arXiv subjects

Lewis Topley

Publications and source records attributed to Lewis Topley.

At least 19 recordsLinked to original sources

arXiv2026

Parabolic induction for modular finite $W$-algebras

We study the modules of minimal dimension for reduced enveloping algebras of Lie algebras of reductive algebraic groups using the theory of modular finite $W$-algebras. First of all we consider the case where the $p$-character lies in a unique sheet, and demonstrate that in classical cases and in most exceptional cases all minimal modules are parabolically induced from a Levi subalgebra and a rigid $p$-character. Secondly we consider the minimal modules which are invariant under twisting by the component group, showing that in classical cases and in most exceptional cases these are also parabolically induced from a Levi subalgebra and a rigid $p$-character.

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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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arXiv2026

Sheets, Jordan classes and induced orbits in the exotic and enhanced modules

Kato developed an exotic Deligne-Langlands correspondence using a geometric model for the multiparameter affine Hecke algebra of type C, based on his exotic nilpotent cone. Achar-Henderson and Springer showed that this exotic nilpotent is intimately related to another, apparently simpler variety called the enhanced nilpotent cone. Each of these is defined as the Hilbert nullcone of a polar module, the exotic Sp(2n)-module and the enhanced GL(n)-module, respectively. In this paper we conduct a detailed study of the geometry of these two modules, by introducing the Jordan stratification, simultaneously generalising classical results on the adjoint representation as well as the symmetric space associated to (gl(2n), sp(2n)). One of the key tools we develop is the theory of induced orbits in the enhanced and exotic nilpotent cones, following the work of Lusztig-Spaltenstein. Our main application is a classification of sheets in these modules, inspired by a theorem of Borho.

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arXiv2025

Shifted twisted Yangians and finite $W$-algebras of classical type

We introduce parabolic presentations of twisted Yangians of types AI and AII, interpolating between the R-matrix presentation and the Drinfeld presentation. Then we formulate and provide parabolic presentations for the shifted twisted Yangians. We define quotient algebras known as truncated shifted twisted Yangians and equip them with baby comultiplications, generalizing the work of Brundan and Kleshchev. PBW bases for all (truncated) shifted twisted Yangians of type AI and AII are established along the way. Applying the theory of universal equivariant quantizations of conic symplectic singularities we show that the truncated twisted shifted Yangian is isomorphic to the finite $W$-algebra which quantizes a suitable Slodowy slice. This provides a presentation of the finite $W$-algebra associated with every even nilpotent element in type {\sf B} and {\sf C}, as well as every nilpotent element with two Jordan blocks in type {\sf D}. Finally we make a conjecture which would supply presentations in the remaining even cases in type {\sf D}.

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arXiv2024

Shifted twisted Yangians and Slodowy slices in classical Lie algebras

In this paper we introduce the shifted twisted Yangian of type {\sf AI}, following the work of Lu--Wang--Zhang, and we study their semiclassical limits, a class of Poisson algebras. We demonstrate that they coincide with the Dirac reductions of the semiclassical shifted Yangian for $\mathfrak{gl}_n$. We deduce that these shifted twisted Yangians admit truncations which are isomorphic to Slodowy slices for many non-rectangular nilpotent elements in types {\sf B}, {\sf C}, {\sf D}. As a direct consequence we obtain parabolic presentations of the semiclassical shifted twisted Yangian, analogous to those introduced by Brundan--Kleshchev for the Yangian of type {\sf A}. Finally we give Poisson presentations of Slodowy slices for all even nilpotent elements in types {\sf B}, {\sf C}, {\sf D}, generalising the recent work of the second author.

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arXiv2024

Modular representations of the Yangian $Y_2$

Let $Y_2$ be the Yangian associated to the general linear Lie algebra $\mathfrak{gl}_2$, defined over an algebraically closed field $\mathbbm{k}$ of characteristic $p > 0$. In this paper, we study the representation theory of the restricted Yangian $Y^{[p]}_2$. This leads to a description of the representations of $\mathfrak{gl}_{2n}$, whose $p$-character is nilpotent with Jordan type given by a two-row partition $(n, n)$.

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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

Modular Representations of Truncated current Lie algebras

In this paper we consider the structure and representation theory of truncated current algebras $\mathfrak{g}_m = \mathfrak{g}[t]/(t^{m+1})$ associated to the Lie algebra $\mathfrak{g}$ of a standard reductive group over a field of positive characteristic. We classify semisimple and nilpotent elements and describe their associated support varieties. Next, we prove various Morita equivalences for reduced enveloping algebras, including a reduction to nilpotent $p$-characters, analogous to a famous theorem of Friedlander--Parshall. We go on to give precise upper bounds for the dimensions of simple modules for all $p$-characters, and give lower bounds on these dimensions for homogeneous $p$-characters. We then develop the theory of baby Verma modules for homogeneous $p$-characters and, whenever the $p$-character has standard Levi type, we give a full classification of the simple modules. In particular we classify all simple modules with homogeneous $p$-characters for $\mathfrak{g}_m$ when $\mathfrak{g} = \mathfrak{gl}_n$. Finally, we compute the Cartan invariants for the restricted enveloping algebra $U_0(\mathfrak{g}_m)$ and show that they can be described by precise formulae depending on decomposition numbers for $U_0(\mathfrak{g})$.

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arXiv2023

Equivariant deformation theory for nilpotent slices in symplectic Lie algebras

The Slodowy slice is a flat Poisson deformation of its nilpotent part, and it was demonstrated by Lehn-Namikawa-Sorger that there is an interesting infinite family of nilpotent orbits in symplectic Lie algebras for which the slice is not the universal Poisson deformation of its nilpotent part. This family corresponds to slices to nilpotent orbits in symplectic Lie algebras whose Jordan normal form has two blocks. We show that the nilpotent Slodowy varieties associated to these orbits are isomorphic as Poisson $\mathbb{C}^\times$-varieties to nilpotent Slodowy varieties in type D. It follows that the universal Poisson deformation in type C is a slice in type D. When both Jordan blocks have odd size the underlying singularity is equipped with a $\mathbb{Z}_2$-symmetry coming from the type D realisation. We prove that the Slodowy slice in type C is the $\mathbb{Z}_2$-equivariant universal Poisson deformation of its nilpotent part. This result also has non-commutative counterpart, identifying the finite W-algebra as the universal equivariant quantization.

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arXiv2023

The centre of the modular affine vertex algebra

The Feigin--Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W.~Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the $p$-centre. They conjectured that at the critical level the centre should be generated by the Feigin--Frenkel centre and the $p$-centre. In this paper we prove the conjecture for classical simple Lie algebras for $p$ larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the $p$-centre.

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arXiv2023

Category $\mathcal{O}$ for truncated current Lie algebras

In this paper we study an analogue of the Bernstein--Gelfand--Gelfand category $\mathcal{O}$ for truncated current Lie algebras $\mathfrak{g}_n$ attached to a complex semisimple Lie algebra. This category admits Verma modules and simple modules, each parametrised by the dual space of the truncated currents on a choice of Cartan subalgebra in $\mathfrak{g}$. Our main result describes an inductive procedure for computing composition multiplicities of simples inside Vermas for $\mathfrak{g}_n$, in terms of similar composition multiplicities for $\mathfrak{l}_{n-1}$ where $\mathfrak{l}$ is a Levi subalgebra. As a consequence, these numbers are expressed as integral linear combinations of Kazhdan--Lusztig polynomials evaluated at 1. This generalises recent work of the first author, where the case $n = 1$ was treated.

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arXiv2021

Parabolic induction for Springer fibres

Let $G$ be a reductive group satisfying the standard hypotheses, with Lie algebra $\mathfrak{g}$. For each nilpotent orbit $\mathcal{O}_0$ in a Levi subalgebra $\mathfrak{g}_0$ we can consider the induced orbit $\mathcal{O}$ defined by Lusztig and Spaltenstein. We observe that there is a natural closed morphism of relative dimension zero from the Springer fibre over a point of $\mathcal{O}_0$ to the Springer fibre over $\mathcal{O}$, which induces an injection on the level of irreducible components. When $G = \operatorname{GL}_N$ the components of Springer fibres was classified by Spaltenstein using standard tableaux. Our main results explains how the Lusztig--Spaltenstein map of Springer fibres can be described combinatorially, using a new associative composition rule for standard tableaux which we call stacking.

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arXiv2021

Modular representations of Lie algebras of reductive groups and Humphreys' conjecture

Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate Ad$(G)$-invariant symmetric bilinear form. Given a linear function $χ$ on $\mathfrak{g}$ we denote by $U_χ(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $χ$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_χ(\mathfrak{g})$-module has dimension divisible by $p^{d(χ)}$ where $2d(χ)$ is the dimension of the coadjoint $G$-orbit containing $χ$. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra $U_χ(\mathfrak{g})$ admits a module of dimension $p^{d(χ)}$.

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arXiv2021

One dimensional representations of finite $W$-algebras, Dirac reduction and the orbit method

In this paper we study the variety of one dimensional representations of a finite $W$-algebra attached to a classical Lie algebra, giving a precise description of the dimensions of the irreducible components. We apply this to prove a conjecture of Losev describing the image of his orbit method map. In order to do so we first establish new Yangian-type presentations of semiclassical limits of the $W$-algebras attached to distinguished nilpotent elements in classical Lie algebras, using Dirac reduction.

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arXiv2020

Universal filtered quantizations of nilpotent Slodowy slices

Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and a universal equivariant quantization with respect to a reductive group acting on it by $\mathbb{C}^\times$-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First we give a complete description of the cases in which the finite $W$-algebra is a universal filtered quantization of the slice, building on the work of Lehn--Namikawa--Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply-laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type we prove that the finite $W$-algebra is a universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowy's theorem. Finally we apply this result to give a presentation of the subregular finite $W$-algebra in type B as a quotient of a shifted Yangian.

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arXiv2019

The Orbit Method for Poisson Orders

A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of $\operatorname{Spec}(Z)$. In this article we consider a Poisson order $A$ over a complex affine Poisson algebra $Z$. We stratify the primitive spectrum $\operatorname{Prim}(A)$ into symplectic cores, which should be thought of as families of non-commutative symplectic leaves. We then introduce a category $A$-$\mathcal{P}$-Mod of $A$-modules adapted to the Poisson structure on $Z$, and we show that when $\operatorname{Spec}(Z)$ is smooth with locally closed symplectic leaves, there is a natural homeomorphism from the spectrum of annihilators of simple objects in $A$-$\mathcal{P}$-Mod to the set of symplectic cores in $\operatorname{Prim}(A)$ with its quotient topology. Several application are given to Poisson representation theory. Our main tool is the Poisson enveloping algebra $A^e$ of a Poisson order $A$, which captures the Poisson representation theory of $A$. For $Z$ regular and affine we prove a PBW theorem for $A^e$ and use this to characterise the annihilators of simple Poisson modules: they coincide with the Poisson weakly locally closed, the Poisson primitive and the Poisson rational ideals. We view this as a generalised weak Poisson Dixmier--Moeglin equivalence.

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arXiv2019

Minimal dimensional representations of reduced enveloping algebras for $\mathfrak{gl}_n$

Let $\mathfrak g = \mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N \in \mathbb Z_{\ge 1}$. Let $χ\in \mathfrak g^*$ and denote by $U_χ(\mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_χ(\mathfrak g)$-module has dimension divisible by $p^{d_χ}$, where $d_χ$ is half the dimension of the coadjoint orbit of $χ$. Our main theorem gives a classification of $U_χ(\mathfrak g)$-modules of dimension $p^{d_χ}$. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for $U_0(\mathfrak h)$ for a certain Levi subalgebra $\mathfrak h$ of $\mathfrak g$; we view this as a modular analogue of Mœglin's theorem on completely primitive ideals in $U(\mathfrak{gl}_N(\mathbb C))$. To obtain these results, we reduce to the case $χ$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra.

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arXiv2019

Restricted shifted Yangians and restricted finite $W$-algebras

We study the truncated shifted Yangian $Y_{n,l}(σ)$ over an algebraically closed field $\mathbb{k}$ of characteristic $p > 0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g}, e)$ associated to a corresponding nilpotent element $e\in \mathfrak{g} = \mathfrak{gl}_N(\mathbb{k})$. We obtain an explicit description of the centre of $Y_{n,l}(σ)$, showing that it is generated by its Harish-Chandra centre and its $p$-centre. We define $Y_{n,l}^{[p]}(σ)$ to be the quotient of $Y_{n,l}(σ)$ by the ideal generated by the kernel of trivial character of its $p$-centre. Our main theorem states that $Y_{n,l}^{[p]}(σ)$ is isomorphic to the restricted finite $W$-algebra $U^{[p]}(\mathfrak{g},e)$. As a consequence we obtain an explicit presentation of this restricted $W$-algebra.

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