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 nil
potent 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 Slodowys work for the regular nilpotent orbit.
We complete the classification of positive rank gradings on Lie algebras of simple algebraic groups over an algebraically closed field k whose characteristic is zero or not too small, and we determine the little Weyl groups in each case. We also clas
sify the stable gradings and prove Popovs conjecture on the existence of a Kostant section.
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 thi
s 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.
We generalize the basic results of Vinbergs theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. W
e deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or good characteristic), confirming a conjecture of Popov in this case.
