Paul Levy

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.

In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq 2,3$, this nilpotent commuting variety has dimension $(r+1)\lfloor \frac{n^2}{4}\rfloor$ for $n\geq 4$, $r\geq 7$. We use this to find the dimension of the (ordinary) $r$-th commuting varieties of $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ for the same range of values of $r$ and $n$. Our principal motivation is the connection between nilpotent commuting varieties and cohomological complexity of finite group schemes, which we exploit in the last section of the paper to obtain explicit values for complexities of a large family of modules over the $r$-th Frobenius kernel $({\rm GL}_n)_{(r)}$. These results indicate an inequality between the complexities of a rational $G$-module $M$ when restricted to $G_{(r)}$ or to $G(\mathbb F_{p^r})$; we subsequently establish this inequality for every simple algebraic group $G$ defined over an algebraically closed field of good characteristic, significantly extending a result of Lin and Nakano.

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 ${\mathfrak{sl}}_d$.

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.

We introduce the notion of a generalized spin representation of the maximal compact subalgebra of a symmetrizable Kac-Moody algebra in order to show that, if defined over a formally real field, every such subalgebra has a non-trivial reductive finite-dimensional quotient. The appendix illustrates how to compute the isomorphism types of these quotients for the real $E_n$ series. In passing this provides an elementary way of determining the isomorphism types of the maximal compact subalgebras of the semisimple split real Lie algebras of types $E_6$, $E_7$, $E_8$.

Let $k$ be an algebraically closed field of characteristic not equal to 2 or 3, let $G$ be an almost simple algebraic group of type $F_4$, $G_2$ or $D_4$ and let $\theta$ be an automorphism of $G$ of finite order, coprime to the characteristic. In this paper we consider the $\theta$-group (in the sense of Vinberg) associated to these choices; we classify the positive rank automorphisms and give their Kac diagrams and we describe the little Weyl group in each case. As a result we show that all such $\theta$-groups have KW-sections, confirming a conjecture of Popov in these cases.