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Heisenberg categorification and Hilbert schemes. (English) Zbl 1263.14020
Let $$\Gamma \subset \mathrm{SL}_2(\mathbb{C})$$ be a nontrivial finite subgroup and the surface $$S = \widehat{\mathbb{C}^2/\Gamma}$$ be the minimal resolution of $$\mathbb{C}/\Gamma$$. Associated to $$\Gamma$$ is a Heisenberg algebra of affine type, $$\mathfrak{h}_\Gamma$$, and the Hilbert schemes of points $$\mathrm{Hilb}^n(S)$$. I. Grojnowski [Math. Res. Lett. 3, No. 2, 275–291 (1996; Zbl 0879.17011)] and H. Nakajima [Ann. Math. (2) 145, No. 2, 379–388 (1997; Zbl 0915.14001)] construct a representation of the Heisenberg algebra (actually a slightly different version from the one considered in this paper) on the cohomology of the Hilbert schemes. Algebraically, I. Frenkel, N. Jing and W. Wang [Int. Math. Res. Not. 2000, No. 4, 195–222 (2000; Zbl 1011.17020)] construct the basic representation of $$\mathfrak{h}_\Gamma$$ on the Grothendieck group of the category of $$\mathbb{C}[\Gamma^n \rtimes S_n]$$-modules.
In this paper, the authors define a 2-category $$\mathcal{H}_\Gamma$$ and their first main result (3.4) states that $$\mathcal{H}_\Gamma$$ categorifies the Heisenberg algebra $$\mathfrak{h}_\Gamma$$.
The second main result of the paper (4.4) is a categorical action of $$\mathcal{H}_\Gamma$$ on a 2-category $$\bigoplus_{n\geq 0} D(A_n^\Gamma -\mathrm{gmod})$$. Here $$D(A_n^\Gamma -\mathrm{gmod})$$ denotes the bounded derived category of finite-dimensional, graded $$A_n^\Gamma$$-modules, where $A_n^\Gamma = [(\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) \otimes \ldots \otimes (\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) ] \rtimes S_n.$
As explained in Section 8, $$D(A_n^\Gamma -\mathrm{gmod})$$ is known to be equivalent to $$D\mathrm{Coh}(\mathrm{Hilb}^n(S))$$ and thus the second main theorem categorifies a representation similar to that of Grojnowski [Zbl 0879.17011] and Nakajima [Zbl 0915.14001].
In Section 9, another 2-representation of $$\mathcal{H}_\Gamma$$ is introduced that is related to the first by Koszul duality. In section 9.6, it is shown that this 2-representation categorifies an action similar to that constructed by Frenkel, Jing and Wang [Zbl 1011.17020].
For the most part geometry appears only in Section 8. The main definitions are algebraic and a number of the proofs are based on graphical calculus. Section 10 contains a description of various connections to other categorical actions and some open problems.
As the case $$\Gamma = \mathbb{Z}/2$$ differs slightly, the necessary modifications are addressed separately in a short appendix.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 19A22 Frobenius induction, Burnside and representation rings 17B99 Lie algebras and Lie superalgebras 14C05 Parametrization (Chow and Hilbert schemes)
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