Stability conditions for noncommutative symplectic resolutions Gufang Zhao Northeastern University Conference on Geometric Methods in Representation Theory 2013 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 1 / 22
Outline Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 2 / 22
Motivation Symplectic resolutions: commutative vs. non-comm. We work over a separably closed field k of characteristic p >> 0. Let Γ n := ( Z r ) n ⋊ S n acting on h = A n in the natural way. Let V = h ⊕ h ∗ � A 2 n be endowed with the diagonal action of Γ n . The action preserves the natural symplectic form on V . A symplectic resolution of A 2 n / Γ n can be given as Hilb n ( � A 2 / Z r ) where � A 2 / Z r is the minimal resolution of A 2 / Z r . Let W ( h ) be the Weyl algebra. The algebra W ( h ) Γ n is a noncommutative desingularization of A 2 n / Γ n . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 3 / 22
Motivation The rational Cherednik algebras The (spherical) Cherednik algebra s H c is a deformation of W ( h ) Γ n . The parameter space of the deformation is spanned by the conjugacy classes of reflections in Γ n . The algebra s H c has a big Frobenius center k [ A 2 n ( 1 ) ] Γ n . For any central character χ , the irreducible objects in the category Mod- χ H c are naturally labeled by Irrep (Γ n ) . If s H c has finite global dimension, then the value c is called spherical value . Otherwise we say c is aspherical . The aspherical values form an affine hyperplane arrangement in the space of parameters. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 4 / 22
Motivation The Localization Theorem Let Hilb ( 1 ) be the Frobenius twist of Hilb := Hilb n ( � A 2 / Z r ) . Let Coh 0 Hilb ( 1 ) be the category of coherent sheaves on Hilb ( 1 ) set-theoretically supported on the zero-fiber of the Hilbert-Chow morphism. Theorem (Bezrukavnikov-Finkelberg-Ginzburg) There is a tilting bundle E c on Hilb ( 1 ) , such that End ( E c ) ˆ 0 � ( s H c ) ˆ 0 In particular, for spherical values c, there is a derived equivalence D b ( Coh 0 Hilb ( 1 ) ) � D b ( Mod - 0 s H c ) . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 5 / 22
Motivation The t -structures For each spherical value c , the derived equivalence D b ( Coh 0 Hilb ( 1 ) ) � D b ( Mod- 0 s H c ) endows D b ( Coh 0 Hilb ( 1 ) ) with a t -structure. Question For two different spherical values c and c ′ , what is the relation between the t-structures on D b ( Coh 0 Hilb ( 1 ) ) ? If c and c ′ are in the same alcove, then the translation functor induces a Morita equivalence; the t -structures are the same. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 6 / 22
The dimension polynomials Table of Contents Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 7 / 22
The dimension polynomials Dimensions of irreducible modules Let L c ( τ ) be the irreducible object in Mod- 0 s H c labeled by τ ∈ Irrep (Γ n ) . Under the BFG-derived equivalence, Mod- 0 s H c ∋ L c ( τ ) ↔ L c ( τ ) ∈ D b ( Coh 0 Hilb ( 1 ) ) ; Mod- End ( E c ) ∋ proj. cover of L c ( τ ) ↔ V τ vector bundle on Hilb. When c + ν is in the same alcove as c , dim L c + ν ( τ ) := χ ( L c ( τ ) ⊗ E 0 ⊗ O ( ν )) is a polynomial in ν . Proposition For a basis { ch O ( b ) } of H ∗ ( Hilb , Q l ) with b = � h b τ ch ( V τ ) , we have χ ( L τ ⊗ O ( b )) = h b τ . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 8 / 22
The dimension polynomials The Chern character problem Theorem (Etingof-Ginzburg, Ginzburg-Kaledin) The algebra H ∗ ( Hilb ; Q ) is isomorphic to the algebra gr Z Q [Γ n ] . The following problem is raised by Etingof, Ginzburg, and Kaledin. Problem Express explicitly the map K 0 (Γ n ) → gr Z Q [Γ n ] induced by the Chern character ch : K 0 ( Hilb ) → H ∗ ( Hilb ; Q ) . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 9 / 22
� The dimension polynomials The Chern character maps: an example Take Γ = Z 2 acting on A 2 . The minimal resolution is T ∗ P 1 , the central fiber is the zero section P 1 . The quiver is the affine A 1 quiver with v = ( 1 , 1 ) and w = ( 1 , 0 ) . X 1 � v 1 v 0 X 2 There is a natural G m -action with fixed points [ 1 , 0 ] and [ 0 , 1 ] . At [ 1 , 0 ] , v 1 has weight 1; at [ 0 , 1 ] , v 1 has weight -1. Therefore, equivariant localization theorem tells us that c 1 ( V 2 ) = − 1. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 10 / 22
The dimension polynomials Dimension polynomials via quasi-invariants For an integral parameter m , let Q m be the m -quasi-invariants in k [ h ] . As Γ n - s H m bimodule, Q m = ⊕ τ ∈ Irrep (Γ n ) τ ∗ ⊗ M m ( τ ) . Let � Q m be the quasi-invariants on the Frobenius neighborhood of 0. Q m : · · · → Q m ⊗ ∧ 2 h ( 1 ) → Q m ⊗ h ( 1 ) → Q m . A resolution of � Theorem (Z., to appear) Fix a character i of Z r . Let τ ( i ) be the 1-dimensional representation of Γ n = ( Z r ) n ⋊ S n on which Z r acts by the character i and S n acts by the sign representation. The Poincaré series of L m ( τ ( i )) is t ni � n − 1 k = 0 ( 1 − t rk + m 0 n + p + 1 + rm i + 1 ) � n . k = 1 ( 1 − t kr ) Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 11 / 22
Real variation of stability conditions Table of Contents Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 12 / 22
Real variation of stability conditions The Central Charge We reparameterize the value c by setting x = cp and define p →∞ p − n dim k L c ( τ ; p ) . Z τ ( x ) = lim We consider the collection of polynomials { Z τ ( x ) | τ ∈ Irrep (Γ n ) } as a polynomial map H 2 ( Hilb ; R ) → Hom Z ( K 0 ( Hilb ) , R ) . This polynomial map is called the central charge . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 13 / 22
Real variation of stability conditions The Main Theorem Assume n = 2. T : { alcoves } → { t -structures } ; A �→ Mod 0 s H c (Γ 2 ) ⊆ D b ( Coh 0 ( Hilb )) for c ∈ A . Z : H 2 ( Hilb ; R ) → Hom Z ( K 0 ( Hilb ) , R ) : the central charge. Theorem (Z., to appear) The pair ( T , Z ) is a real variation of stability conditions. More concretely, for any alcove A, let A := heart of T ( A ) . We have, for any x ∈ A, Z L ( x ) > 0 for any simple object L ∈ A ; 1 for any A ′ , sharing a codim. 1 wall H with A. 2 Let A � A i := � L ∈ A | Z L ( x ) vanishes of order ≥ i on H � . Then, ◮ the T ( A ′ ) is compatible with the filtration on T ( A ) ; ◮ on gr i ( A ) = A i / A i + 1 , φ ( A ′ ) differs by [ i ] from φ ( A ) . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 14 / 22
The t -structures associated to alcoves Table of Contents Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 15 / 22
The t -structures associated to alcoves The hyperplane arrangement for Γ 1 = Z r and n = 2 The t -structure t 0 is generated by the Procesi bundle on Hilb. t 1 is obtained from t 0 by a P 2 -semi-reflection. b t 2 t 1 t 0 a Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 16 / 22
The t -structures associated to alcoves The P n -semi-reflections A = an abelian category, of finite length, having finitely many simples; either A has enough projective objects; or A = Mod- χ H , for some algebra H , finitely generated over its center Z ( H ) , and a central character χ . R S A := the (right) tilting of A with respect to a simple S . Let S θ be a simple object with Ext 1 ( S θ , S θ ) = 0. Proposition Assume the projective objects in R S θ [ n − 1 ] R S θ [ n − 2 ] · · · R S θ ( A ) are concentrated in degree zero. Then for all 0 ≤ i ≤ n, R S θ [ i − 1 ] R S θ [ i − 2 ] · · · R S θ ( A ) has a set of projectives consisting of objects lying in A , is derived equivalent to A . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 17 / 22
The t -structures associated to alcoves An example of P n -semi-reflection Endow P n with the standard Bruhat stratification. A = { perverse sheaves on P n } . S n := C P n [ n ] , simple object in A . It is an P n object. The semi-reflection of A with respect to S n can be obtained by iterated tilting. The projective generators consist of perverse sheaves. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 18 / 22
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