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Daniel Juteau

Publications and source records attributed to Daniel Juteau.

At least 19 recordsLinked to original sources
arXiv2026

Lusztig's special pieces conjecture

Let $\mathcal O$ be a special nilpotent orbit in the Lie algebra $\mathfrak g$ of a simple algebraic group $G$. We give two proofs of the result that every special piece ${\mathcal P}(\mathcal O)$ in $\mathfrak g$ is the quotient of a smooth $G$-variety $X$ by the action of a certain finite group $H$. We first deduce the result from a similar result for transverse slices, established in earlier work of the first three authors and Fu. Then we give a more explicit construction of $X$, as a subvariety of the closure of a $G$-orbit in the direct sum of $\mathfrak g$ and some fundamental weight representations of $G$. Both methods apply to classical $\mathfrak g$, where we give new proofs of this result, which was first proved by Kraft and Procesi. The result in the exceptional groups was conjectured by Lusztig. Our first proof shows that there can be several $G$-varieties $X$ that satisfy the conjecture, related to a natural embedding of $H$ in the fundamental group of $\mathcal O$. In an appendix, we relate this natural embedding to Lusztig's definition of $H$ that arises from the family in the Weyl group of $G$ attached to $\mathcal O$ and from the Springer correspondence.

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arXiv2024

Frobenius extensions and the exotic nilCoxeter algebra for $G(m,m,3)$

In a previous paper of the first author, the type A affine Cartan matrix was q-deformed to produce a deformation of the reflection representation of the affine Weyl group. This deformation plays a role in the quantum geometric Satake equivalence. In this paper we introduce the study of q-deformed divided difference operators. When q is specialized to a primitive 2m-th root of unity, this affine reflection representation factors through a quotient, the complex reflection group $G(m,m,n)$. The divided difference operators now generate a finite-dimensional algebra we call the exotic nilCoxeter algebra. This algebra is new and has surprising features. In addition to the usual braid relations, we prove a new relation called the roundabout relation. A classic result of Demazure, for Weyl groups, states that the polynomial ring of the reflection representation is a Frobenius extension over its subring of invariant polynomials, and describes how the Frobenius trace can be constructed within the nilCoxeter algebra. We study the analogous Frobenius extension for $G(m,m,n)$, and identify the Frobenius trace within the exotic nilCoxeter algebra for $G(m,m,3)$.

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arXiv2024

A closed formula in the deformed affine nilHecke algebra

There is a q-deformation of the reflection representation of the affine symmetric group, which arises in the quantum geometric Satake equivalence, and in the study of the complex reflection groups $G(m,m,n)$. Demazure operators (often called divided difference operators) act on the polynomial ring of this deformed representation. When $n=3$ we prove an explicit closed formula for the scalar one obtains when applying a degree $-k$ Demazure operator to a monomial of degree $k$. We also prove a simpler formula for the scalar obtained after specializing q to a root of unity.

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arXiv2024

Local geometry of special pieces of nilpotent orbits

The nilpotent cone of a simple Lie algebra is partitioned into locally closed subvarieties called special pieces, each containing exactly one special orbit. Lusztig conjectured that each special piece is the quotient of some smooth variety by a precise finite group $H$, a result proved for the classical types by Kraft and Procesi. The present work is about exceptional types. Our main result is a local version of Lusztig's conjecture: the intersection of a special piece with a Slodowy slice transverse to the minimal orbit in the piece is isomorphic to the quotient of a vector space by $H$. Along the way, we complete our previous work on the generic singularities of nilpotent orbit closures, by providing proofs for the last two `exotic' singularities. Four further, non-isolated, exotic singularities are studied: we show that quotients $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{so}_8)}/\mathfrak{S}_4$, $S^2({\mathbb C}^2/μ_3)$, $S^3({\mathbb C}^2/μ_2)$ and $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{sl}_3)}/\mathfrak{S}_4$ occur as Slodowy slice singularities between nilpotent orbits in types $F_4$, $E_6$, $E_7$ and $E_8$ respectively. We also extend, to fields other than ${\mathbb C}$, the results of Brylinski and Kostant on shared orbit pairs. In the course of our analysis, we discover a shared pair which is missing from Brylinski and Kostant's classification.

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arXiv2023

Minimal special degenerations and duality

This paper includes the classification, in a simple Lie algebra, of the singularities of Slodowy slices between special nilpotent orbits that are adjacent in the partial order on nilpotent orbits. The irreducible components of most singularities are (up to normalization) either a simple surface singularity or the closure of a minimal special nilpotent orbit in a smaller rank Lie algebra. Besides those cases, there are some exceptional cases that arise as certain quotients of the closure of a minimal orbit in types $A_2$ and $D_n$. We also consider the action on the slice of the fundamental group of the smaller orbit. With this action, we observe that under Lusztig-Spaltenstein duality, in most cases, a simple surface singularity is interchanged with the closure of a minimal special orbit of Langlands dual type (or a cover of it with action). This empirical observation generalizes an observation of Kraft and Procesi in type $A_n$, where all nilpotent orbits are special. We also resolve a conjecture of Lusztig that concerns the intersection cohomology of slices between special nilpotent orbits.

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arXiv2021

A new family of isolated symplectic singularities with trivial local fundamental group

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of $\mathbb{C}^4$ by the dihedral group of order $2d$, (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order $2d$ at equal parameters, (3) as singularities of a certain Slodowy slice in the $d$-fold cover of the nilpotent cone in ${\mathfrak{sl}}_d$.

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arXiv2021

Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic $p$, and with coefficients in a field of odd characteristic $\ell\neq p$): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We also provide a quicker proof, by sorting characters according to the dimension of the corresponding Springer fiber, an invariant which is directly computable from symbols.

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arXiv2020

The localization spectral sequence in the motivic setting

We construct and study a motivic lift of a spectral sequence associated to a stratified scheme, recently discovered by Petersen in the context of mixed Hodge theory and $\ell$-adic Galois representations. The original spectral sequence expresses the compactly supported cohomology of an open stratum in terms of the compactly supported cohomology of the closures of strata and the combinatorics of the poset underlying the stratification. Some of its special cases are classical tools in the study of arrangements of subvarieties and configuration spaces. Our motivic lift lives in the triangulated category of \'{e}tale motives and takes the shape of a Postnikov system. We describe its connecting morphisms and study some of its functoriality properties.

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arXiv2016

Modular generalized Springer correspondence: an overview

This is an overview of our series of papers on the modular generalized Springer correspondence. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture `Algebraic Groups and their Representations' Workshop honouring G. Lusztig. The material that has not appeared in print before includes some discussion of the motivating idea of modular character sheaves, and heuristic remarks about geometric functors of parabolic induction and restriction.

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arXiv2016

Modular generalized Springer correspondence III: exceptional groups

We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type $G_2$.

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arXiv2016

Generic singularities of nilpotent orbit closures

According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(\mathbb C)$-variety whose normalization is ${\mathbb A}^2$, an $Sp_4(\mathbb C)$-variety whose normalization is ${\mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowy's work for the regular nilpotent orbit.

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arXiv2015

Constructible sheaves on nilpotent cones in rather good characteristic

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.

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arXiv2015

Parabolic degeneration of rational Cherednik algebras

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category O may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group.

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arXiv2015

Modular generalized Springer correspondence II: classical groups

We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-$0$ coefficients. We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for $\mathrm{SL}(n)$ with coefficients of arbitrary characteristic and for $\mathrm{SO}(n)$ and $\mathrm{Sp}(2n)$ with characteristic-$2$ coefficients.

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arXiv2014

Modular Springer correspondence, decomposition matrices and basic sets

The Springer correspondence makes a link between the characters of a Weyl group and the geometry of the nilpotent cone of the corresponding semisimple Lie algebra. In this article, we consider a modular version of the theory, and show that the decomposition numbers of a Weyl group are particular cases of decomposition numbers for equivariant perverse sheaves on the nilpotent cone. We give some decomposition numbers which can be obtained geometrically. In the case of the symmetric group, we show that James' row and column removal rule for the symmetric group can be derived from a smooth equivalence between nilpotent singularities proved by Kraft and Procesi. We give the complete structure of the Springer and Grothendieck sheaves in the case of $SL_2$. Finally, we determine explicitly the modular Springer correspondence for exceptional types.

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arXiv2014

Modular generalized Springer correspondence I: the general linear group

We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the `recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups.

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arXiv2014

Parity sheaves and tilting modules

We show that tilting modules and parity sheaves on the affine Grassmannian are related through the geometric Satake correspondence, when the characteristic is bigger than an explicit bound.

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