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 [1]. 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.
Highlights
Zeta distributions or zeta integrals are studied extensively by many authors
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 [1]
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
Summary
Zeta distributions or zeta integrals are studied extensively by many authors. Most classical one is the Tate’s zeta integral: for a Schwarz function φ ∈ S (R),. Oi (φ ∈ S (VR), s ∈ C), regarded as a tempered distribution For these zeta integrals, Sato and Shintani proved a fundamental theorem which states (1) existence of a meromorphic continuation of the distribution in the parameter s ∈ C; (2) duality with respect to the Fourier transform, which is called functional equations. We will study the above two problems for a prehomogeneous vector space called an “enhanced symmetric space” associated with a double flag variety (see [1]) In this case, there are two fundamental relative invariants so that we consider a zeta integral in two complex variables s = (s1, s2).
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