A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group $K={{O}_{p}}\left( C \right)\times {{O}_{q}}\left( C \right)$ acts on the space ${{M}_{p,\,q}}\,\text{of}\,p\,\times \,q$ complex matrices by $\left( a,b \right)\cdot x=ax{{b}^{-1}}$, and so does its identity component ${{K}^{0}}=\text{S}{{\text{O}}_{p}}\left( \text{C} \right)\times \text{S}{{\text{O}}_{\text{q}}}\left( \text{C} \right)$. A $K$-orbit (or ${{K}^{0}}$-orbit) in ${{M}_{p,q}}$ is said to be nilpotent if its closure contains the zero matrix. The closure, $\bar{\mathcal{O}}$, of a nilpotent $K$-orbit (resp. ${{K}^{0}}$-orbit) $\mathcal{O}$ in ${{M}_{p,q}}$ is a union of $\mathcal{O}$ and some nilpotent $K$-orbits (resp. ${{K}^{0}}$-orbits) of smaller dimensions. The description of the closure of nilpotent $K$-orbits has been known for some time, but not so for the nilpotent ${{K}^{0}}$-orbits. A conjecture describing the closure of nilpotent ${{K}^{0}}$-orbits was proposed in $[11]$ and verified when $\min \left( p,\,q \right)\le 7$. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to $\mathcal{O}$ and determination of the basic relative invariants of these spaces.

The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra $\mathfrak{s}\mathfrak{o}\left( p,\,q \right)$ under the adjoint action of the identity component of the real orthogonal group $\text{O}\left( p,\,q \right)$.

The purpose of this paper is to determine the b-functions of all the prehomogeneous vector spaces associated to nilpotent orbits in the Dynkin–Kostant theory for all the complex simple algebraic groups of exceptional type. Our method to calculate b-functions is based on the structure theorem for b-functions of several variables and the functional equations for them.

Let G be a complex linear algebraic group and ρ:G → GL(V) a finite dimensional rational representation. Assume that G is connected and reductive, and that V has an open G-orbit. Let f in C[V] be a non-zero relative invariant with character ϕ ∈ Hom (G, C$^×$), meaning that f ˆ ρ (g) =ϕ (g) f for all g in G. Choose a non-zero relative invariant f$^v$ in C[V$^v$], with character ϕ$^-1$, for the dual representation ρ$^v$:G → GL(V$^v$). Roughly, the fundamental theorem of the theory of prehomogeneous vector spaces due to M. Sato says that the Fourier transform of |f|$^s$equals |f$^v$|$^-s$ up to some factors. The purpose of the present paper is to study a finite field analogue of Sato‘s theorem and to give a completely explicit description of the Fourier transform assuming that the characteristic of the base field F$_q$ is large enough. Now |f|$^s$ is replaced by χ (f), with χ in Hom (F$_q$$^×$, C$^×$), and the factors involve Gauss sums, the Bernstein–Sato polynomial b(s) of f, and the parity of the split rank of the isotropy group at v$^v$∈ V$^v(F<math>_q$). We also express this parity in terms of the quadratic residue of the discriminant of the Hessian of log f$^v$ (v$^v$). Moreover we prove a conjecture of N. Kawanaka on the number of integer roots of b(s).