# Enhanced zeta distributions and its functional equations

## TL;DR

This paper studies an enhanced symmetric space related to prehomogeneous vector spaces, establishing a meromorphic continuation of associated zeta integrals, and deriving explicit functional equations with gamma factors.

## Contribution

It introduces a new framework for enhanced symmetric spaces, providing explicit formulas for zeta integrals' continuation and their functional equations.

## Key findings

- Meromorphic continuation of zeta integrals with two complex variables.
- Explicit formulas for the location of poles and residues.
- Functional equations involving gamma factors.

## Abstract

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 cone'', we consider a zeta integral with two complex variables, which is analytically continued to meromorphic family of tempered distributions. One of the main results of this paper is to establish a precise formula for the meromorphic continuation which clarifies the location of poles (and may be useful to obtain residues). We also compute the Fourier transform of the zeta distribution and obtain a functional equation with explicit gamma factors.