Prehomogeneous space and zeta integrals

# Prehomogeneous space and zeta integrals

## Basic definitions 
 Let $\rho: G\rightarrow GL_(V)$ is a rational representation of a complex connected algebraic group $G$ on a finite dimensional complex vector space $V$,  if $V$ has a dense $G$ orbits $\rho(G)v_{0}=\{ \rho(g)v_0\in V; g\in G \}$ with respect to the Zariski topology. Then we call this triple $(G,\rho, V)$ a prehomogeneous space. 
 
 A relative invariant of $(G, \rho, V)$ is a nonzero rational function $f(x)$ on $V$ satisfying $f(\rho(g)x)=\chi(g)f(x)$ with a rational character $\chi(g)$. We say that $f$ is a relative invariant  corresponding to the character $\chi$ 
A absolute invariant of $(G, \rho, V)$ is a nonzero rational function $f(x)$ on $V$ satisfying
 $f(\rho(g)x)=f(x)
 $. Because $f$ takes constant value in a Zariski dense subspace $\rho(g)x_0$, so $f$ is equal to a constant function on $V$. 
 For two relative invariants $f_1, f_2$ corresponding to the same character, $f_1/f_2$ is an absolute invariant, hence it is a constant and we have
 $f_2(x)=cf_1(x)$. Thus we see that

> Relative invariants of a prehomegeneous vector space corresponding to the same characters are identical up to a constant

When $G$ is a reductive algrabraic group, we call $(G,\rho,V)$ a reductive prehomogeneous space. Sato and Kimura got a classification of irreducible reductive prehomogeneous space. We will study the reductive prehomogeneous in the next section.

## Zeta integrals associates to a relative invariant
Here comes the fundamental theorem of reductive prehomogenous vector space, concerning the Fourier transform of a complex power of a relative invariant. 
Suppose that $(G,\rho, V)$ is a reductive prehomogeneous space defined over $\mathbb{R}$ and the complement $S= V /\rho(G)v_0 $  of a dense $G$ orbit $\rho(G)v_0$ is the zeros $S=\{ x\in V; f(x)=0\} $ of a irreducible polynomial $f(x)$. Then it is know that $f(x)$ is a relative invariant corresponding to a character $\chi$, and $\chi$ will satifying $\chi(g)^{m}=\det(g)^{2}$, where $m=2n/d, d=\deg f, n=\dim V$
In this case, the dual $(G, \rho^{*}, V^{*})$ is also a prehomogeneous space satisfying similar properties with a irreducible relative invariant $f^{*}(y)$ satisfying $f^{*}(\rho^{*}(g)y)=\chi^{-1}(g)f^{*}(y)$
Then $V_{\mathbb{R}}-S_{\mathbb R}$ and  $V^{*}_{\mathbb{R}}-S^{*}_{\mathbb R}$ are decomposed into the same number of connected components:
$V_{\mathbb{R}}-S_{\mathbb R}= V_1\cup V_2 \cdots \cup V_l$ and $V^{*}_{\mathbb{R}}-S^{*}_{\mathbb R}= V^{*}_1\cup V^{*}_2 \cdots \cup V^{*}_l$. For any functions $\Phi$ and $\Phi^{*}$ in $\mathcal S(V_{\mathbb{R}})$ and $\mathcal S(V^{*}_{\mathbb{R}})$,  the schwartz space on $V_{\mathbb{R}}$ and $V^{*}_{\mathbb R}$, define the zeta integrals:
$$
Z_i(\Phi,f,s)=\int_{V_i}|f(x)|^{s}\Phi(x)dx
$$
and 
$$
Z^{*}_i(\Phi^{*},f,s)=\int_{V_i}|f^{*}(y)|^{s}\Phi^{*}(y)dx
$$
Both intergals converges for $Re(s)>0$ and defines a distributions on $\mathcal S(V_{\mathbb{R}})$ and $\mathcal S(V^{*}_{\mathbb{R}})$.
For $\Phi \in \mathcal S(V_{\mathbb{R}})$, the Fourier transform of $\Phi$ is defined by $\hat{\Phi}(y)=\int_{V_{\mathbb{R}}}\Phi(x)e^{2\pi i<x,y>}dx$ for $y\in V^{*}_{\mathbb{R}}$. 
Here is the fundamental theorem

> Main Lemma:  $Z_i(\Phi,f,s)$ and $Z^{*}_i(\Phi^{*},f,s)$ can extended analytically to the whole $s$ plane. Furthermore
$$
Z^{*}_i(\hat{\Phi},f,s)=\gamma(s-n/d).\sum_{j=1}^{l}c_{ij}(s)Z_j(\Phi,f,s)
$$
where $r(s)=\Gamma(s+a_i)$ and $a_i$ are some real numbers which do not depend on $\Phi$, $c_{ij}(s)$ are entire functions which also does not depend on $\Phi$

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