Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves

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

81 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 gr U_c = \mathbb{C} [ \mathfrac{h} ⊕ \mathfrac{h}^* ]^W, where is the n-th symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered H_c - or U_c - module \hat{Φ}(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and we relate it to Hilb(n) and to the resolution of singularities τ : Hilb(n) → \mathfrac{h} ⊕ \mathfrac{h}^* / W. For example, we prove the following. • If c=1/n so that L_c(triv) is the unique one-dimensional simple H_c-module, then \hat{Φ}(e L_c(triv)) ≅ \mathcal{O}_{Z_n}, where Z_n = τ^{-1}(0) is the punctual Hilbert scheme. • If c = 1/n+k for k \in \mathbb{N}, then under a canonical filtration on the finite-dimensional module L_c(triv), gr e L_c(triv) has a natural bigraded structure that coincides with that on H^0( Z_n, \mathscr{L}^k), where \mathscr{L} ≅ \mathcal{O}_{Hilb(n)}(1); this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3]. • Under mild restrictions on c, the characteristic cycle of \hat{Φ}(e Δ_c(μ)) equals \sum_λ K_{μλ}[Z_λ], where K_{μλ} are Kostka numbers and the Z_λ are (known) irreducible components of τ^{-1}(\mathfrak{h}/W)

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

View Paper: https://www.semanticscholar.org/paper/6ac1c0fa4def1e3ad0c4b99665902b201bbad7d2

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In Algebra and Number Theory

1. I. Gordon
2. E. Gorsky
3. Andrei Neguct
4. T. Nevins
5. Kevin McGerty
6. J. T. Stafford
7. V. Ginzburg
8. I. Losev

In Geometry and Topology

1. Andrei Neguct
2. I. Gordon
3. E. Gorsky
4. J. T. Stafford
5. T. Nevins
6. Kevin McGerty
7. V. Ginzburg
8. Stephen Griffeth

In Mathematical Physics

1. I. Gordon
2. E. Gorsky
3. Andrei Neguct
4. J. T. Stafford
5. T. Nevins
6. Kevin McGerty
7. V. Ginzburg
8. Stephen Griffeth

Citations in the corpus

• Symplectic reflection algebras (2007, background): “Not every finitely generated U1,c-module has a natural filtration on it, but for some that do the corresponding coherent sheaf on Hilb C(2) can be calculated, [79]: • If M = U1,c with its differential operators filtration then the corresponding sheaf isOHilbn C2 .”
• Refined knot invariants and Hilbert schemes (2013, background): “s bee mainly motivated by [35] where L m n has been related to the Hilbert schemes on the singular curve {xm = yn} and to the knot homology. For m = kn ± 1, the conjecture follows from the results of [27, 28]. The structure of this paper is the following: in Section 2, we recall the basics on symmet”
• Derived equivalence for quantum symplectic resolutions (2011, background): “[6,14,29,30,39,46]) toward a theory of microlocalization, i.”
• Rational Cherednik algebras and Hilbert schemes (2004, methodology): “These modules have a natural good filtration and we mention a couple of illustrative results from [ GS ].”