arXiv subjects

Filippo Ambrosio

Publications and source records attributed to Filippo Ambrosio.

10 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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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

Hyperplane arrangements and Vinberg's $\theta$-groups

Let $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$ be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. L\^e that the hyperplane arrangement determined by the restrictions of the roots of $\mathfrak{g}$ to a Cartan subspace $\mathfrak{c} \subset \mathfrak{g}_1$ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$.

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arXiv2024

\'Etale geometry of closures of Jordan classes

Let $G$ be a connected reductive algebraic group with simply connected derived subgroup. Over the complex numbers there exists a local method to study the geometric properties of a point $g$ in the closure of a Jordan class of $G$ in terms of Jordan classes of a maximal rank reductive subgroup $M \leq G$ depending on the point $g$, and further to the closures of certain decomposition classes in Lie($M$). We adapt this method to the case of an algebraically closed field of characteristic $p$, and we give sufficient restrictions on $p$ for it to hold.

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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

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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arXiv2022

A parametrization of sheets of conjugacy classes in bad characteristic

Let G be a simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

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arXiv2020

Local geometry of Jordan classes in semisimple algebraic groups

We prove that the closure of every Jordan class J in a semisimple simply connected complex group G at a point x with Jordan decomposition x = rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r that are contained in J and contain x in their closure. For x unipotent we also show that the closure of J around x is smoothly equivalent to the closure of a Jordan class in Lie(G) around exp^{-1}x. For G simple we apply these results in order to determine a (non-exhaustive) list of smooth sheets in G, the complete list of regular Jordan classes whose closure is normal and Cohen-Macaulay, and to prove that all sheets and Lusztig's strata in SL(n,C) are smooth.

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arXiv2020

Spherical birational sheets in reductive groups

We classify the spherical birational sheets in a complex simple simply-connected algebraic group. We use the classification to show that, when $G$ is a connected reductive complex algebraic group with simply-connected derived subgroup, two conjugacy classes $\mathcal{O}_1$, $\mathcal{O}_2$ of $G$ lie in the same birational sheet, up to a shift by a central element of $G$, if and only if the coordinate rings of $\mathcal{O}_1$ and $\mathcal{O}_2$ are isomorphic as $G$-modules. As a consequence, we prove a conjecture of Losev for the spherical subvariety of the Lie algebra of $G$.

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arXiv2020

Birational sheets in reductive groups

We define the group analogue of birational sheets, a construction performed by Losev for reductive Lie algebras. For G semisimple simply connected, we describe birational sheets in terms of Lusztig-Spaltenstein induction and we prove that they form a partition of G, and that they are unibranch varieties with smooth normalization by means of a local study.

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