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Eric Sommers

Publications and source records attributed to Eric Sommers.

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

Automorphisms and deformations of regular semisimple Hessenberg varieties

We show that regular semisimple Hessenberg varieties can have moduli. To be precise, suppose $X$ is a regular semisimple Hessenberg variety of codimension $1$ in the flag variety $G/B$, where $G$ is a simple algebraic group of rank $r$ over $\mathbb{C}$ and $B$ is a Borel subgroup. We show that the space~$\mathrm{H}^1(X,TX)$ of first order deformations of $X$ has dimension $r-1$ except in type $A_2$. (In type $A_2$, the Hessenberg varieties in question are all isomorphic to the permutohedral toric surface, and $\dim\mathrm{H}^1(X,TX) = 0$.) Moreover, we show that the Kodaira--Spencer map $\mathfrak{g}\to \mathrm{H}^1(X,TX)$ is onto, that the identity component of the automorphism group of $X$ is a maximal torus of $G$, and that $\mathrm{H}^i(X,TX) = 0$ for $i \geq 2$. Along the way, we prove several theorems of independent interest about the cohomology of homogeneous vector bundles on~$G/B$. In type $A$, we can give an even more precise statement determining when two codimension $1$ regular semisimple Hessenberg varieties in $G/B$ are isomorphic. We also compute the automorphism groups explicitly in type~$A_{n-1}$ in the terms of stabilizer subgroups of the action of the symmetric group $S_{n}$ on the moduli space $M_{0,n+1}$ of smooth genus $0$ curves with $n + 1$ marked points. Using this, we describe the moduli stack of the regular semisimple Hessenberg varieties $X$ explicitly as a quotient stack of $M_{0,n+1}$. We prove several analogous results for Hessenberg varieties in generalized flag varieties $G/P$, where $P$ is a parabolic subgroup of $G$. In type $A$, these results are used in the proofs of the results for $G/B$, but they are also of independent interest because the associated moduli stacks are related directly to the action of $S_n$ on $M_{0,n}$.

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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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arXiv2023

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 $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{so}_8)}/\mathfrak{S}_4$, $S^2({\mathbb C}^2/\mu_3)$, $S^3({\mathbb C}^2/\mu_2)$ 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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arXiv2022

Perverse sheaves, nilpotent Hessenberg varieties, and the modular law

We consider generalizations of the Springer resolution of the nilpotent cone of a simple Lie algebra by replacing the cotangent bundle with certain other vector bundles over the flag variety. We show that the analogue of the Springer sheaf has as direct summands only intersection cohomology sheaves that arise in the Springer correspondence. The fibers of these general maps are nilpotent Hessenberg varieties, and we build on techniques established by De Concini, Lusztig, and Procesi to study their geometry. For example, we show that these fibers have vanishing cohomology in odd degrees. This leads to several implications for the dual picture, where we consider maps that generalize the Grothendieck-Springer resolution of the whole Lie algebra. In particular we are able to prove a conjecture of Brosnan. As we vary the maps, the cohomology of the corresponding nilpotent Hessenberg varieties often satisfy a relation we call the geometric modular law, which also has origins in the work on De Concini, Lusztig, and Procesi. We connect this relation in type $A$ with a combinatorial modular law defined by Guay-Paquet that is satisfied by certain symmetric functions and deduce some consequences of that connection.

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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

A transitivity result for ad-nilpotent ideals in type A

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal meets a unique largest nilpotent orbit in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals with the same largest nilpotent orbit. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan-Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.

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arXiv2019

Invariant theory for coincidental complex reflection groups

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family $G(d,1,n)$. We highlight consequences for the $q$-Narayana and $q$-Kirkman polynomials, giving simple product formulas for both, and give a $q$-analogue of the identity transforming the $h$-vector to the $f$-vector for the coincidental finite type cluster/Cambrian complexes of Fomin--Zelevinsky and Reading.

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arXiv2018

Equations for some nilpotent varieties

Let $\mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ over $\mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{ \mathcal{O}}$ in $S \mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broer's result for the subregular nilpotent orbit. Along the way we give another proof of Broer's result that $\overline{ \mathcal{O}}$ is normal. We also prove a result connecting a property of invariants related to flat bases to the question of when one copy of the adjoint representation is in the ideal in $S \mathfrak{g}^*$ generated by another copy of the adjoint representation and the fundamental invariants.

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arXiv2016

Irreducible local systems on nilpotent orbits

Let G be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $\mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit O through a nilpotent element $e \in \mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of G associated to e. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis. In the last section of the article, we state two applications of this result, whose details will appear elsewhere: to answering a question of Lusztig regarding special pieces in the exceptional groups (joint work with Fu, Juteau, and Levy); and to computing the G-module structure of the sections of an irreducible local system on O. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author.

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arXiv2016

Weyl group $q$-Kreweras numbers and cyclic sieving

The paper concerns a definition for $q$-Kreweras numbers for finite Weyl groups $W$, refining the $q$-Catalan numbers for $W$, and arising from work of the second author. We give explicit formulas in all types for the $q$-Kreweras numbers. In the classical types $A, B, C$, we also record formulas for the $q$-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types $B$ and $C$). In addition we verify that in the classical types $A,B,C,D$ that the $q$-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

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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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arXiv2011

Exterior powers of the reflection representation in Springer theory

We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shoji's approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using a result from rational Cherednik algebras.

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arXiv2010

Pieces of nilpotent cones for classical groups

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.

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arXiv2004

Exponents for B-stable ideals

Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers $m_1, m_2, ..., m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A_n, B_n, C_n and some other types. When I is zero, we recover the usual exponents of G by Kostant and one of our conjectures reduces to a well-known factorization of the Poincare polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

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arXiv2003

B-stable ideals in the nilradical of a Borel subalgebra

We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also compute the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

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arXiv2003

Normality of very even nilpotent varieties in D_2l

For the classical groups, Kraft and Procesi have resolved the question of which nilpotent orbits have closures which are normal and which are not, with the exception of the very even orbits in $D_{2l}$ which have partition of the form $(a^{2k}, b^2)$ for $a, b$ distinct even natural numbers with $a k + b = 2 l$. In this article, we show that these orbits do have normal closure.

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