Rational Cherednik algebras and Hilbert schemes
The article strengthens the connections between Cherednik algebras and geometry by showing that they can be regarded as noncommutative deformations of Hilbert schemes of points in the plane. The article explicitly defines the rational Cherednik algebra \(H_c\) of type \(A_{n-1}\), and its spherical subalgebra \(U_c\). Let \(W=\mathfrak S_n\) be the symmetry group on \(n\) letters, regarded as the Weyl group of type \(A_{n-1}\) acting on its \((n-1)\)-dimensional representation \(\mathfrak h\in\mathbb{C}^n\) by permutations. Then \(H_c\) may be regarded as a deformation of the twisted group ring \(D(\mathfrak h)\ast W\), where \(D(\mathfrak h)\) is the ring of differential operators on \(\mathfrak h \) with the natural action of the symmetric group \(W=\mathfrak S_n.\) The algebra \(U_c\) is then the corresponding deformation of the fixed ring \(D(\mathfrak h)^W.\) \(U_c\) and \(H_c\) both have a natural filtration by order of differential operators, with associated graded rings \(\operatorname{gr}U_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W\) and \(\operatorname{gr}H_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]\ast W\), and so \(U_c\) may be regarded as a deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W.\) The article is mostly concerned about \(U_c\), but the authors prove that \(U_c\) and \(H_c\) are Morita equivalent. It is known that the map \(\tau:\text{Hilb}^n\mathbb{C}\rightarrow \mathbb{C}^{2n}/W\) is a resolution of singularities, and Haiman has described \(\text{Hilb}^n\) as ``Proj'' of Rees rings. This article proves that for \(\text{Hilb}(n)=\tau^{-1}(\mathfrak h\oplus\mathfrak h^\ast/W)\), \(\tau:\text{Hilb}(n)\rightarrow \mathfrak h\oplus\mathfrak h^\ast/W \) is a crepant resolution of singularities. The ring \(U_c\) has finite global homological dimension so one should expect the properties of a smooth deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W,\) that is, its properties should be more closely related to those of \(\text{Hilb}(n)\) than to \(\mathfrak h\oplus\mathfrak h^\ast/W\). The main ideal of this article is to formalize this idea by showing that there exists a second way of passing to associated graded objects that maps \(U_c\)-modules precisely to coherent modules on \(\text{Hilb}(n).\) The main result of this article gives an affirmative answer to the question of whether the following diagram can be completed: \[ \begin{tikzcd} {?} \ar[d,"\mathrm{gr}" '] & U_c \ar[l,"\sim" ']\ar[d,"\mathrm{gr}"] \\ \mathcal O_{\mathrm{Hilb}(n)} & \mathcal O(\mathfrak h\oplus\mathfrak h^\ast/W) \ar[l,"\tau" ']\rlap{\,.} \end{tikzcd} \] Given a graded ring \(R\), write \(R\)-qgr for the quotient category of noetherian graded \(R\)-modules modulo those of finite length. The main result then says that There exists a graded ring \(B\), filtered by order of differential operators, such that (1) there is an equivalence of categories \(U_c\text{-mod}\cong B\text{-qgr};\) (2) there is an equivalence of categories \(\text{gr }B\text{-qgr}\cong\text{Coh}(\text{Hilb}(n))\). The construction of \(B\) is not the same as Haiman's construction of the Rees ring, but a noncommutative deformation. This leads to the theory of \(\mathbb{Z}\)-algebras and Poincaré series in this setting. The authors then need results concerning Morita equivalence of the Cherednik algebras. The authors give explicit definitions and results about the Cherednik algebras and its representations. They study thoroughly the Morita equivalence of these algebras. Then Haiman's constructions of the Hilbert scheme and the resulting formulas for the Poincaré series are derived, and a geometric interpretation is given. Finally, \((\mathfrak h\oplus\mathfrak h^\ast)/W)\) is blown up. In the noncommutative case, the Rees rings have to be replaced by noncommutative analogs. These are called \(\mathbb{Z}\)-algebras and a nice introduction is given here. The main theorem is then proved by introducing the noncommutative Poincaré series, and this is higly nontrivial. This article is a very nice introduction to the theory and problems of noncommutative (graded) projective geometry.
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