Geometric representations of graded and rational Cherednik algebras

By Alexei Oblomkov, Zhiwei Yun. 2014

77 citations recorded by the source corpus.

We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_\nu$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_\nu$ attached to a simple algebraic group $\mathbb{G}$ together with a pinned automorphism $\theta$. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit $\mathbb{C}^*$-actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When $\theta$ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules $\mathfrak{L}_\nu(triv)$ of $\mathfrak{H}^{gr}_\nu$ and of $\mathfrak{H}^{rat}_\nu$. We give a formula for the dimension of $\mathfrak{L}_\nu(triv)$ and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7aec90b1f380076ac3cf17d74f0df37d399ea70c

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Citations in the corpus

The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “The precise relation between the ﬁltration in the construction of [57] and the ﬁltration of [28; 29] is such that the equivariant variables T 1 and T 2 are related to the perverse and homological q and t as follows: (16) T 1 WD q 2 ; T 2 WD 1 q 2 t 2 : Combining Proposition 4 and Conjecture 20 we…”
Rational Parking Functions and Catalan Numbers (2014, background): “We do not know any natural construction of this bi-graded Sa-module analogous to the diagonal coinvariant ring, but see Oblomkov and Yun [35].”