Exotic t-structures for two-block Springer fibres Vinoth Nandakumar - PowerPoint PPT Presentation

Exotic t-structures for two-block Springer fibres Vinoth Nandakumar Massachusetts Institute of Technology July 20, 2012 Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 1 / 15

Outline Two-block Springer fibres 1 Affine tangles 2 The exotic t-structure 3 Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 2 / 15

Two-block Springer fibres Two-block Springer fibres: the definitions Fix m ∈ Z ≥ 0 , and let n ∈ Z ≥ 0 . Recall the following definitions: Definition Let G = SL m +2 n ( C ) , g = sl m +2 n ( C ), B ⊂ G be the Borel subgroup of upper triangular matrices; and G / B the flag variety: B n = G / B = { 0 ⊂ V 1 ⊂ · · · ⊂ V m +2 n − 1 ⊂ V m +2 n = C m +2 n | dim V i = i } Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 3 / 15

Two-block Springer fibres Two-block Springer fibres: the definitions Fix m ∈ Z ≥ 0 , and let n ∈ Z ≥ 0 . Recall the following definitions: Definition Let G = SL m +2 n ( C ) , g = sl m +2 n ( C ), B ⊂ G be the Borel subgroup of upper triangular matrices; and G / B the flag variety: B n = G / B = { 0 ⊂ V 1 ⊂ · · · ⊂ V m +2 n − 1 ⊂ V m +2 n = C m +2 n | dim V i = i } Definition Let N n = { x ∈ g | x nilpotent } be the nilpotent cone, � N n it’s Springer resolution (with the natural map π n : � N n → N n ): � N n := T ∗ B n = { (0 ⊂ V 1 ⊂ · · · ⊂ V m +2 n − 1 ⊂ V m +2 n ) , x | x ( V i ) ⊆ V i − 1 } Let z n ∈ N n be the standard nilpotent with Jordan type ( m + n , n ), and let B m + n , n = π − 1 n ( z n ) be the corresponding Springer fiber. Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 3 / 15

Two-block Springer fibres Two-block Springer fibres: transverse slices Definition Define the Mirkovic-Vybornov transverse slice as follows: � � S n = { z n + a i e m + n , i + b j e m + n , m + n + j 1 ≤ i ≤ m + n 1 ≤ j ≤ n � � + c j e m +2 n , j + d j e m +2 n , m + n + j } 1 ≤ j ≤ n 1 ≤ j ≤ m +2 n Let U n = π − 1 n ( S n ∩ N n ) denote the resolution of the variety S n ∩ N n . Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 4 / 15

Two-block Springer fibres Two-block Springer fibres: transverse slices Definition Define the Mirkovic-Vybornov transverse slice as follows: � � S n = { z n + a i e m + n , i + b j e m + n , m + n + j 1 ≤ i ≤ m + n 1 ≤ j ≤ n � � + c j e m +2 n , j + d j e m +2 n , m + n + j } 1 ≤ j ≤ n 1 ≤ j ≤ m +2 n Let U n = π − 1 n ( S n ∩ N n ) denote the resolution of the variety S n ∩ N n . Note that: B m + n , n = { (0 ⊂ V 1 ⊂ · · · ⊂ V m +2 n − 1 ⊂ V m +2 n ) | z n V i ⊆ V i − 1 } U n = { (0 ⊂ V 1 ⊂ · · · ⊂ V m +2 n − 1 ⊂ V m +2 n ) , x | x ∈ S n , xV i ⊆ V i − 1 } D n := D b (Coh B m + n , n ( U n )), the bounded derived category of coherent sheaves on U n supported on B m + n , n , will be our primary object of interest. Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 4 / 15

Two-block Springer fibres Two-block Springer fibres: some identities Define the partial flag variety P k , n (and T ∗ P k , n ) as follows: P k , n = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n = C m +2 n ) } T ∗ P k , n = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n = C m +2 n ) , x | x ∈ gl m +2 n , xV k +1 ⊂ V k − 1 , xV i ⊂ V i − 1 for i � = k , k + 1 } Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 5 / 15

Two-block Springer fibres Two-block Springer fibres: some identities Define the partial flag variety P k , n (and T ∗ P k , n ) as follows: P k , n = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n = C m +2 n ) } T ∗ P k , n = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n = C m +2 n ) , x | x ∈ gl m +2 n , xV k +1 ⊂ V k − 1 , xV i ⊂ V i − 1 for i � = k , k + 1 } Proposition We have: S n +1 × gl m +2 n +2 T ∗ P k , n +1 ≃ U n . Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 5 / 15

Two-block Springer fibres Two-block Springer fibres: some identities Define the partial flag variety P k , n (and T ∗ P k , n ) as follows: P k , n = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n = C m +2 n ) } T ∗ P k , n = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n = C m +2 n ) , x | x ∈ gl m +2 n , xV k +1 ⊂ V k − 1 , xV i ⊂ V i − 1 for i � = k , k + 1 } Proposition We have: S n +1 × gl m +2 n +2 T ∗ P k , n +1 ≃ U n . Proof (Sketch): Below, xV k +1 = V k − 1 ; ∃ a canonical isomorphism φ x : xV m +2 n +2 ≃ V m +2 n inducing φ ( x ) ∈ End( C m +2 n ). S n +1 × gl m +2 n +2 T ∗ P k , n +1 = { (0 ⊂ V 1 ⊂ · · · ⊂ � V k ⊂ · · · ⊂ V m +2 n +2 ) | x ∈ S n +1 , xV k +1 ⊆ V k − 1 , xV i ⊂ V i − 1 } Map this to { (0 ⊂ V 1 ⊂ · · · V k − 1 ⊂ xV k +2 ⊂ · · · xV m +2 n +2 ) , φ ( x ) } ∈ U n . Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 5 / 15

Two-block Springer fibres Two-block Springer fibres: some identities Let V m , n = C { e i , f i } 1 ≤ i ≤ m +2 n ; define z by ze i = e i − 1 , zf i = f i − 1 . Let W m , n = C { e k , f l } 1 ≤ k ≤ m + n , 1 ≤ l ≤ n , and P : V m , n → W m , n be the natural projection. Y m +2 n = { ( L 1 ⊂ · · · ⊂ L m +2 n ⊂ V m , n ) | dim L i = i , zL i ⊂ L i − 1 } � U m +2 n = { ( L 1 ⊂ · · · ⊂ L m +2 n ) ∈ Y m +2 n | P ( L m +2 n ) = W m , n } Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 6 / 15

Two-block Springer fibres Two-block Springer fibres: some identities Let V m , n = C { e i , f i } 1 ≤ i ≤ m +2 n ; define z by ze i = e i − 1 , zf i = f i − 1 . Let W m , n = C { e k , f l } 1 ≤ k ≤ m + n , 1 ≤ l ≤ n , and P : V m , n → W m , n be the natural projection. Y m +2 n = { ( L 1 ⊂ · · · ⊂ L m +2 n ⊂ V m , n ) | dim L i = i , zL i ⊂ L i − 1 } � U m +2 n = { ( L 1 ⊂ · · · ⊂ L m +2 n ) ∈ Y m +2 n | P ( L m +2 n ) = W m , n } The categories � D n := D b (Coh( Y m +2 n )) have been studied by Cautis and Kamnitzer. The below fact allows us to apply their results to study D n . Proposition There is a closed embedding U n → � U m +2 n (hence U n is locally closed in Y m +2 n ). Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 6 / 15

Two-block Springer fibres Two-block Springer fibres: some identities Let V m , n = C { e i , f i } 1 ≤ i ≤ m +2 n ; define z by ze i = e i − 1 , zf i = f i − 1 . Let W m , n = C { e k , f l } 1 ≤ k ≤ m + n , 1 ≤ l ≤ n , and P : V m , n → W m , n be the natural projection. Y m +2 n = { ( L 1 ⊂ · · · ⊂ L m +2 n ⊂ V m , n ) | dim L i = i , zL i ⊂ L i − 1 } � U m +2 n = { ( L 1 ⊂ · · · ⊂ L m +2 n ) ∈ Y m +2 n | P ( L m +2 n ) = W m , n } The categories � D n := D b (Coh( Y m +2 n )) have been studied by Cautis and Kamnitzer. The below fact allows us to apply their results to study D n . Proposition There is a closed embedding U n → � U m +2 n (hence U n is locally closed in Y m +2 n ). n = { z n + � Proof (Sketch): Let S ′ 1 ≤ i ≤ m +2 n ( u i e m + n , i + v i e m +2 n , i ) } . It suffices to show that � U m +2 n ≃ S ′ n × gl m +2 n T ∗ B n , since U n = S n × gl m +2 n T ∗ B n is closed in S ′ n × gl m +2 n T ∗ B n . Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 6 / 15

Affine tangles Affine tangles: definitions Definition If p ≡ q (mod 2), a ( p , q ) affine tangle is an embedding of p + q arcs and a 2 finite number of circles into the region { ( x , y ) ∈ C × R | 1 ≤ | x | ≤ 2 } , such that the end-points of the arcs are given { (1 , 0) , ( ζ p , 0) , · · · , ( ζ p − 1 , 0) , p 2 π i (2 , 0) , (2 ζ q , 0) , · · · , (2 ζ q − 1 k . , 0) } ; where ζ k = e q Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 7 / 15

Affine tangles Affine tangles: definitions Definition If p ≡ q (mod 2), a ( p , q ) affine tangle is an embedding of p + q arcs and a 2 finite number of circles into the region { ( x , y ) ∈ C × R | 1 ≤ | x | ≤ 2 } , such that the end-points of the arcs are given { (1 , 0) , ( ζ p , 0) , · · · , ( ζ p − 1 , 0) , p 2 π i (2 , 0) , (2 ζ q , 0) , · · · , (2 ζ q − 1 k . , 0) } ; where ζ k = e q Definition Let g i n denote the ( n − 2 , n ) tangle with an arc connecting (2 ζ i n , 0) to (2 ζ i +1 , 0). Let f i n denote the ( n , n − 2) tangle with an arc connecting n ( ζ i n , 0) and ( ζ i +1 , 0). n Let t i n (1) denote the ( n , n ) tangle with a strand connecting ( ζ i n , 0) to (2 ζ i +1 , 0) passes beneath a strand connecting ( ζ i +1 , 0) to (2 ζ i n , 0). n n Let r n denote the ( n , n ) tangle connecting ( ζ j n , 0) to (2 ζ j +1 , 0) for each n n (1) − 1 . 1 ≤ j ≤ n . Also let r ′ n := r − 1 n , t i n (2) := t i Vinoth Nandakumar Exotic t-structures for two-block Springer fibres July 20, 2012 7 / 15