ξ ′ ] = 0 , ∀ ξ, ξ ′ ∈ h ∗ . They generate a subalgebra C [ h ] = Sym h ∗ ⊂ H trig .

RDAHAs dDAHAs Algebraic KZ dDAHAs : structure Dunkl operator : � h α � ξ, α ∨ � (1 − z − α ∨ ) − 1 (1 − s α )+ � ξ, ρ ∨ h � ∈ D ( T ∨ D ξ := ∂ ξ − ◦ ) ⋊ C W α ∈ ∆ + [ D ξ , D ξ ′ ] = 0 , ∀ ξ, ξ ′ ∈ h ∗ . They generate a subalgebra C [ h ] = Sym h ∗ ⊂ H trig . There is a subalgebra H gr − aff ∼ = C W ⊗ C [ h ] ⊂ H trig

RDAHAs dDAHAs Algebraic KZ dDAHAs : structure Dunkl operator : � h α � ξ, α ∨ � (1 − z − α ∨ ) − 1 (1 − s α )+ � ξ, ρ ∨ h � ∈ D ( T ∨ D ξ := ∂ ξ − ◦ ) ⋊ C W α ∈ ∆ + [ D ξ , D ξ ′ ] = 0 , ∀ ξ, ξ ′ ∈ h ∗ . They generate a subalgebra C [ h ] = Sym h ∗ ⊂ H trig . There is a subalgebra H gr − aff ∼ = C W ⊗ C [ h ] ⊂ H trig Triangular decomposition H trig = C [ T ∨ ] ⊗ C W ⊗ C [ h ]

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦ Via ( ∗ ), M ◦ is W -equivariant coherent D ( T ∨ ◦ ) -module

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦ Via ( ∗ ), M ◦ is W -equivariant coherent D ( T ∨ ◦ ) -module Suppose D ξ acts locally finitely on M for ξ ∈ h ∗ .

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦ Via ( ∗ ), M ◦ is W -equivariant coherent D ( T ∨ ◦ ) -module Suppose D ξ acts locally finitely on M for ξ ∈ h ∗ . M ◦ is a W -equivariant local system on T ∨ ◦

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦ Via ( ∗ ), M ◦ is W -equivariant coherent D ( T ∨ ◦ ) -module Suppose D ξ acts locally finitely on M for ξ ∈ h ∗ . M ◦ is a W -equivariant local system on T ∨ ◦ flat sections of M ◦ form a π 1 ([ T ∨ ◦ /W ]) -module, denoted M ∇

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦ Via ( ∗ ), M ◦ is W -equivariant coherent D ( T ∨ ◦ ) -module Suppose D ξ acts locally finitely on M for ξ ∈ h ∗ . M ◦ is a W -equivariant local system on T ∨ ◦ flat sections of M ◦ form a π 1 ([ T ∨ ◦ /W ]) -module, denoted M ∇ ◦ /W ]) ∼ = ˜ orbifold fundamental group π 1 ([ T ∨ B W , (extended) affine braid group

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition Isomorphism ◦ ] ⊗ C [ T ∨ ] H trig ∼ C [ T ∨ = D ( T ∨ ◦ ) ⋊ C W =: H trig ( ∗ ) ◦ Given M : coherent H trig -module M ◦ := H trig ⊗ H trig M : coherent H trig -module ◦ ◦ Via ( ∗ ), M ◦ is W -equivariant coherent D ( T ∨ ◦ ) -module Suppose D ξ acts locally finitely on M for ξ ∈ h ∗ . M ◦ is a W -equivariant local system on T ∨ ◦ flat sections of M ◦ form a π 1 ([ T ∨ ◦ /W ]) -module, denoted M ∇ ◦ /W ]) ∼ = ˜ orbifold fundamental group π 1 ([ T ∨ B W , (extended) affine braid group B W -action on M ∇ factorises through H aff , (extended) ˜ affine Hecke algebra for ∆

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition (cont.) O ( H trig ) : category of coherent H trig -modules on which D ξ acts locally finitely for ξ ∈ h ∗

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition (cont.) O ( H trig ) : category of coherent H trig -modules on which D ξ acts locally finitely for ξ ∈ h ∗ O ( H aff ) : category of f.d. H aff -modules

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition (cont.) O ( H trig ) : category of coherent H trig -modules on which D ξ acts locally finitely for ξ ∈ h ∗ O ( H aff ) : category of f.d. H aff -modules KZ functor [Varagnolo–Vasserot ’04] The KZ functor V : O ( H trig ) → O ( H aff ) is defined by the assignement O ( H trig ) ∋ M �→ M ∇ ∈ O ( H aff ) .

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition (cont.) O ( H trig ) : category of coherent H trig -modules on which D ξ acts locally finitely for ξ ∈ h ∗ O ( H aff ) : category of f.d. H aff -modules KZ functor [Varagnolo–Vasserot ’04] The KZ functor V : O ( H trig ) → O ( H aff ) is defined by the assignement O ( H trig ) ∋ M �→ M ∇ ∈ O ( H aff ) . Parameters are given by v α = exp( π √− 1 h α )

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : definition (cont.) O ( H trig ) : category of coherent H trig -modules on which D ξ acts locally finitely for ξ ∈ h ∗ O ( H aff ) : category of f.d. H aff -modules KZ functor [Varagnolo–Vasserot ’04] The KZ functor V : O ( H trig ) → O ( H aff ) is defined by the assignement O ( H trig ) ∋ M �→ M ∇ ∈ O ( H aff ) . Parameters are given by v α = exp( π √− 1 h α ) This also works for non-reduced root systems

RDAHA v.s. dDAHA

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig ∂ ξ ( f ) − � α h α � ξ, α ∨ � 1 − s α KZ eq α ∨ f = 0 ∂ ξ ( f ) + ξ · f − � 1 − s α α h α � ξ, α ∨ � 1 − z − α ∨ f + � ξ, ρ ∨ h � f = 0

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig ∂ ξ ( f ) − � α h α � ξ, α ∨ � 1 − s α KZ eq α ∨ f = 0 ∂ ξ ( f ) + ξ · f − � 1 − s α α h α � ξ, α ∨ � 1 − z − α ∨ f + � ξ, ρ ∨ h � f = 0 f.d. H gr − aff -mod coeffs f.d. C W -mod

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig ∂ ξ ( f ) − � α h α � ξ, α ∨ � 1 − s α KZ eq α ∨ f = 0 ∂ ξ ( f ) + ξ · f − � 1 − s α α h α � ξ, α ∨ � 1 − z − α ∨ f + � ξ, ρ ∨ h � f = 0 f.d. H gr − aff -mod coeffs f.d. C W -mod h ∗ [ T ∨ over ◦ /W ◦ /W ]

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig ∂ ξ ( f ) − � α h α � ξ, α ∨ � 1 − s α KZ eq α ∨ f = 0 ∂ ξ ( f ) + ξ · f − � 1 − s α α h α � ξ, α ∨ � 1 − z − α ∨ f + � ξ, ρ ∨ h � f = 0 f.d. H gr − aff -mod coeffs f.d. C W -mod h ∗ [ T ∨ over ◦ /W ◦ /W ] f.d. H aff -mod mndrmy f.d. H W -mod

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig ∂ ξ ( f ) − � α h α � ξ, α ∨ � 1 − s α KZ eq α ∨ f = 0 ∂ ξ ( f ) + ξ · f − � 1 − s α α h α � ξ, α ∨ � 1 − z − α ∨ f + � ξ, ρ ∨ h � f = 0 f.d. H gr − aff -mod coeffs f.d. C W -mod h ∗ [ T ∨ over ◦ /W ◦ /W ] f.d. H aff -mod mndrmy f.d. H W -mod h ∗ -loc. nilp. h ∗ -loc. fin. O

RDAHAs dDAHAs Algebraic KZ Recapitulation H rat H trig ∂ ξ ( f ) − � α h α � ξ, α ∨ � 1 − s α KZ eq α ∨ f = 0 ∂ ξ ( f ) + ξ · f − � 1 − s α α h α � ξ, α ∨ � 1 − z − α ∨ f + � ξ, ρ ∨ h � f = 0 f.d. H gr − aff -mod coeffs f.d. C W -mod h ∗ [ T ∨ over ◦ /W ◦ /W ] f.d. H aff -mod mndrmy f.d. H W -mod h ∗ -loc. nilp. h ∗ -loc. fin. O O ( H rat ) → H W -mod O ( H trig ) → O ( H aff ) V :

Coming back to dDAHAs...

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : properties KZ functor V : O ( H trig ) → O ( H aff ) V ( M ) = M ∇

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : properties KZ functor V : O ( H trig ) → O ( H aff ) V ( M ) = M ∇ Theorem [L.] 1 V is a quotient functor of abelian categories, inducing equivalence O ( H trig ) / ker V ∼ = O ( H aff )

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : properties KZ functor V : O ( H trig ) → O ( H aff ) V ( M ) = M ∇ Theorem [L.] 1 V is a quotient functor of abelian categories, inducing equivalence O ( H trig ) / ker V ∼ = O ( H aff ) 2 V satisfies the double centraliser property, i.e. V is fully faithful on projective objects of O ( H trig )

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : properties KZ functor V : O ( H trig ) → O ( H aff ) V ( M ) = M ∇ Theorem [L.] 1 V is a quotient functor of abelian categories, inducing equivalence O ( H trig ) / ker V ∼ = O ( H aff ) 2 V satisfies the double centraliser property, i.e. V is fully faithful on projective objects of the completion of O ( H trig )

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : properties KZ functor V : O ( H trig ) → O ( H aff ) V ( M ) = M ∇ Theorem [L.] 1 V is a quotient functor of abelian categories, inducing equivalence O ( H trig ) / ker V ∼ = O ( H aff ) 2 V satisfies the double centraliser property, i.e. V is fully faithful on projective objects of the completion of O ( H trig ) 3 L ∈ O ( H trig ) : simple module, P L ∈ O ( H trig ) : projective cover of L . Then L ∈ ker V ⇔ P L is not injective

RDAHAs dDAHAs Algebraic KZ KZ functor for dDAHAs : properties KZ functor V : O ( H trig ) → O ( H aff ) V ( M ) = M ∇ Theorem [L.] 1 V is a quotient functor of abelian categories, inducing equivalence O ( H trig ) / ker V ∼ = O ( H aff ) 2 V satisfies the double centraliser property, i.e. V is fully faithful on projective objects of the completion of O ( H trig ) 3 L ∈ O ( H trig ) : simple module, P L ∈ Pro O ( H trig ) : projective cover of L . Then L ∈ ker V ⇔ P L is not relatively injective / categorical centre

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 ,

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 , Q ∨ = Z · (2 ϵ ) , T ∨ = C [ z ± 2 ϵ ] . P = Z · ( α/ 2) ,

RDAHAs dDAHAs Algebraic KZ Example : SL2 W = � s ; s 2 = e � . h R = R · ϵ , ∆ = {± α } , � α, ϵ � = 1 , Q ∨ = Z · (2 ϵ ) , T ∨ = C [ z ± 2 ϵ ] . P = Z · ( α/ 2) , Let y = z 2 ϵ . AKZ : y d dyf + α · f − 2 h 1 − s 1 − y − 1 f − h · f = 0