arXiv subjects

Matthew Westaway

Publications and source records attributed to Matthew Westaway.

14 recordsLinked to original sources

arXiv2026

Quantization of nilpotent coadjoint $GL_N$-orbit closures in positive characteristics

Let $G$ be a reductive group over an algebraically closed field of positive characteristic $p$, good for the root system of $G$. The closures of $G$-orbits in the Hilbert nullcone of the coadjoint representation are conical affine Poisson varieties, generically of full rank, known as {\em nilpotent coadjoint orbits}. In this paper, we classify the filtered Hamiltonian quantizations of these orbit closures for $G = GL_N$ and any $p > 0$. Our main new technique is a construction of quantizations from certain primitive quotients of the enveloping algebra, inducing them from the stabiliser in $G$ of the Frobenius twisted $p$-character.

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arXiv2024

Two problems in the representation theory of reduced enveloping algebras

In this paper we consider two problems relating to the representation theory of Lie algebras ${\mathfrak g}$ of reductive algebraic groups $G$ over algebraically closed fields ${\mathbb K}$ of positive characteristic $p>0$. First, we consider the tensor product of two baby Verma modules $Z_{\chi}(\lambda)\otimes Z_{\chi'}(\mu)$ and show that it has a filtration of baby Verma modules of a particular form. Secondly, we consider the minimal-dimension representations of a reduced enveloping algebra $U_\chi({\mathfrak g})$ for a nilpotent $\chi\in{\mathfrak g}^{*}$. We show that under certain assumptions in type $A$ we can obtain the minimal-dimensional modules as quotients of certain modules obtained by base change from simple highest weight modules over ${\mathbb C}$.

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arXiv2023

Birational induction of nilpotent orbit covers in exceptional types

Let $G$ be a semisimple simply connected algebraic group over ${\mathbb C}$ of exceptional type. For each $G$-equivariant nilpotent cover of a nilpotent coadjoint $G$-orbit ${\mathbb O}$, we determine the unique birationally rigid induction datum from which it is birationally induced.

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arXiv2023

On induced completely prime primitive ideals in enveloping algebras of classical Lie algebras

A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ are those ideals constructed from one-dimensional representations of finite $W$-algebras. We refer to these ideals as Losev--Premet ideals. For ${\mathfrak g}$ simple of classical type, we prove that for a Losev-Premet ideal $I$ in $U({\mathfrak g})$, there exists a Losev-Premet ideal $I_0$ for a certain Levi subalgebra ${\mathfrak g}_0$ of ${\mathfrak g}$ such that associated variety of $I_0$ is the closure of a rigid nilpotent orbit in ${\mathfrak g}_0$ and $I$ is obtained from $I_0$ by parabolic induction; in turn, this gives a classification of rigid Losev-Premet ideals in $U({\mathfrak g})$. This is deduced from the corresponding statement about one-dimensional representations of finite $W$-algebras.

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arXiv2023

Tilting modules and highest weight theory for reduced enveloping algebras

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic $p>0$, and let ${\mathfrak g}$ be its Lie algebra. Given $χ\in{\mathfrak g}^{*}$ in standard Levi form, we study a category ${\mathscr C}_χ$ of graded representations of the reduced enveloping algebra $U_χ({\mathfrak g})$. Specifically, we study the effect of translation functors and wall-crossing functors on various highest-weight-theoretic objects in ${\mathscr C}_χ$, including tilting modules. We also develop the theory of canonical $Δ$-flags and $\overline{\nabla}$-sections of $Δ$-flags, in analogy with similar concepts for algebraic groups studied by Riche and Williamson.

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arXiv2022

A note on Humphreys' conjecture on blocks

Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping algebras $U_χ({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a reductive algebraic group over an algebraically closed field of characteristic $p>0$ and $χ\in{\mathfrak g}^{*}$. It is well-known to hold under Jantzen's standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result for ${\mathfrak g}$ of type $G_2$ in characteristic 3, a case in which the previously mentioned weaker assumptions do not hold. This approach requires some dimensional calculations for certain centralisers, which we conduct in the Appendix for all the exceptional Lie algebras in bad characteristic.

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arXiv2022

Higher Deformations of Lie Algebra Representations II

Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper we prove that Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

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arXiv2021

On graded representations of modular Lie algebras over commutative algebras

We develop the theory of a category ${\mathscr C}_A$ which is a generalisation to non-restricted ${\mathfrak g}$-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted ${\mathfrak g}$-modules, where ${\mathfrak g}$ is the Lie algebra of a reductive group $G$ over an algebraically closed field ${\mathbb K}$ of characteristic $p>0$. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra $U_χ$ associated to ${\mathfrak g}$ and to $χ\in{\mathfrak g}^{*}$ in standard Levi form. On the right, they are modules over a commutative Noetherian $S({\mathfrak h})$-algebra $A$, where ${\mathfrak h}$ is the Lie algebra of a maximal torus of $G$. We develop here certain important modules $Z_{A,χ}(λ)$, $Q_{A,χ}^I(λ)$ and $Q_{A,χ}(λ)$ in ${\mathscr C}_A$ which generalise familiar objects when $A={\mathbb K}$, and we prove some key structural results regarding them.

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arXiv2021

Integration of Modules II: Exponentials

We continue our exploration of various approaches to integration of representations from a Lie algebra $\mbox{Lie} (G)$ to an algebraic group $G$ in positive characteristic. In the present paper we concentrate on an approach exploiting exponentials. This approach works well for over-restricted representations, introduced in this paper, and takes no note of $G$-stability.

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arXiv2021

Centrification of Algebras and Hopf Algebras

We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.

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arXiv2020

Higher Deformations of Lie Algebra Representations I

In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.

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arXiv2019

Covering Groups of Nonconnected Topological Groups and 2-Groups

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are inverse of each other.

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arXiv2019

Integration of Modules I: Stability

We explore the integration of representations from a Lie algebra to its algebraic group in positive characteristic. An integrable module is stable under the twists by group elements. Our aim is to investigate cohomological obstructions for passing from stability to an algebraic group action. As an application, we prove integrability of bricks for a semisimple algebraic group.

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arXiv2017

$K_2$ of Kac-Moody Groups

Ulf Rehmann and Jun Morita, in their 1989 paper "A Matsumoto-type theorem for Kac-Moody groups", gave a presentation of $K_2(A,F)$ for any generalised Cartan matrix $A$ and field $F$. The purpose of this paper is to use this presentation to compute $K_2(A,F)$ more explicitly in the case when $A$ is hyperbolic. In particular, we shall show that these $K_2(A,F)$ can always be expressed as a product of quotients of $K_2(F)$ and $K_2(2,F)$. Along the way, we shall also prove a similar result in the case when $A$ has an odd entry in each column.

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