[Paper Review] Rational Cherednik algebras and Hilbert schemes II: representations and sheaves

Iain J. Gordon, J. T. Stafford|ArXiv.org|Oct 12, 2004

Algebraic structures and combinatorial models17 references4 citations

TL;DR

This paper establishes a deep connection between rational Cherednik algebras and the Hilbert scheme of points in the plane via a $Δ$-functor construction that associates coherent sheaves on $̅{Hilb}(n)$ to filtered modules over the spherical Cherednik algebra $U_c$. It proves that for $c = 1/n$, the sheaf associated to the trivial module is the structure sheaf of the punctual Hilbert scheme, and confirms conjectures on bigraded structures of finite-dimensional modules matching cohomology of line bundles on the Hilbert scheme.

ABSTRACT

Let H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c=eH_ce. Then U_c is filtered by order of differential operators with associated graded ring gr U_c=C[h + h*]^W, where W is the n-th symmetric group. Using the Z-algebra construction from our earlier paper (math.RA/0407516) it is also possible to associate to a filtered H_c- or U_c-module M a coherent sheaf on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and relate it to Hilb(n) and to the resolution of singularities from Hilb(n) to h+h*/W. For example, we prove: (1) If c=1/n, so that L_c(triv) is the unique one-dimensional simple H_c-module, then L_c(triv) corresponds to the structure sheaf of the punctual Hilbert scheme. (2) If c=1/n+k (for k some natural number) then, under a canonical filtration on the finite dimensional module L_c(triv), gr eL_{c}(triv) has a natural bigraded structure which coincides with that on the global sections of certain ample line bundles on the punctual Hilbert scheme; this confirms conjectures of Berest, Etingof and Ginzburg, and relates representations of H_c and U_c with Haiman's combinatorial work on the Hilbert scheme. (3) Under mild restrictions on c, the characteristic cycle of the standard H_c-modules are described in terms of certain irreducible subvarieties of the Hilbert scheme (appearing originally in work of Grojnowski) with multiplicities given by Kostka numbers.

Motivation & Objective

• To establish a functorial correspondence between filtered modules over the spherical Cherednik algebra $U_c$ and coherent sheaves on the Hilbert scheme $\operatorname{Hilb}(n)$.
• To relate the representation theory of $H_c$ and $U_c$ to geometric invariants of $\operatorname{Hilb}(n)$, particularly its resolution of singularities $\tau: \operatorname{Hilb}(n) \to \mathfrak{h} \oplus \mathfrak{h}^* / W$.
• To verify conjectures by Berest, Etingof, and Ginzburg on the bigraded structure of $\operatorname{gr} eL_c(\operatorname{triv})$ matching $H^0(\operatorname{Z}_n, \mathcal{L}^k)$ for $c = 1/n + k$.
• To compute characteristic cycles of sheaves associated to standard modules and relate them to Kostka numbers and irreducible components of $\tau^{-1}(0)$.

Proposed method

• Utilizes the $\mathbb{Z}$-algebra construction from [GS] to associate a filtered $\mathbb{Z}$-algebra $B$ to $U_c$, with $\operatorname{gr} B \cong \bigoplus_{k \geq 0} H^0(\operatorname{Hilb}(n), \mathcal{L}^k)$.
• Constructs a sheaf $\widehat{\Phi}(N)$ on $\operatorname{Hilb}(n)$ for any filtered $H_c$-module $N$ via tensor product filtration on $eH_c \otimes_{H_c} N$.
• Applies the associated graded functor to the $\mathbb{Z}$-algebra module to obtain a graded $\operatorname{gr} B$-module, which corresponds to a coherent sheaf on $\operatorname{Hilb}(n)$.
• Employs the Morita equivalence between $H_c$ and $U_c$ via the idempotent $e$, allowing the functor to be defined on $U_c$-modules.
• Uses the bigrading on $\operatorname{gr} U_c$ induced by the Euler operator $\mathbf{E}$ and the $W$-action to analyze the structure of $\operatorname{gr} eL_c(\operatorname{triv})$.
• Applies the Conze embedding and analysis of $D(\mathfrak{h}/W)$ to prove that $D(\mathfrak{h}/W)$ is a flat left $U_c$-module, supporting the geometric interpretation.

Experimental results

Research questions

• RQ1 How can filtered modules over the spherical Cherednik algebra $U_c$ be systematically associated to coherent sheaves on the Hilbert scheme $\operatorname{Hilb}(n)$?
• RQ2 What is the geometric interpretation of the associated graded module $\operatorname{gr} eL_c(\operatorname{triv})$ for $c = 1/n$?
• RQ3 Does the bigraded structure of $\operatorname{gr} eL_c(\operatorname{triv})$ for $c = 1/n + k$ match the cohomology $H^0(\operatorname{Z}_n, \mathcal{L}^k)$ as conjectured by Berest, Etingof, and Ginzburg?
• RQ4 Can the characteristic cycle of the sheaf $\widehat{\Phi}(e\Delta_c(\mu))$ be expressed in terms of Kostka numbers and irreducible components of $\tau^{-1}(\mathfrak{h}/W)$?
• RQ5 What is the role of the $\mathbb{Z}$-algebra $B$ in realizing $U_c$-modules as sheaves on $\operatorname{Hilb}(n)$, and how does its graded structure reflect the geometry of the Hilbert scheme?

Key findings

• For $c = 1/n$, the sheaf $\widehat{\Phi}(eL_c(\operatorname{triv}))$ is isomorphic to $\mathcal{O}_{\operatorname{Z}_n}$, the structure sheaf of the punctual Hilbert scheme $\operatorname{Z}_n = \tau^{-1}(0)$.
• When $c = 1/n + k$ for $k \in \mathbb{N}$, the associated graded module $\operatorname{gr} eL_c(\operatorname{triv})$ carries a natural bigrading that matches the bigraded structure of $H^0(\operatorname{Z}_n, \mathcal{L}^k)$, confirming the conjecture of Berest, Etingof, and Ginzburg.
• Under mild conditions on $c$, the characteristic cycle of $\widehat{\Phi}(e\Delta_c(\mu))$ is $\sum_{\lambda} K_{\mu\lambda} [Z_\lambda]$, where $K_{\mu\lambda}$ are Kostka numbers and $Z_\lambda$ are irreducible components of $\tau^{-1}(\mathfrak{h}/W)$.
• The ring $D(\mathfrak{h}/W)$ is a flat left $U_c$-module, as shown via the Conze embedding and bigrading arguments on $\operatorname{gr} D(\mathfrak{h}/W)$.
• The $\mathbb{Z}$-algebra $B$ provides a noncommutative deformation of the homogeneous coordinate ring of $\operatorname{Hilb}(n)$, with $\operatorname{gr} B \cong \bigoplus_{k \geq 0} H^0(\operatorname{Hilb}(n), \mathcal{L}^k)$.
• The construction of $\widehat{\Phi}$ via tensor product filtrations on $eH_c \otimes_{H_c} N$ yields a well-defined functor from filtered $H_c$-modules to coherent sheaves on $\operatorname{Hilb}(n)$, compatible with Morita equivalence.