Towards a functor between affine and finite Hecke categories in type - PowerPoint PPT Presentation

Towards a functor between affine and finite Hecke categories in type A Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 1 / 19

Braid groups B n ≃ π 1 ( Conf n ( C ) , ζ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 2 / 19

Braid groups B n ≃ π 1 ( Conf n ( C ) , ζ ) ≃ π 1 ( Conf n ( C ∗ ) , ζ ) B aff n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 2 / 19

Braid groups B n ≃ π 1 ( Conf n ( C ) , ζ ) ≃ π 1 ( Conf n ( C ∗ ) , ζ ) B aff n C ∗ ֒ → C � B aff → B n n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 2 / 19

Finite Hecke algebra W = S n – symmetric group. I = { (1 2) , (2 3) , ..., ( n − 1 n ) } ⊂ S n . s i = ( i i + 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 3 / 19

Finite Hecke algebra W = S n – symmetric group. I = { (1 2) , (2 3) , ..., ( n − 1 n ) } ⊂ S n . s i = ( i i + 1). H ( W ) = H n – unital algebra over Z [ v , v − 1 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 3 / 19

Finite Hecke algebra W = S n – symmetric group. I = { (1 2) , (2 3) , ..., ( n − 1 n ) } ⊂ S n . s i = ( i i + 1). H ( W ) = H n – unital algebra over Z [ v , v − 1 ]. Generators: { t s , s ∈ I } ; t i := t s i . Relations: 1. t i t i +1 t i = t i +1 t i t i +1 . 2. t i t j = t j t i , | i − j | > 1 . i = 1 + ( v − 1 − v ) t i . 3. t 2 H n has a basis { t w , w ∈ W } , defined by t w = t s 1 . . . t s k for a reduced expression w = s 1 . . . s k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 3 / 19

Extended affine Hecke algebra ( X ∗ , Φ , X ∗ , Φ ∨ ) – root datum of GL n . X ∗ = X ∗ =: X ≃ Z n = span Z { e 1 , . . . , e n } , Φ ∨ = Φ = { e i − e j } i � = j ⊂ X . W acts on X and Φ permuting e i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

Extended affine Hecke algebra ( X ∗ , Φ , X ∗ , Φ ∨ ) – root datum of GL n . X ∗ = X ∗ =: X ≃ Z n = span Z { e 1 , . . . , e n } , Φ ∨ = Φ = { e i − e j } i � = j ⊂ X . W acts on X and Φ permuting e i . ∆ = { e i − e i +1 } n − 1 i =1 – simple roots. X + = { ( λ 1 , . . . , λ n ) , λ k ≥ λ k +1 for all k } – dominant weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

Extended affine Hecke algebra ( X ∗ , Φ , X ∗ , Φ ∨ ) – root datum of GL n . X ∗ = X ∗ =: X ≃ Z n = span Z { e 1 , . . . , e n } , Φ ∨ = Φ = { e i − e j } i � = j ⊂ X . W acts on X and Φ permuting e i . ∆ = { e i − e i +1 } n − 1 i =1 – simple roots. X + = { ( λ 1 , . . . , λ n ) , λ k ≥ λ k +1 for all k } – dominant weights. ˜ – unital algebra over Z [ v , v − 1 ]. H aff n Generators: { t s , s ∈ I , θ x , x ∈ X } ; t i := t s i , θ i := θ e i . Relations: 1. t i t i +1 t i = t i +1 t i t i +1 . 2. t i t j = t j t i , | i − j | > 1 . i = 1 + ( v − 1 − v ) t i . 3. t 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

Extended affine Hecke algebra ( X ∗ , Φ , X ∗ , Φ ∨ ) – root datum of GL n . X ∗ = X ∗ =: X ≃ Z n = span Z { e 1 , . . . , e n } , Φ ∨ = Φ = { e i − e j } i � = j ⊂ X . W acts on X and Φ permuting e i . ∆ = { e i − e i +1 } n − 1 i =1 – simple roots. X + = { ( λ 1 , . . . , λ n ) , λ k ≥ λ k +1 for all k } – dominant weights. ˜ – unital algebra over Z [ v , v − 1 ]. H aff n Generators: { t s , s ∈ I , θ x , x ∈ X } ; t i := t s i , θ i := θ e i . Relations: 1. t i t i +1 t i = t i +1 t i t i +1 . 2. t i t j = t j t i , | i − j | > 1 . i = 1 + ( v − 1 − v ) t i . 3. t 2 4. θ x θ y = θ x + y . 5. t i θ j = θ j t i if j � = i , i + 1. 6. t i θ i t i = θ i +1 . 7. θ 0 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 4 / 19

The homomorphism Π : ˜ H aff n → H n . Definition Π( t i ) = t i , Π( θ 1 ) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 5 / 19

The homomorphism Π : ˜ H aff n → H n . Definition Π( t i ) = t i , Π( θ 1 ) = 1 . This defines Π uniquely. Π( θ k ) = t k − 1 t k − 2 . . . t 2 t 2 1 t 2 ... t k − 2 t k − 1 =: JM k are called (multiplicative) Jucys-Murphy elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 5 / 19

The homomorphism Π : ˜ H aff n → H n . Definition Π( t i ) = t i , Π( θ 1 ) = 1 . This defines Π uniquely. Π( θ k ) = t k − 1 t k − 2 . . . t 2 t 2 1 t 2 ... t k − 2 t k − 1 =: JM k are called (multiplicative) Jucys-Murphy elements. λ ∈ X � W λ – W-orbit. Theorem (Bernstein) The center of ˜ H aff is a free Z [ v , v − 1 ] -module with a basis given by n elements { z λ , λ ∈ X + } , ∑ z λ := θ µ . µ ∈ W λ Theorem (Dipper-James, Francis-Graham) Set of symmetric polynomials in { JM i } is the center of H n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 5 / 19

Categorification. Finite side G = GL n ( C ), B – Borel subgroup, U ⊂ B – unipotent radical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 6 / 19

Categorification. Finite side G = GL n ( C ), B – Borel subgroup, U ⊂ B – unipotent radical. B = G / B – flag variety, Y = G / U – base affine space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 6 / 19

Categorification. Finite side G = GL n ( C ), B – Borel subgroup, U ⊂ B – unipotent radical. B = G / B – flag variety, Y = G / U – base affine space. Y × Y is a T × T -torsor over B × B . D fin := ˆ ˆ D b c , G , mon ( Y × Y ) – finite Hecke category. ˆ D bc , G , mon ( Y × Y ) – completed monodromic (with unipotent monodromy) bounded G -equivariant derived category of constructible sheaves on Y × Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 6 / 19

Categorification. Finite side G = GL n ( C ), B – Borel subgroup, U ⊂ B – unipotent radical. B = G / B – flag variety, Y = G / U – base affine space. Y × Y is a T × T -torsor over B × B . D fin := ˆ ˆ D b c , G , mon ( Y × Y ) – finite Hecke category. ˆ D bc , G , mon ( Y × Y ) – completed monodromic (with unipotent monodromy) bounded G -equivariant derived category of constructible sheaves on Y × Y . G × Y a ( g , x )=( x , gx ) π ( g , x )= g G Y × Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kostiantyn Tolmachov partly joint with Roman Bezrukavnikov Towards a functor between Hecke categories 6 / 19