Related papers: On the convergence of the zeta function for certai…

Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial \delta(x) on V is called a relative…

In this note, we prove that if $(G,V)$ is a prehomogeneous vector space over any field $k$ such that the stabilizer of a generic point is reductive, the set of semi-stable points is a single orbit over the separable closure of $k$.

In this paper, we give an introduction to the rationality of the equivariant Morse stratification, and state the author's results on zeta functions of prehomogeneous vector spaces.

We prove a general convergence result for zeta functions of prehomogeneous vector spaces extending results of H. Saito, F. Sato and Yukie. Our analysis points to certain subspaces which yield boundary terms. We study it further in the setup…

Let $X$ be a real prehomogeneous vector space under a reductive group $G$, such that $X$ is an absolutely spherical $G$-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions…

In this paper, we consider the prehomogeneous vector space for pair of simple algebras which are $k$-forms of the $D_4$ type and the $E_6$ type. We mainly study the non-split cases. The main purpose of this paper is to determine the…

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group…

For absolutely simple, finite-dimensional Lie algebras g of rank at least 2, defined over a local field of characteristic 0 and admitting a graduation: g=g(-2)+g(-1)+g(0)+g(1)+g(2) given by an element H such that 2H is simple, we construct…

This is an extended version of the first part of a forthcoming paper where we will study the local Zeta functions of the minimal spherical series for the symmetric spaces arising as open orbits of the parabolic prehomogeneous spaces of…

Let $X$ be a prehomogeneous vector space under a connected reductive group $G$ over $\mathbb{R}$. Assume that the open $G$-orbit $X^+$ admits a finite covering by a symmetric space. We study certain zeta integrals involving (i) Schwartz…

We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector…

We consider a $G$-function $F(z)=\sum_{k=0}^{\infty} A_k z^k \in \mathbb{K}[[z]]$, where $\mathbb{K}$ is a number field, of radius of convergence $R$ and annihilated by the $G$-operator $L \in \mathbb{K}(z)[\mathrm{d}/\mathrm{d}z]$, and a…

In this paper we study the zeta functions associated to the minimal spherical principal series of representations for a class of reductive p-adic symmetric spaces, which are realized as open orbits of some prehomogeneous spaces. These…

Some zeta functions which are naturally attached to the locally homogeneous vector bundles over compact locally symmetric spaces of rank one are investigated. We prove that such functions can be expressed in terms of entire functions whose…

In this paper, we consider the most non-split parabolic D_4 type prehomogeneous vector space. The vector space is an analogue of the space of Hermitian forms. We determine the principal part of the zeta function.

A 'prehomogeneous vector space' is a rational representation $\rho:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. Mikio Sato showed that the hypersurface components of…

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts…

Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary…

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of…

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary,…