Unramified affine Springer fibers and isospectral Hilbert schemes
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Selecta Mathematica New Series
Unramified affine Springer fibers and isospectral Hilbert schemes Oscar Kivinen1
© Springer Nature Switzerland AG 2020
Abstract For any connected reductive group G over C, we revisit Goresky–Kottwitz– MacPherson’s description of the torus equivariant Borel–Moore homology of affine Springer fibers Spγ ⊂ Gr G , where γ = zt d and z is a regular semisimple element in the Lie algebra of G. In the case G = G L n , we relate the equivariant cohomology of Spγ to Haiman’s work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of (n, dn)-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve {x n = y dn }.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Anti-invariants and subspace arrangements . . . . . . . . . 1.2 Relation to braids . . . . . . . . . . . . . . . . . . . . . . 1.3 Hilbert schemes of points on curves . . . . . . . . . . . . 1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . 2 Affine Springer fibers . . . . . . . . . . . . . . . . . . . . . . 3 Equivariant Borel–Moore homology of affine Springer fibers . . 3.1 Borel–Moore homology . . . . . . . . . . . . . . . . . . . 3.2 The S L 2 case . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The general case . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The affine flag variety . . . . . . . . . . . . . . . . 3.3.2 Equivariant K-homology . . . . . . . . . . . . . . . 4 The isospectral Hilbert scheme . . . . . . . . . . . . . . . . . 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diagonal coinvariants and a conjecture on the lattice action 4.3 Rational and elliptic versions . . . . . . . . . . . . . . . . 4.4 Other root data . . . . . . . . . . . . . . . . . . . . . . . 5 Relation to knot homology . . . . . . . . . . . . . . . . . . . . 6 Hilbert schemes of points on planar curves . . . . . . . . . . .
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Oscar Kivinen [email protected] Department of Mathematics, California Institute of Technology, Pasadena, USA 0123456789().: V,-vol
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6.1 Hilbert schemes on curves and compactified Jacobians 6.2 Conjectural description in the case C = {x
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