KnizhnikZamolodchikov functor for degenerate double affine Hecke - - PowerPoint PPT Presentation
RDAHAs dDAHAs Algebraic KZ KnizhnikZamolodchikov functor for degenerate double affine Hecke algebras Wille Liu Universit de Paris 30 June 2020 RDAHAs dDAHAs Algebraic KZ Plan 1 RDAHAs KZ equations RDAHAs KZ functor 2 dDAHAs AKZ
RDAHAs dDAHAs Algebraic KZ
Wille Liu
Université de Paris
30 June 2020
RDAHAs dDAHAs Algebraic KZ
1 RDAHAs
KZ equations RDAHAs KZ functor
2 dDAHAs
AKZ equations dDAHAs KZ functor Issues
3 Algebraic KZ
Another presentation Blocks of O and block algebras Block algebra Generalisation
RDAHAs dDAHAs Algebraic KZ
Sn : symmetric group on n elements, M f.d. CSn-module
RDAHAs dDAHAs Algebraic KZ
Sn : symmetric group on n elements, M f.d. CSn-module Knizhnik–Zamolodchikov (KZ) equations for M are the following PDEs for f with values in M : ∂ ∂zi f − h
i−1
1 − si,k zi − zk f + h
n
1 − si,k zi − zk f = 0, i ∈ [1, n] (∗) h ∈ C : parameter
RDAHAs dDAHAs Algebraic KZ
Sn : symmetric group on n elements, M f.d. CSn-module Knizhnik–Zamolodchikov (KZ) equations for M are the following PDEs for f with values in M : ∂ ∂zi f − h
i−1
1 − si,k zi − zk f + h
n
1 − si,k zi − zk f = 0, i ∈ [1, n] (∗) h ∈ C : parameter (∗)
Schur–Weyl
= = = = = = = ⇒ KZ equations for GLm on P1 (WZW conformal blocks).
RDAHAs dDAHAs Algebraic KZ
hR : euclidean vector space, ∆ ⊂ h∗
R reduced root system,
Π ⊂ ∆ base, ∆+ ⊂ ∆ positive roots, h = hR ⊗R C
RDAHAs dDAHAs Algebraic KZ
hR : euclidean vector space, ∆ ⊂ h∗
R reduced root system,
Π ⊂ ∆ base, ∆+ ⊂ ∆ positive roots, h = hR ⊗R C W : Weyl group of ∆, M : f.d. CW-module
RDAHAs dDAHAs Algebraic KZ
hR : euclidean vector space, ∆ ⊂ h∗
R reduced root system,
Π ⊂ ∆ base, ∆+ ⊂ ∆ positive roots, h = hR ⊗R C W : Weyl group of ∆, M : f.d. CW-module KZ equations Knizhnik–Zamolodchikov (KZ) equations are the following PDEs for analytic functions f : h∗ → M ∂ξ(f) −
hαξ, α∨1 − sα α∨ f = 0, ξ ∈ h∗ (∗∗) sα ∈ W : reflection w.r.t. α, α∨ ∈ h : dual root of α, hα ∈ C : parameters
RDAHAs dDAHAs Algebraic KZ
KZ equations : ∂ξ(f) −
hαξ, α∨1 − sα α∨ f = 0, ξ ∈ h∗
RDAHAs dDAHAs Algebraic KZ
KZ equations : ∂ξ(f) −
hαξ, α∨1 − sα α∨ f = 0, ξ ∈ h∗ Regular part : h∗
RDAHAs dDAHAs Algebraic KZ
KZ equations : ∂ξ(f) −
hαξ, α∨1 − sα α∨ f = 0, ξ ∈ h∗ Regular part : h∗
D(h∗
variety h∗
RDAHAs dDAHAs Algebraic KZ
KZ equations : ∂ξ(f) −
hαξ, α∨1 − sα α∨ f = 0, ξ ∈ h∗ Regular part : h∗
D(h∗
variety h∗
Dunkl operator Dξ ∈ D(h∗
Dξ := ∂ξ −
hαξ, α∨(α∨)−1(1 − sα) The rational double affine Hecke algebra (RDAHA), is the subalgebra Hrat ⊂ D(h∗
and Dξ for ξ ∈ h∗.
RDAHAs dDAHAs Algebraic KZ
Dunkl operators : Dξ := ∂ξ −
hαξ, α∨(α∨)−1(1 − sα) ∈ D(h∗
RDAHAs dDAHAs Algebraic KZ
Dunkl operators : Dξ := ∂ξ −
hαξ, α∨(α∨)−1(1 − sα) ∈ D(h∗
[Dξ, Dξ′] = 0, ∀ξ, ξ′ ∈ h∗. They generate a subalgebra C[h] = Sym h∗ ⊂ Hrat.
RDAHAs dDAHAs Algebraic KZ
Dunkl operators : Dξ := ∂ξ −
hαξ, α∨(α∨)−1(1 − sα) ∈ D(h∗
[Dξ, Dξ′] = 0, ∀ξ, ξ′ ∈ h∗. They generate a subalgebra C[h] = Sym h∗ ⊂ Hrat. Triangular decomposition Hrat = C[h∗] ⊗ CW ⊗ C[h]
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
Via (∗), M◦ is W-equivariant coherent D(h∗
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
Via (∗), M◦ is W-equivariant coherent D(h∗
Suppose Dξ acts locally nilpotently on M for ξ ∈ h∗.
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
Via (∗), M◦ is W-equivariant coherent D(h∗
Suppose Dξ acts locally nilpotently on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on h∗
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
Via (∗), M◦ is W-equivariant coherent D(h∗
Suppose Dξ acts locally nilpotently on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on h∗
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
Via (∗), M◦ is W-equivariant coherent D(h∗
Suppose Dξ acts locally nilpotently on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on h∗
π1 (h∗
= BW , braid group of W
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[h∗
= D(h∗
Given M : coherent Hrat-module M◦ := Hrat
Via (∗), M◦ is W-equivariant coherent D(h∗
Suppose Dξ acts locally nilpotently on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on h∗
π1 (h∗
= BW , braid group of W BW -action on M ∇ factorises through HW , Iwahori–Hecke algebra of W
RDAHAs dDAHAs Algebraic KZ
O(Hrat) : category of coherent Hrat-modules on which Dξ acts locally nilpotently for ξ ∈ h∗
RDAHAs dDAHAs Algebraic KZ
O(Hrat) : category of coherent Hrat-modules on which Dξ acts locally nilpotently for ξ ∈ h∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O(Hrat) → HW -mod is defined by the assignement O(Hrat) ∋ M → M∇ ∈ HW -mod .
RDAHAs dDAHAs Algebraic KZ
O(Hrat) : category of coherent Hrat-modules on which Dξ acts locally nilpotently for ξ ∈ h∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O(Hrat) → HW -mod is defined by the assignement O(Hrat) ∋ M → M∇ ∈ HW -mod . Recall the Iwahori–Hecke algebra HW is
generated by Tα for simple roots α ∈ Π
RDAHAs dDAHAs Algebraic KZ
O(Hrat) : category of coherent Hrat-modules on which Dξ acts locally nilpotently for ξ ∈ h∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O(Hrat) → HW -mod is defined by the assignement O(Hrat) ∋ M → M∇ ∈ HW -mod . Recall the Iwahori–Hecke algebra HW is
generated by Tα for simple roots α ∈ Π modulo the braid relations TαTβTα · · · = TβTαTβ · · ·
RDAHAs dDAHAs Algebraic KZ
O(Hrat) : category of coherent Hrat-modules on which Dξ acts locally nilpotently for ξ ∈ h∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O(Hrat) → HW -mod is defined by the assignement O(Hrat) ∋ M → M∇ ∈ HW -mod . Recall the Iwahori–Hecke algebra HW is
generated by Tα for simple roots α ∈ Π modulo the braid relations TαTβTα · · · = TβTαTβ · · · and the quadratic relations (Tα − vα)(Tα + v−1
α ) = 0
RDAHAs dDAHAs Algebraic KZ
O(Hrat) : category of coherent Hrat-modules on which Dξ acts locally nilpotently for ξ ∈ h∗ KZ functor [Guay–Ginzburg–Opdam–Rouquier ’03] The KZ functor V : O(Hrat) → HW -mod is defined by the assignement O(Hrat) ∋ M → M∇ ∈ HW -mod . Recall the Iwahori–Hecke algebra HW is
generated by Tα for simple roots α ∈ Π modulo the braid relations TαTβTα · · · = TβTαTβ · · · and the quadratic relations (Tα − vα)(Tα + v−1
α ) = 0
Parameters are given by vα = exp(π√−1hα)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Hrat) → HW -mod V(M) = M∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03]
1 O(Hrat) is a highest weight category with index set
Irrep(W)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Hrat) → HW -mod V(M) = M∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03]
1 O(Hrat) is a highest weight category with index set
Irrep(W)
2 V is a quotient functor of abelian categories, inducing
equivalence O(Hrat)/ ker V ∼ = HW -mod
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Hrat) → HW -mod V(M) = M∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03]
1 O(Hrat) is a highest weight category with index set
Irrep(W)
2 V is a quotient functor of abelian categories, inducing
equivalence O(Hrat)/ ker V ∼ = HW -mod
3 V satisfies the double centraliser property, i.e. V is fully
faithful on projective objects of O(Hrat)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Hrat) → HW -mod V(M) = M∇ Theorem [Guay–Ginzburg–Opdam–Rouquier ’03]
1 O(Hrat) is a highest weight category with index set
Irrep(W)
2 V is a quotient functor of abelian categories, inducing
equivalence O(Hrat)/ ker V ∼ = HW -mod
3 V satisfies the double centraliser property, i.e. V is fully
faithful on projective objects of O(Hrat)
4 L ∈ O(Hrat) : simple module,
PL ∈ O(Hrat) : projective cover of L. Then L ∈ ker V ⇔ PL is not injective
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e.
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. KZ : d dz f − h1 − s z f = 0
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. KZ : d dz f − h1 − s z f = 0 Dunkl : D = d
dz − hz−1(1 − s)
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. KZ : d dz f − h1 − s z f = 0 Dunkl : D = d
dz − hz−1(1 − s)
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs).
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. KZ : d dz f − h1 − s z f = 0 Dunkl : D = d
dz − hz−1(1 − s)
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C.
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module Ltriv = C · u0.
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module Ltriv = C · u0. zu0 = Du0 = 0, su0 = u0.
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module Ltriv = C · u0. zu0 = Du0 = 0, su0 = u0. In this case,
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module Ltriv = C · u0. zu0 = Du0 = 0, su0 = u0. In this case, v = exp(π√−1h) = √−1
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module Ltriv = C · u0. zu0 = Du0 = 0, su0 = u0. In this case, v = exp(π√−1h) = √−1 HW = C[T]/(T − √−1)2
RDAHAs dDAHAs Algebraic KZ
RDAHA : Hrat = Cz, s, D / (sz = −zs, sD = −Ds, [D, z] = 1 − 2hs). Iwahori–Hecke HW = C[T]/(T − v)(T + v−1), v = exp(π√−1h). T = monodromy of half-turn around 0 ∈ C. Consider h = 1/2. Then Hrat has a one-dimensional module Ltriv = C · u0. zu0 = Du0 = 0, su0 = u0. In this case, v = exp(π√−1h) = √−1 HW = C[T]/(T − √−1)2 ker V = Ltriv.
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn,
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn,
generated by subalgebras C[x1, · · · , xn] and CSn
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn,
generated by subalgebras C[x1, · · · , xn] and CSn modulo relations : [si, xj] = 0 for j / ∈ {i, i + 1}, sixi − xi+1si = h and sixi+1 − xisi = −h.
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn,
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn, M f.d. Hgr−aff
n
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn, M f.d. Hgr−aff
n
Affine Knizhnik–Zamolodchikov (AKZ) equations for M are the following PDEs for f with values in M : zi ∂ ∂zi f + xi · f − h
i−1
1 − si,k 1 − zk/zi f (∗) − h
n
1 − si,k 1 − zi/zk f − hρ, ω∨
i = 0,
i ∈ [1, n] h ∈ C : parameter
RDAHAs dDAHAs Algebraic KZ
n
Hgr−aff
n
: graded affine Hecke algebra for GLn, M f.d. Hgr−aff
n
Affine Knizhnik–Zamolodchikov (AKZ) equations for M are the following PDEs for f with values in M : zi ∂ ∂zi f + xi · f − h
i−1
1 − si,k 1 − zk/zi f (∗) − h
n
1 − si,k 1 − zi/zk f − hρ, ω∨
i = 0,
i ∈ [1, n] h ∈ C : parameter (∗)
Schur–Weyl
= = = = = = = ⇒ KZ equations for GLm on P1 (WZW conformal blocks).
RDAHAs dDAHAs Algebraic KZ
∆ ⊂ h∗
R reduced root system as above,
P ⊂ h∗
R weight
lattice, Q∨ = Z∆∨ dual root lattice, h = hR ⊗R C
RDAHAs dDAHAs Algebraic KZ
∆ ⊂ h∗
R reduced root system as above,
P ⊂ h∗
R weight
lattice, Q∨ = Z∆∨ dual root lattice, h = hR ⊗R C T ∨ = P ⊗ C× dual torus, C[T ∨] = C[zµ ; µ ∈ Q∨]
RDAHAs dDAHAs Algebraic KZ
∆ ⊂ h∗
R reduced root system as above,
P ⊂ h∗
R weight
lattice, Q∨ = Z∆∨ dual root lattice, h = hR ⊗R C T ∨ = P ⊗ C× dual torus, C[T ∨] = C[zµ ; µ ∈ Q∨] W : Weyl group of ∆, Hgr−aff : graded affine Hecke algebra for ∆, M : f.d. Hgr−aff-module
RDAHAs dDAHAs Algebraic KZ
∆ ⊂ h∗
R reduced root system as above,
P ⊂ h∗
R weight
lattice, Q∨ = Z∆∨ dual root lattice, h = hR ⊗R C T ∨ = P ⊗ C× dual torus, C[T ∨] = C[zµ ; µ ∈ Q∨] W : Weyl group of ∆, Hgr−aff : graded affine Hecke algebra for ∆, M : f.d. Hgr−aff-module AKZ equations Affine Knizhnik–Zamolodchikov (AKZ) equations are the following PDEs for analytic functions f : T ∨ → M ∂ξ(f) + ξ · f −
hαξ, α∨ 1 − sα 1 − z−α∨ f − ξ, ρ∨
h f = 0
for ξ ∈ h∗. hα ∈ C : parameters, ρ∨
h = (1/2) α∈∆+ hαα∨
RDAHAs dDAHAs Algebraic KZ
AKZ equations : ∂ξ(f) + ξ · f −
hαξ, α∨ 1 − sα 1 − z−α∨ f − ξ, ρ∨
h f = 0
RDAHAs dDAHAs Algebraic KZ
AKZ equations : ∂ξ(f) + ξ · f −
hαξ, α∨ 1 − sα 1 − z−α∨ f − ξ, ρ∨
h f = 0
Regular part : T ∨
RDAHAs dDAHAs Algebraic KZ
AKZ equations : ∂ξ(f) + ξ · f −
hαξ, α∨ 1 − sα 1 − z−α∨ f − ξ, ρ∨
h f = 0
Regular part : T ∨
variety T ∨
RDAHAs dDAHAs Algebraic KZ
AKZ equations : ∂ξ(f) + ξ · f −
hαξ, α∨ 1 − sα 1 − z−α∨ f − ξ, ρ∨
h f = 0
Regular part : T ∨
variety T ∨
Dunkl operator Dξ ∈ D(T ∨
Dξ := ∂ξ −
hαξ, α∨(1 − z−α∨)−1(1 − sα) + ξ, ρ∨
h
RDAHAs dDAHAs Algebraic KZ
AKZ equations : ∂ξ(f) + ξ · f −
hαξ, α∨ 1 − sα 1 − z−α∨ f − ξ, ρ∨
h f = 0
Regular part : T ∨
variety T ∨
Dunkl operator Dξ ∈ D(T ∨
Dξ := ∂ξ −
hαξ, α∨(1 − z−α∨)−1(1 − sα) + ξ, ρ∨
h
The degenerate double affine Hecke algebra (dDAHA), is the subalgebra Htrig ⊂ D(T ∨
C[T ∨],CW and Dξ for ξ ∈ h∗.
RDAHAs dDAHAs Algebraic KZ
Dunkl operator : Dξ := ∂ξ−
hαξ, α∨(1−z−α∨)−1(1−sα)+ξ, ρ∨
h ∈ D(T ∨
RDAHAs dDAHAs Algebraic KZ
Dunkl operator : Dξ := ∂ξ−
hαξ, α∨(1−z−α∨)−1(1−sα)+ξ, ρ∨
h ∈ D(T ∨
[Dξ, Dξ′] = 0, ∀ξ, ξ′ ∈ h∗. They generate a subalgebra C[h] = Sym h∗ ⊂ Htrig.
RDAHAs dDAHAs Algebraic KZ
Dunkl operator : Dξ := ∂ξ−
hαξ, α∨(1−z−α∨)−1(1−sα)+ξ, ρ∨
h ∈ D(T ∨
[Dξ, Dξ′] = 0, ∀ξ, ξ′ ∈ h∗. They generate a subalgebra C[h] = Sym h∗ ⊂ Htrig. There is a subalgebra Hgr−aff ∼ = CW ⊗ C[h] ⊂ Htrig
RDAHAs dDAHAs Algebraic KZ
Dunkl operator : Dξ := ∂ξ−
hαξ, α∨(1−z−α∨)−1(1−sα)+ξ, ρ∨
h ∈ D(T ∨
[Dξ, Dξ′] = 0, ∀ξ, ξ′ ∈ h∗. They generate a subalgebra C[h] = Sym h∗ ⊂ Htrig. There is a subalgebra Hgr−aff ∼ = CW ⊗ C[h] ⊂ Htrig Triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h]
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
Via (∗), M◦ is W-equivariant coherent D(T ∨
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
Via (∗), M◦ is W-equivariant coherent D(T ∨
Suppose Dξ acts locally finitely on M for ξ ∈ h∗.
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
Via (∗), M◦ is W-equivariant coherent D(T ∨
Suppose Dξ acts locally finitely on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on T ∨
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
Via (∗), M◦ is W-equivariant coherent D(T ∨
Suppose Dξ acts locally finitely on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on T ∨
M ∇
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
Via (∗), M◦ is W-equivariant coherent D(T ∨
Suppose Dξ acts locally finitely on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on T ∨
M ∇
= ˜ BW , (extended) affine braid group
RDAHAs dDAHAs Algebraic KZ
Isomorphism C[T ∨
= D(T ∨
Given M : coherent Htrig-module M◦ := Htrig
Via (∗), M◦ is W-equivariant coherent D(T ∨
Suppose Dξ acts locally finitely on M for ξ ∈ h∗.
M◦ is a W-equivariant local system on T ∨
M ∇
= ˜ BW , (extended) affine braid group ˜ BW -action on M ∇ factorises through Haff, (extended) affine Hecke algebra for ∆
RDAHAs dDAHAs Algebraic KZ
O(Htrig) : category of coherent Htrig-modules on which Dξ acts locally finitely for ξ ∈ h∗
RDAHAs dDAHAs Algebraic KZ
O(Htrig) : category of coherent Htrig-modules on which Dξ acts locally finitely for ξ ∈ h∗ O(Haff) : category of f.d. Haff-modules
RDAHAs dDAHAs Algebraic KZ
O(Htrig) : category of coherent Htrig-modules on which Dξ acts locally finitely for ξ ∈ h∗ O(Haff) : category of f.d. Haff-modules KZ functor [Varagnolo–Vasserot ’04] The KZ functor V : O(Htrig) → O(Haff) is defined by the assignement O(Htrig) ∋ M → M∇ ∈ O(Haff).
RDAHAs dDAHAs Algebraic KZ
O(Htrig) : category of coherent Htrig-modules on which Dξ acts locally finitely for ξ ∈ h∗ O(Haff) : category of f.d. Haff-modules KZ functor [Varagnolo–Vasserot ’04] The KZ functor V : O(Htrig) → O(Haff) is defined by the assignement O(Htrig) ∋ M → M∇ ∈ O(Haff). Parameters are given by vα = exp(π√−1hα)
RDAHAs dDAHAs Algebraic KZ
O(Htrig) : category of coherent Htrig-modules on which Dξ acts locally finitely for ξ ∈ h∗ O(Haff) : category of f.d. Haff-modules KZ functor [Varagnolo–Vasserot ’04] The KZ functor V : O(Htrig) → O(Haff) is defined by the assignement O(Htrig) ∋ M → M∇ ∈ O(Haff). Parameters are given by vα = exp(π√−1hα) This also works for non-reduced root systems
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig KZ eq ∂ξ(f) −
α hαξ, α∨1−sα α∨ f = 0
∂ξ(f) + ξ · f −
α hαξ, α∨ 1−sα 1−z−α∨ f
+ξ, ρ∨
h f = 0
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig KZ eq ∂ξ(f) −
α hαξ, α∨1−sα α∨ f = 0
∂ξ(f) + ξ · f −
α hαξ, α∨ 1−sα 1−z−α∨ f
+ξ, ρ∨
h f = 0
coeffs f.d. CW-mod f.d. Hgr−aff-mod
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig KZ eq ∂ξ(f) −
α hαξ, α∨1−sα α∨ f = 0
∂ξ(f) + ξ · f −
α hαξ, α∨ 1−sα 1−z−α∨ f
+ξ, ρ∨
h f = 0
coeffs f.d. CW-mod f.d. Hgr−aff-mod
h∗
[T ∨
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig KZ eq ∂ξ(f) −
α hαξ, α∨1−sα α∨ f = 0
∂ξ(f) + ξ · f −
α hαξ, α∨ 1−sα 1−z−α∨ f
+ξ, ρ∨
h f = 0
coeffs f.d. CW-mod f.d. Hgr−aff-mod
h∗
[T ∨
mndrmy f.d. HW -mod f.d. Haff-mod
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig KZ eq ∂ξ(f) −
α hαξ, α∨1−sα α∨ f = 0
∂ξ(f) + ξ · f −
α hαξ, α∨ 1−sα 1−z−α∨ f
+ξ, ρ∨
h f = 0
coeffs f.d. CW-mod f.d. Hgr−aff-mod
h∗
[T ∨
mndrmy f.d. HW -mod f.d. Haff-mod O h∗-loc. nilp. h∗-loc. fin.
RDAHAs dDAHAs Algebraic KZ
Hrat Htrig KZ eq ∂ξ(f) −
α hαξ, α∨1−sα α∨ f = 0
∂ξ(f) + ξ · f −
α hαξ, α∨ 1−sα 1−z−α∨ f
+ξ, ρ∨
h f = 0
coeffs f.d. CW-mod f.d. Hgr−aff-mod
h∗
[T ∨
mndrmy f.d. HW -mod f.d. Haff-mod O h∗-loc. nilp. h∗-loc. fin. V : O(Hrat) → HW -mod O(Htrig) → O(Haff)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Htrig) → O(Haff) V(M) = M∇
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Htrig) → O(Haff) V(M) = M∇ Theorem [L.]
1 V is a quotient functor of abelian categories, inducing
equivalence O(Htrig)/ ker V ∼ = O(Haff)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Htrig) → O(Haff) V(M) = M∇ Theorem [L.]
1 V is a quotient functor of abelian categories, inducing
equivalence O(Htrig)/ ker V ∼ = O(Haff)
2 V satisfies the double centraliser property, i.e. V is fully
faithful on projective objects of O(Htrig)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Htrig) → O(Haff) V(M) = M∇ Theorem [L.]
1 V is a quotient functor of abelian categories, inducing
equivalence O(Htrig)/ ker V ∼ = O(Haff)
2 V satisfies the double centraliser property, i.e. V is fully
faithful on projective objects of the completion of O(Htrig)
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Htrig) → O(Haff) V(M) = M∇ Theorem [L.]
1 V is a quotient functor of abelian categories, inducing
equivalence O(Htrig)/ ker V ∼ = O(Haff)
2 V satisfies the double centraliser property, i.e. V is fully
faithful on projective objects of the completion of O(Htrig)
3 L ∈ O(Htrig) : simple module,
PL ∈ O(Htrig) : projective cover of L. Then L ∈ ker V ⇔ PL is not injective
RDAHAs dDAHAs Algebraic KZ
KZ functor V : O(Htrig) → O(Haff) V(M) = M∇ Theorem [L.]
1 V is a quotient functor of abelian categories, inducing
equivalence O(Htrig)/ ker V ∼ = O(Haff)
2 V satisfies the double centraliser property, i.e. V is fully
faithful on projective objects of the completion of O(Htrig)
3 L ∈ O(Htrig) : simple module,
PL ∈ Pro O(Htrig) : projective cover of L. Then L ∈ ker V ⇔ PL is not relatively injective / categorical centre
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e.
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. P = Z · (α/2), Q∨ = Z · (2ϵ), T ∨ = C[z±2ϵ].
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. P = Z · (α/2), Q∨ = Z · (2ϵ), T ∨ = C[z±2ϵ]. Let y = z2ϵ. AKZ : y d dyf + α · f − 2h 1 − s 1 − y−1 f − h · f = 0
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. P = Z · (α/2), Q∨ = Z · (2ϵ), T ∨ = C[z±2ϵ]. Let y = z2ϵ. AKZ : y d dyf + α · f − 2h 1 − s 1 − y−1 f − h · f = 0 Dunkl : D = y d
dy − h(1 − y−1)−1(1 − s) + h/2
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. P = Z · (α/2), Q∨ = Z · (2ϵ), T ∨ = C[z±2ϵ]. Let y = z2ϵ. AKZ : y d dyf + α · f − 2h 1 − s 1 − y−1 f − h · f = 0 Dunkl : D = y d
dy − h(1 − y−1)−1(1 − s) + h/2
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)).
RDAHAs dDAHAs Algebraic KZ
hR = R · ϵ, ∆ = {±α}, α, ϵ = 1, W = s ; s2 = e. P = Z · (α/2), Q∨ = Z · (2ϵ), T ∨ = C[z±2ϵ]. Let y = z2ϵ. AKZ : y d dyf + α · f − 2h 1 − s 1 − y−1 f − h · f = 0 Dunkl : D = y d
dy − h(1 − y−1)−1(1 − s) + h/2
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0.
RDAHAs dDAHAs Algebraic KZ
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0. Consider h = 1/2. Then Htrig has a one-dimensional module Ltriv = C · u0.
RDAHAs dDAHAs Algebraic KZ
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0. Consider h = 1/2. Then Htrig has a one-dimensional module Ltriv = C · u0. yu0 = u0, Du0 = u0/4, su0 = u0.
RDAHAs dDAHAs Algebraic KZ
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0. Consider h = 1/2. Then Htrig has a one-dimensional module Ltriv = C · u0. yu0 = u0, Du0 = u0/4, su0 = u0. In this case,
RDAHAs dDAHAs Algebraic KZ
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0. Consider h = 1/2. Then Htrig has a one-dimensional module Ltriv = C · u0. yu0 = u0, Du0 = u0/4, su0 = u0. In this case, v = exp(π√−1h) = √−1
RDAHAs dDAHAs Algebraic KZ
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0. Consider h = 1/2. Then Htrig has a one-dimensional module Ltriv = C · u0. yu0 = u0, Du0 = u0/4, su0 = u0. In this case, v = exp(π√−1h) = √−1 Haff = CX, T/((T − √−1)2 = 0, TX − X−1T = 2√−1X)
RDAHAs dDAHAs Algebraic KZ
dDAHA : Htrig = Cy, s, D / (sy = y−1s, sD + Ds = h, [D, y] = y − h(y + 1)). Affine Hecke Haff = CT, X/((T − v)(T + v−1) = 0, TX − X−1T = (v − v−1)X), v = exp(π√−1h). T = monodromy of half-turn around 1 ∈ C, X = monodromy around 0. Consider h = 1/2. Then Htrig has a one-dimensional module Ltriv = C · u0. yu0 = u0, Du0 = u0/4, su0 = u0. In this case, v = exp(π√−1h) = √−1 Haff = CX, T/((T − √−1)2 = 0, TX − X−1T = 2√−1X) ker V = Ltriv.
RDAHAs dDAHAs Algebraic KZ
There are several issues :
RDAHAs dDAHAs Algebraic KZ
There are several issues : Unlike O(Hrat), we need to complete O(Htrig) to get projective objects.
RDAHAs dDAHAs Algebraic KZ
There are several issues : Unlike O(Hrat), we need to complete O(Htrig) to get projective objects. The construction of V is analytic.
RDAHAs dDAHAs Algebraic KZ
There are several issues : Unlike O(Hrat), we need to complete O(Htrig) to get projective objects. The construction of V is analytic. The monodromy representation of Haff on M∇ is not easily computable.
RDAHAs dDAHAs Algebraic KZ
There are several issues : Unlike O(Hrat), we need to complete O(Htrig) to get projective objects. The construction of V is analytic. The monodromy representation of Haff on M∇ is not easily computable. − → A more algebraic approach to O(Htrig) and V is needed.
RDAHAs dDAHAs Algebraic KZ
Recall triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h].
RDAHAs dDAHAs Algebraic KZ
Recall triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h]. We have C[T ∨] = CQ∨ ; Q∨ ⊂ hR : dual root lattice
RDAHAs dDAHAs Algebraic KZ
Recall triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h]. We have C[T ∨] = CQ∨ ; Q∨ ⊂ hR : dual root lattice C W = C[T ∨] ⊗ CW ⊂ Htrig ;
RDAHAs dDAHAs Algebraic KZ
Recall triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h]. We have C[T ∨] = CQ∨ ; Q∨ ⊂ hR : dual root lattice C W = C[T ∨] ⊗ CW ⊂ Htrig ;
Π ;
∆ base of affine root system
RDAHAs dDAHAs Algebraic KZ
Recall triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h]. We have C[T ∨] = CQ∨ ; Q∨ ⊂ hR : dual root lattice C W = C[T ∨] ⊗ CW ⊂ Htrig ;
Π ;
∆ base of affine root system
RDAHAs dDAHAs Algebraic KZ
Recall triangular decomposition Htrig = C[T ∨] ⊗ CW ⊗ C[h]. We have C[T ∨] = CQ∨ ; Q∨ ⊂ hR : dual root lattice C W = C[T ∨] ⊗ CW ⊂ Htrig ;
Π ;
∆ base of affine root system
Another presentation of dDAHA Htrig = C W ⊗ C[h] satisfies the following relations for f ∈ C[h] and α ∈ Π : sα f − sα(f) sα = hα f − sα(f) α .
RDAHAs dDAHAs Algebraic KZ
Let M ∈ O(Htrig), so that Dξ acts locally finitely for ξ ∈ h∗
RDAHAs dDAHAs Algebraic KZ
Let M ∈ O(Htrig), so that Dξ acts locally finitely for ξ ∈ h∗ M decomposes into generalised eigenspaces of C[h] : M =
Mλ, Mλ =
RDAHAs dDAHAs Algebraic KZ
Let M ∈ O(Htrig), so that Dξ acts locally finitely for ξ ∈ h∗ M decomposes into generalised eigenspaces of C[h] : M =
Mλ, Mλ =
Wλ0 ⊂ h denote the W-orbit of λ0. Define Oλ0(Htrig) =
Wλ0
Mλ
RDAHAs dDAHAs Algebraic KZ
Let M ∈ O(Htrig), so that Dξ acts locally finitely for ξ ∈ h∗ M decomposes into generalised eigenspaces of C[h] : M =
Mλ, Mλ =
Wλ0 ⊂ h denote the W-orbit of λ0. Define Oλ0(Htrig) =
Wλ0
Mλ
O(Htrig) =
W
Oλ0(Htrig).
RDAHAs dDAHAs Algebraic KZ
We want to study the block Oλ0(Htrig) =
Wλ0
Mλ
RDAHAs dDAHAs Algebraic KZ
We want to study the block Oλ0(Htrig) =
Wλ0
Mλ
A result of Rouquier and Brundan–Kleshchev : each block
a quiver Hecke algebra for the linear quiver
RDAHAs dDAHAs Algebraic KZ
We want to study the block Oλ0(Htrig) =
Wλ0
Mλ
A result of Rouquier and Brundan–Kleshchev : each block
a quiver Hecke algebra for the linear quiver This result has a double-affine analogue.
RDAHAs dDAHAs Algebraic KZ
For a fixed λ0 ∈ h, we want to study the block Oλ0(Htrig) =
Wλ0
Mλ
RDAHAs dDAHAs Algebraic KZ
For a fixed λ0 ∈ h, we want to study the block Oλ0(Htrig) =
Wλ0
Mλ
There is a non-unital associative algebra Hλ0 generated by
RDAHAs dDAHAs Algebraic KZ
For a fixed λ0 ∈ h, we want to study the block Oλ0(Htrig) =
Wλ0
Mλ
There is a non-unital associative algebra Hλ0 generated by
Wλ0,
RDAHAs dDAHAs Algebraic KZ
For a fixed λ0 ∈ h, we want to study the block Oλ0(Htrig) =
Wλ0
Mλ
There is a non-unital associative algebra Hλ0 generated by
Wλ0, a polynomial subalgebra C[h]e(λ) ⊂ Hλ0 for each λ ∈ Wλ0,
RDAHAs dDAHAs Algebraic KZ
For a fixed λ0 ∈ h, we want to study the block Oλ0(Htrig) =
Wλ0
Mλ
There is a non-unital associative algebra Hλ0 generated by
Wλ0, a polynomial subalgebra C[h]e(λ) ⊂ Hλ0 for each λ ∈ Wλ0, for α ∈ Π, a Hecke operator ταe(λ) = e(sαλ)τα ∈ Hλ0 for each λ ∈ Wλ0.
RDAHAs dDAHAs Algebraic KZ
For a fixed λ0 ∈ h, we want to study the block Oλ0(Htrig) =
Wλ0
Mλ
There is a non-unital associative algebra Hλ0 generated by
Wλ0, a polynomial subalgebra C[h]e(λ) ⊂ Hλ0 for each λ ∈ Wλ0, for α ∈ Π, a Hecke operator ταe(λ) = e(sαλ)τα ∈ Hλ0 for each λ ∈ Wλ0. Hλ0 satisfies relations similar to those of the quiver Hecke algebras (Khovanov–Lauda–Rouquier algebras)
RDAHAs dDAHAs Algebraic KZ
Let Hλ0 -mod0 be the category of Hλ0-modules M such that
RDAHAs dDAHAs Algebraic KZ
Let Hλ0 -mod0 be the category of Hλ0-modules M such that M =
λ∈ Wλ0 Mλ
RDAHAs dDAHAs Algebraic KZ
Let Hλ0 -mod0 be the category of Hλ0-modules M such that M =
λ∈ Wλ0 Mλ
dim Mλ < ∞ for all λ ∈ Wλ0.
RDAHAs dDAHAs Algebraic KZ
Let Hλ0 -mod0 be the category of Hλ0-modules M such that M =
λ∈ Wλ0 Mλ
dim Mλ < ∞ for all λ ∈ Wλ0. Generalisation of [Rouquier ’08] and [Brundan–Kleshchev ’09] There is an equivalence of categories Oλ0(Htrig) ∼ = Hλ0 -mod0
RDAHAs dDAHAs Algebraic KZ
Let Hλ0 -mod0 be the category of Hλ0-modules M such that M =
λ∈ Wλ0 Mλ
dim Mλ < ∞ for all λ ∈ Wλ0. Generalisation of [Rouquier ’08] and [Brundan–Kleshchev ’09] There is an equivalence of categories Oλ0(Htrig) ∼ = Hλ0 -mod0 Thus Hλ0 is the “block algebra” of Oλ0(Htrig).
RDAHAs dDAHAs Algebraic KZ
Advantages of Hλ0 over Htrig :
RDAHAs dDAHAs Algebraic KZ
Advantages of Hλ0 over Htrig : As Hλ0 has a Z-grading, we can consider its graded modules.
RDAHAs dDAHAs Algebraic KZ
Advantages of Hλ0 over Htrig : As Hλ0 has a Z-grading, we can consider its graded modules. Hλ0 has enough graded projective modules.
RDAHAs dDAHAs Algebraic KZ
Advantages of Hλ0 over Htrig : As Hλ0 has a Z-grading, we can consider its graded modules. Hλ0 has enough graded projective modules. The centre of Hλ0 is a graded polynomial ring.
RDAHAs dDAHAs Algebraic KZ
Advantages of Hλ0 over Htrig : As Hλ0 has a Z-grading, we can consider its graded modules. Hλ0 has enough graded projective modules. The centre of Hλ0 is a graded polynomial ring. Hλ0 has a lot of idempotents.
RDAHAs dDAHAs Algebraic KZ
Similarly, there is a block decomposition for affine Hecke algebras O(Haff) =
Oℓ0(Haff)
RDAHAs dDAHAs Algebraic KZ
Similarly, there is a block decomposition for affine Hecke algebras O(Haff) =
Oℓ0(Haff) and a graded algebra Kℓ0 with an equivalence of category Oℓ0(Haff) ∼ = Kℓ0 -mod0 .
RDAHAs dDAHAs Algebraic KZ
Similarly, there is a block decomposition for affine Hecke algebras O(Haff) =
Oℓ0(Haff) and a graded algebra Kℓ0 with an equivalence of category Oℓ0(Haff) ∼ = Kℓ0 -mod0 . Theorem [L.] There exists an idempotent e ∈ Hλ0 of degree 0 and a graded ring isomorphism eHλ0e ∼ = Kℓ0 with ℓ0 = exp(λ0) ∈ T.
RDAHAs dDAHAs Algebraic KZ
Using the isomorphism eHλ0e ∼ = Kℓ0, we define a functor on ungraded modules V : Hλ0 -mod0 → Kλ0 -mod0, M → eM.
RDAHAs dDAHAs Algebraic KZ
Using the isomorphism eHλ0e ∼ = Kℓ0, we define a functor on ungraded modules V : Hλ0 -mod0 → Kλ0 -mod0, M → eM. Question Does the following diagram commutes ? Oλ0(Htrig) Oℓ0(Haff) Hλ0 -mod0 Kλ0 -mod0
V ∼ = ∼ = V
RDAHAs dDAHAs Algebraic KZ
Using the isomorphism eHλ0e ∼ = Kℓ0, we define a functor on ungraded modules V : Hλ0 -mod0 → Kλ0 -mod0, M → eM. Question Does the following diagram commutes ? Oλ0(Htrig) Oℓ0(Haff) Hλ0 -mod0 Kλ0 -mod0
V ∼ = ∼ = V
Proposition [L.] ker V = ker V
RDAHAs dDAHAs Algebraic KZ
Hλ0 and Kℓ0 have are defined in a similar way as quiver Hecke algebras
RDAHAs dDAHAs Algebraic KZ
Hλ0 and Kℓ0 have are defined in a similar way as quiver Hecke algebras The definition can be easily generalised, just as the quiver Hecke algebras are defined for all quivers.
RDAHAs dDAHAs Algebraic KZ
Hλ0 and Kℓ0 have are defined in a similar way as quiver Hecke algebras The definition can be easily generalised, just as the quiver Hecke algebras are defined for all quivers. The isomorphism eHλ0e ∼ = Kℓ0 generalises.
RDAHAs dDAHAs Algebraic KZ
Hλ0 and Kℓ0 have are defined in a similar way as quiver Hecke algebras The definition can be easily generalised, just as the quiver Hecke algebras are defined for all quivers. The isomorphism eHλ0e ∼ = Kℓ0 generalises. The KZ functor V : M → eM satisfies the double centraliser property : it is fully faithful on graded projective modules
RDAHAs dDAHAs Algebraic KZ
Hλ0 and Kℓ0 have are defined in a similar way as quiver Hecke algebras The definition can be easily generalised, just as the quiver Hecke algebras are defined for all quivers. The isomorphism eHλ0e ∼ = Kℓ0 generalises. The KZ functor V : M → eM satisfies the double centraliser property : it is fully faithful on graded projective modules The kernel ker V is characterised as follows : L ∈ Hλ0 -gmod : simple module, PL ∈ Hλ0 -gmod : projective cover of L. Then L ∈ ker V ⇔ PL is not relatively injective over Z(Hλ0)