The purpose of this note is to give a short derivation of the finite field analogue of Sato’s functional equation for the zeta function associated with a prehomogeneous vector space (see [16]). We restrict ourselves to the case of a regular prehomogeneous vector space, however, we allow to twist our character sums by local systems associated to arbitrary representations of the component group o...

suppose $g$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$f,$ $p=mn$ is a maximal parabolic subgroup of $g,$ $frak{n}$ is‎ ‎the lie algebra of the unipotent radical $n.$ under the adjoint‎ ‎action of its stabilizer in $m,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

We prove the properties of induced conjugacy classes, without using the original proof by Lusztig and Spaltenstein in the unipotent case, by adapting Borho’s simpler arguments for induced adjoint orbits. We study properties of equivariant fibrations of prehomogeneous affine spaces, especially the existence of relative invariants. We also detect prehomogeneous affine spaces as subquotients of ca...

We introduce the concept of a prehomogeneous determinant as a possibly nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the b-function are symmetric about −1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under whi...

In this paper, we construct a new series of prehomogeneous vector spaces from figures made up triangles, which call triangle arrangements. Our main theorem states that, under suitable assumptions, are able to space obtained arrangement by attaching two arrangements corresponding at vertex. We also give examples Many them seem be new.

In this paper, we shall construct explicitly irreducible relative invariants of two 2-simple prehomogeneous vector spaces. Together with a preprint by the same authors, this completes the list of all relative invariants of regular 2-simple prehomogeneous vector spaces of type I.

We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coinci...

For a number field K with ring of integers O K , we prove an analogue over finite rings of the form O K /P m of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where P is a big enough prime ideal of O K and m > 1. In the appendix, F. to the functional equation of L-functions of Dirichlet type associated with prehomogeneous vector spaces.