finite commutative group acting on a finite dimensional $\C$-vector space $V$, and if $\chi$ is a linear character of $H$, we denote by $V_\chi$ the $\chi$-isotypical component, namely

$$V_\chi=\{f\in V\,|\, h\cdot f=\chi(h)\, f\text{ for all }h\in H\}.$$

Let $\xi\in\g(\R)^\vee$. Then $$\C[\V(\R)\times\V(\R)^\vee]_{\varpi\circ\xi}=\C[\mu_\R^{-1}(\xi)],$$ where we identify $\C[\mu_\R^{-1}(\xi)]$ with the $\C$-vector subspace of $\C[\V(\R)\times\V(\R)^\vee]$ of functions supported on $\mu_\R^{-1}(\xi)$. \[lemmaiso\]

Suppose $f\in\C[\V(\R)\times\V(\R)^\vee]_{\varpi\circ\xi}$, and let $(x_1,x_3)\in\V(\R)\times\V(\R^\vee)$. For any $x\in\g(\R)$ we have

$$(x\cdot f)(x_1,x_3)=(\varpi\circ\xi)(x)\, f(x_1,x_3).$$ But also by (\[action\]) we have

$$(x\cdot f)(x_1,x_3)=\varpi(\mu_\R(x_1,x_3)(x))\, f(x_1,x_3).$$ If $f(x_1,x_3)\neq 0$, then for all $x\in\g(\R)$ we have $$\varpi(\xi(x))=\varpi(\mu_\R(x_1,x_3)(x)),$$ i.e. $$\xi(x)-\mu_\R(x_1,x_3)(x)\in{\rm Ker}(\varpi).$$ As $\xi(x)-\mu_\R(x_1,x_3)(x)$ is $\R$-linear in $x$ the image $\{\xi(x)-\mu_\R(x_1,x_3)(x) : x\in \g(\R)\}$ is an ideal of $\R$. Since the kernel of $\varpi$ does not contain non-trivial ideals of $\R$ we must have $$\xi(x)-\mu_\R(x_1,x_3)(x)=0$$ for all $x\in\g(\R),$ that is $(x_1,x_3)\in\mu_\R^{-1}(\xi)$. We proved that $\C[\V(\R)\times\V(\R)^\vee]_{\varpi\circ\xi}\subset\C[\mu^{-1}_\R(\xi)]$ and the other inclusion is straigthforward.

From Proposition \[mainprop\] and Lemma \[lemmaiso\] we deduce

Let $\xi\in\g(\R)^\vee$. Then $\mathcal{F}$ restricts to an isomorphism of ${\rm Stab}_{\G(\R)}(\xi)$-modules

$$\C[\V(\R)\times\V(\R)]_{\varpi\circ\xi}\simeq \C[\mu_\R^{-1}(\xi)].\label{mainiso}$$

If $\xi$ is central, then

$$\#\left(\mu_\R^{-1}(\xi)/\G(\R)\right)={\rm dim}\, \C[\V(\R)\times\V(\R)]_{\varpi\circ \xi}^{\G(\R)},
\label{mainfor1}$$

where the right hand side is the dimension of the $\G(\R)$-invariant subspace of $\C[\V(\R)\times\V(\R)]_{\varpi\circ \xi}$.

In particular for $\xi=0$, we have

$$\#\left(\mu_\R^{-1}(0)/\G(\R)\right)=\# \left(\V(\R[\epsilon])/\G(\R[\epsilon])\right),
\label{mainfor}$$

\[maincoro\]

Formula (\[mainfor1\]) is 