$x\in\g(\R)$, $g\in\G(\R)$, $x_1\in\V(\R)$ and $x_3\in\V(\R)^\vee$, where $\langle\,,\,\rangle:\V(\R)\times\V(\R)^\vee\rightarrow\R$ is the natural pairing.

We now assume that $\k$ is a finite field and we fix a non-trivial additive character $\psi:\k\rightarrow\C^\times$ and we put $\varpi=\psi\circ\lambda:\R\rightarrow\C^\times$. Define an action of $\G(\R[\epsilon])$ on the space of functions $\C[\V(\R)\times\V(\R)^\vee]$ by

((x,g)\^[-1]{}f)(x\_1,x\_3):=(-\_(x\_1,x\_3)([Ad]{}(g\^[-1]{})(x)))f(gx\_1,gx\_3), for all $(x,g)\in\G(\R[\epsilon])$, $(x_1,x_3)\in\V(\R)\times\V(\R)^\vee$ and $f\in\C[\V(\R)\times\V(\R)^\vee]$.

We now consider the Fourier transform $\mathcal{F}:\C[\V(\R)\times\V(\R)]\rightarrow\C[\V(\R)\times\V(\R)^\vee]$ as defined in §\[Fouriergeneral\] with $X=\V(\R)$ and $\M=\V(\R)$. Namely

$$\mathcal{F}(f)(x_1,x_3)=\sum_{x_2\in\V(\R)}\varpi\left(\langle x_2,x_3\rangle\right)f(x_1,x_2),$$ for all $f\in\C[\V(\R)\times\V(\R)]$ and $(x_1,x_3)\in\V(\R)\times\V(\R)^\vee$.

$\mathcal{F}$ is an $\G(\R[\epsilon])$-equivariant isomorphism. \[mainprop\]

We have

$$\begin{aligned}
\mathcal{F}\left((x,g)^{-1}\cdot f\right)(x_1,x_3)&=\sum_{x_2}\varpi(\langle x_2,x_3\rangle)((x,g)^{-1}\cdot f)(x_1,x_2)\\
&=\sum_{x_2}\varpi(\langle x_2,x_3\rangle)f(g\cdot x_1,\varrho_\R(x)(g\cdot x_1)+g\cdot x_2)\end{aligned}$$

After the change of variables

$$x'_2=\varrho_\R(x)(g\cdot x_1)+g\cdot x_2,$$ we find

$$\begin{aligned}
\mathcal{F}\left((x,g)^{-1}\cdot f\right)(x_1,x_3)&=\sum_{x_2'}\varpi\left(\langle x'_2-\varrho_\R(x)(g\cdot x_1),g\cdot x_3\rangle\right) f(g\cdot x_1,x'_2)\\
&=\varpi\left(\langle-\varrho_\R(x)(g\cdot x_1),g\cdot x_3\rangle\right)\sum_{x'_2}\varpi(\langle x'_2,g\cdot x_3\rangle)f(g\cdot x_1,x_2')\\
&=\varpi(-\mu_\R(x_1,x_3)({\rm Ad}(g^{-1}(x))))\,\mathcal{F}(f)(g\cdot x_1,g\cdot x_3)\\
&=\left((x,g)^{-1}\cdot\mathcal{F}(f)\right)(x_1,x_3).\end{aligned}$$

The third identity follows from (\[momentfor\]).

We regard $\g(\R)$ as a subgroup of $\G(\R[\epsilon])$ from which we get an action of $\g(\R)$ on $\C[\V(\R)\times\V(\R)^\vee]$ by restricting the action . The next result relates the isotypical components of $\C[\V(\R)\times\V(\R)^\vee]$ with respect to linear characters of $\g(\R)$ and the fibers of the moment map $\mu_\R$.

First of all notice that any linear (additive) character $\g(\R)\rightarrow\C^\times$ can be written uniquely in the form $\varpi\circ \xi$ for some $\xi\in\g(\R)^\vee$. If $H$ is a 