Rational Cherednik algebras and Hilbert schemes

By I. Gordon, J. T. Stafford. 2004

110 citations recorded by the source corpus.

Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c = e H_c e$. Then $U_c$ is filtered by order of differential operators, with associated graded ring $\mbox{gr} U_c = \mathbb{C} [ \mathfrak{h} \oplus \mathfrak{h}^*]$ where $W$ is the $n$th symmetric group. We construct a filtered $\mathbb{Z}$-algebra $b$ such that, under mild conditions on $c$: • the category $B$-qgr of graded noetherian $B$-modules modulo torsion is equivalent to $U_c$-mod; • the associated graded $\mathbb{Z}$-algebra $\mbox{gr}B$ has $\mbox{gr}B-lqgr \simeq \coh \Hilb(n)$ This can be regarded as saying that $U_c$ simultaneously gives a non-commutative deformation of $\mathfrak{h} \oplus \mathfrak{h}^* / W$ and of its resolution of singularities $\Hilb(n) \rightarrow \mathfrak{h} \oplus \mathfrak{h}^*$. As we show elsewhere, this result is a powerful tool for studying the representation theory of $H_c$ and its relationship to $\Hilb(n)$.

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

View Paper: https://www.semanticscholar.org/paper/ff3688f2e7a66423a7cbe3f440a73c91b6def129

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In Algebra and Number Theory

1. Stephen Griffeth
2. Masaki Kashiwara
3. Jacob Rasmussen
4. Vivek V. Shende
5. T. Nevins
6. Kevin McGerty
7. Raphaël Rouquier
8. Andrei Neguct

In Geometry and Topology

1. E. Gorsky
2. Andrei Neguct
3. Stephen Griffeth
4. Vivek V. Shende
5. Alexei Oblomkov
6. Masaki Kashiwara
7. Jacob Rasmussen
8. Charles F. Dunkl

In Mathematical Physics

1. E. Gorsky
2. Andrei Neguct
3. Vivek V. Shende
4. Alexei Oblomkov
5. Jacob Rasmussen
6. Stephen Griffeth
7. Charles F. Dunkl
8. Masaki Kashiwara

Citations in the corpus

• q-Schur algebras and complex reflection groups, I (2005, background): “When W has type A n − 1 , Gordon and Stafford proved that the shift functor is an equivalence (parameter 6∈ + Z ) [GoSt1, Proposition 3.16].”
• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, methodology): “J / is compatible with the natural bigrading on H sph n .k=n/ ; ie it is a good filtration in the sense of Gordon and Stafford [28; 29].”
• Refined knot invariants and Hilbert schemes (2013, methodology): “, the analogous construction has been carried out by Bezrukavnikov-Finkelberg-Ginzburg ([7]). We will only be concerned with characteristic 0, in which case the initial result of Gordon and Stafford ([27]) claims the existence of a map: (47) DbRep(H c) −→ DbCoh(Hilbn) for all c. The category on the ”
• Microlocalization of rational Cherednik algebras (2007, background): “Gordon and Stafford [8, 9] construct a one-parameter family of graded (Z)-algebras Bc that quantize (a graded (Z)-algebra Morita-equivalent to) the homogeneous coordinate ring of HilbC2.”
• Derived equivalence for quantum symplectic resolutions (2011, background): “[6,14,29,30,39,46]) toward a theory of microlocalization, i.”
• Generalized Jack polynomials and the representation theory of rational Cherednik algebras (2010, methodology): “We hope that by combining these equivalences with the techniques of the papers [7, 8 ] and [5], one may obtain a tighter relationship between the Cherednik algebra of type G(r, 1, n) and certain quiver varieties, including the Hilbert schemes of points on resolutions of type A surface singularities.”