arXiv subjects

J. T. Stafford

Publications and source records attributed to J. T. Stafford.

At least 19 recordsLinked to original sources

arXiv2024

Invariant holonomic systems on symmetric spaces and other polar representations

Let $V$ be a symmetric space over a connected reductive Lie algebra $G$, with Lie algebra $\mathfrak{g}$ and discriminant $δ\in \mathbb{C}[V]$. A fundamental object is the invariant holonomic system $\mathcal{G} =\mathcal{D}(V)\Big/ \Bigl(\mathcal{D}(V)\mathfrak{g}+ \mathcal{D}(V)(\mathrm{Sym}\, V)^G_+ \Bigr) $ over the ring of differential operators $\mathcal{D}(V)$. Jointly with Levasseur we have shown that there exists a surjective radial parts map $\mathrm{rad}$ from $ \mathcal{D}(V)^G$ to the spherical subalgebra $A_κ$ of a Cherednik algebra. When $A_κ$ is simple we show that $\mathcal{G}$ has no $δ$-torsion submodule nor factor module and we determine when $\mathcal{G}$ is semisimple, thereby answering questions of Sekiguchi, respectively Levasseur-Stafford. In the diagonal case when $V=\mathfrak{g}$, these results reduce to fundamental theorems of Harish-Chandra and Hotta-Kashiwara. We generalise these results to polar representations $V$ satisfying natural conditions. By twisting the radial parts map, we obtain families of invariant holonomic systems. We introduce shift functors between the different twists. We show that the image of the simple summands of $\mathcal{G} $ under these functors is described by Opdam's KZ-twist.

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arXiv2024

Quantum Hamiltonian Reduction for Polar Representations

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_κ$ from the $G$-invariant differential operators on $V$ to the spherical subalgebra $A_κ$ of a rational Cherednik algebra. Under mild hypotheses $\mathrm{rad}$ is shown to be surjective. If $V$ is a symmetric space, then $\mathrm{rad}$ is always surjective, and we determine exactly when $A_κ$ is a simple ring. When $A_κ$ is simple, we also show that the kernel of $\mathrm{rad}$ is $\left(\mathcal{D}(V)τ(\mathfrak{g}\right)^G$, where $τ:\mathfrak{g}\to \mathcal{D}(V)$ is the differential of the $G$-action.

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arXiv2023

The prime spectrum of the Drinfeld double of the Jordan plane

The Hopf algebra $\mathcal{D}$ which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime and primitive spectra are completely determined. As a corollary of this analysis it is shown that $\mathcal{D}$ satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.

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arXiv2021

Ring-theoretic blowing down II: Birational transformations

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). In a companion paper the authors described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surfaces containing an elliptic curve. Notably we show that Van den Bergh's quadrics can be obtained from the Sklyanin algebra by suitably blowing up and down, and we also provide a noncommutative analogue of the classical Cremona transform. This extends and amplifies earlier work of Presotto and Van den Bergh.

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arXiv2020

Some Noncommutative Minimal Surfaces

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra R) as well as noncommutative analogues of P^1 x P^1 and of the Hirzebruch surface F_2 (arising from Van den Bergh's quadrics R) satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following theorem. Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and let A be any connected graded noetherian maximal order containing R, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R. Secondly, let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules).

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arXiv2017

Higher symmetries of powers of the Laplacian and rings of differential operators

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the \emph{algebra of symmetries} $\mathscr{S}(\Box^r)$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of highest weight representation of $\mathfrak{g}$ via the ring of differential operators $\mathcal{D}(X)$ on the singular scheme $X=(F^r=0)\subset \mathbb{C}^n$, where $F$ is the sum of squares. In particular we prove that $ \mathscr{S}(\Box^r)\cong \mathcal{D}(X)$ is isomorphic to a primitive factor ring of $U(\mathfrak{g})$. Interestingly, if (and only if) $n$ is even with $2r\geq n$ then both $\mathcal{D}(X)$ and its natural module $\mathcal{O}(X)$ have a finite dimensional factor. These results all have real analogues, with $\Box$ replaced by the d'Alembertian on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

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arXiv2016

Ring-theoretic blowing down: I

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo's classic theorem that (-1)-lines on a smooth surface can be contracted. The resulting noncommutative blown-down algebra has pleasant properties; in particular it is always noetherian and is smooth if the original noncommutative surface is smooth. In a companion paper we will use this technique to construct explicit birational transformations between various noncommutative surfaces which contain an elliptic curve.

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arXiv2015

Noncommutative Blowups of Elliptic Algebras

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T(d) is quite rigid. Our results generalise those of the first author. In the companion paper "Classifying Orders in the Sklyanin Algebra", we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.

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arXiv2013

Classifying Orders in the Sklyanin Algebra

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem. Specifically, let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative P^2.) Let A be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S^(3n). Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S^(3n) at a (possibly non-effective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example it is automatically noetherian, indeed strongly noetherian, and has a dualizing complex.

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arXiv2013

Algebras in which every subalgebra is noetherian

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property.

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arXiv2013

The Auslander-Gorenstein property for Z-algebras

We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors.

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arXiv2008

Differential operators and Cherednik algebras

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result, [GS, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].

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arXiv2006

Naive noncommutative blowups at zero-dimensional schemes

In an earlier paper (D. S. Keeler, D. Rogalski, and J. T. Stafford, ``Naive noncommutative blowing up,'' Duke Math. J., 126 (2005), 491-546), we defined and investigated the properties of the naive blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naively blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despite the fact that the category of torsion modules in the quotient category qgr A is equivalent to the category of torsion coherent sheaves over X. These results are used in the companion paper ``A class of noncommutative projective surfaces'' to prove that a large class of noncommutative surfaces can be written as naive blowups.

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arXiv2006

A class of noncommutative projective surfaces

Let A=k+A_1+A_2.... be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A)=k(Y)[t,t^{-1},sigma], where sigma is an automorphism of the integral projective surface Y. Then we prove that A can be written as a naive blowup algebra of a projective surface X birational to Y. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in the quotient category qgr A will always be in (1-1) correspondence with the closed points of the scheme X.

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arXiv2006

Noncommutative resolutions and rational singularities

Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.

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arXiv2004

Rational Cherednik algebras and Hilbert schemes

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. We construct a filtered Z-algebra B such that, under mild conditions on c: (1) The category B-qgr of graded noetherian B-modules modulo torsion is equivalent to U_c-mod; (2) The associated graded Z-algebra gr(B) has gr(B)-qgr equivalent to Coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane. This can be regarded as saying that U_c simultaneously gives a noncommutative deformation both of (h+h*)/W and of its resolution of singularities Hilb(n) --> (h+h*)/W. As our forthcoming companion paper [GS] shows, this result is a powerful tool for studying the representation theory of H_c and its relationship to Hilb(n).

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arXiv2004

Rational Cherednik algebras and Hilbert schemes II: representations and sheaves

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.

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arXiv2004

Differential Operators and Cohomology Groups on the Basic Affine Space

We study the ring of differential operators D(X) on the basic affine space X=G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H^*(X,O_X) decomposes as a finite direct sum of non-isomorphic simple X-modules, each of which is isomorphic to a twist of O(X) by an automorphism of D(X). We also use D(X) to study the properties of D(Y) for highest weight varieties Y. For example we prove under mild hypotheses that Y is D-simple in the sense that O(Y) is a simple D(Y)-module and produce an irreducible G-module of differential operators on Y of degree -1 and specified order.

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