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تسجيل مستخدم جديدCategorification of DAHA and Macdonald polynomials
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We describe a categorification of the Double Affine Hecke Algebra ${mathcal{H}kern -.4emmathcal{H}}$ associated with an affine Lie algebra $widehat{mathfrak{g}}$, a categorification of the polynomial representation and a categorification of Macdonald polynomials. All categorification results are given in the derived setting. That is, we consider the derived category associated with graded modules over the Lie superalgera ${mathfrak I}[xi]$, where ${mathfrak I}subsetwidehat{mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $xi$ is a formal odd variable. The Euler characteristic of graded characters of a complex of ${mathfrak I}[xi]$-modules is considered as an element of a polynomial representation. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${mathfrak{p}}_{i}$ categorify Demazure operators $T_i+1in{mathcal{H}kern -.4emmathcal{H}}$, meaning that all algebraic relations of $T_i$ have categorical meanings. Second, we describe a natural collection of complexes ${mathbb{EM}}_{lambda}$ of ${mathfrak I}[xi]$-modules whose Euler characteristic is equal to nonsymmetric Macdonald polynomials $E_lambda$ for dominant $lambda$ and a natural collection of complexes of $mathfrak{g}[z,xi]$-modules ${mathbb{PM}}_{lambda}$ whose Euler characteristic is equal to the symmetric Macdonald polynomial $P_{lambda}$. We illustrate our theory with the example $mathfrak{g}=mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${mathfrak I}[xi]$ such that their supercharacters coincide with renormalizations of nonsymmetric Macdonald polynomials.
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