Modules over plane curve singularities in any ranks and DAHA

We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks; explicit formulas are obtained for torus knots. The new feature is the definition of counterparts of Jacobian factors (directly related to compactifie...

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Bibliographic Details
Published inJournal of algebra Vol. 520; pp. 186 - 236
Main Authors Cherednik, Ivan, Philipp, Ian
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.02.2019
Subjects
Algebraic knot
Compactified Jacobian
Hecke algebra
Khovanov–Rozansky homology
Macdonald polynomial
Orbital integral
Plane curve singularity
Puiseux expansion
Algebraic knot
33D52
14H50
57M25
Puiseux expansion
30F10
17B22
17B45
Hecke algebra
Khovanov–Rozansky homology
22E50
Macdonald polynomial
20C08
33D80
Compactified Jacobian
20F36
22E57
Plane curve singularity
Orbital integral
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Summary:We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks; explicit formulas are obtained for torus knots. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible sheaves to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials for any algebraic knots colored by columns.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2018.11.006