Related papers: On the Shintani zeta function for the space of bin…

We consider an ``enhanced symmetric space'', which is a prehomogeneous vector space. This vector space is intimately related to a double flag variety studied in \cite{NO.2018}. On a distinguished open orbit called ``enhanced positive…

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…

We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the…

We refine the classification of prehomogeneous vector spaces, provided by Sato-Kimura, in the case of tensor spaces, presenting a quick way to check whether a given tensor space is prehomogeneous or not.

Let $(\tilde Q,g) $ be a para-quaternionic Hermitian structure on the real vector space $V$. By referring to the tensorial presentation $(V, \tilde{Q},g) \simeq (H^2 \otimes E^{2n}, \mathfrak{sl}(H),\omega^H \otimes \omega^E)$, we give an…

We describe an approach to express the geometric side of the Arthur-Selberg trace formula in terms of zeta integrals attached to prehomogeneous vector spaces. This will provide explicit formulas for weighted orbital integrals and for the…

A functional equation between the zeta distributions can be obtained from the theory of prehomogeneous vector spaces. We show that the functional equation can be extended from the Schwartz space to certain degenerate principal series.

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 compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are…

In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the…

We construct polynomials of degree 4 that can not be obtained from prehomogeneous vector spaces, but, for which one can associate local zeta functions satisfying functional equations.

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy,…

A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level $N$ is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass forms.

We construct the so-called theta vectors on noncommutative T^4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and…

We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time…

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…

Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most…

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta…

The O(N) non-linear sigma model in a $D$-dimensional space of the form ${\bf R}^{D-M} \times {\bf T}^M$, ${\bf R}^{D-M} \times {\bf S}^M$, or ${\bf T}^M \times {\bf S}^P$ is studied, where ${\bf R}^M$, ${\bf T}^M$ and ${\bf S}^M$ correspond…

We study the behaviour of motivic zeta functions of prehomogeneous vector spaces under castling transformations. In particular we deduce how the motivic Milnor fibre and the Hodge spectrum at the origin behave under such transformations.