Abstract. For algebraic varieties defined by hyperkähler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the "hyperkähler Kirwan map" on cohomology is surjective. We resolve this question in the affirmative for Nakajima quiver varieties. We also establish similar results for other cohomology theories and for the derived category. Our proofs use only classical topological and geometric arguments.
Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson-Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon-Stafford [GS1, GS2] and Kashiwara-Rouquier [KR] as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities. M.S.C.: 14H60, 18E30.
We give a geometric interpretation of the inner product on the modified quantum group of sln. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.
ABSTRACT. We give a representation-theoretic interpretation of the Langlands character duality of [FH], and show that the "Langlands branching multiplicites" for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.
The quantum Frobenius map and it splitting are shown to descend to maps between generalized q-Schur algebras at a root of unity. We also define analogs of q-Schur algebras for any affine algebra, and prove the corresponding results for them.
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