obtained from the isomorphism (\[mainiso\]) by taking the $\G(\R)$-invariant parts.

We can compute the dimension of $\C[\V(\R)\times\V(\R)]_{\varpi\circ\xi}$ in by taking inner product of the character $\chi_{\bar{\varrho}_{R}}$ of the representation $\bar{\varrho}_{R}$ of the finite additive group $\g(R)$ on $\C[\V(\R)\times\V(\R)]$ with the character ${\varpi\circ\xi}$. Recalling that the action $\bar{\varrho}_{\R}$ is induced by restricting to $\g(\R)\subset \G(\R[\epsilon])$, i.e. setting $g=1$, we obtain (\_)&=&\_[|\_[R]{}]{},\
& =& \_[x() ]{}([|\_[R]{}]{}(-x))( )(x)\
&=& \_[x() ]{}|(R)||(\_(x))|( )(x).Thus implies that $$\label{dimension}\dim(\C[\mu_\R^{-1}(\xi)])=|\mu_\R^{-1}(\xi)|=\frac{1}{|\g(\R)|}\sum_{x\in \g(\R)  }|\V(\R)||\ker(\varrho_\R(x))|( \varpi\circ\xi)(x).$$ When $\R=\k$ a finite field is [@hausel-kac Proposition 2], which was the count formula used to determine the Poincaré polynomials of Nakajima quiver varieties. Thus could be considered a categorification of [@hausel-kac Proposition 2].

A counter-example {#counterex}
-----------------

We give counter-examples to Formula (\[mainfor\]) when $\R$ is not a Frobenius $\k$-algebra.

Consider $\R=\F_q[t_1,\dots,t_n]/(t_1,\dots,t_n)^2$. It is not a Frobenius $\F_q$-algebra for $n>1$. We put $$\G:=\mathbb{G}_m={\rm Spec}(\F_q[t,t^{-1}]),$$ and we consider the action of $\G$ on $\mathbb{V}=\mathbb{A}^1={\rm Spec}(\F_q[t])$ by multiplication.

Then $\V(\R[\epsilon])/\G(\R[\epsilon])$ is the set of $\R[\epsilon]^\times$-orbits on $\R[\epsilon]$ for the multiplication action. Then

$$A_n(q):=\#\,\R[\epsilon]/\R[\epsilon]^\times=\frac{1}{|\R[\epsilon]^\times|}\#\{(a,b)\in\R[\epsilon]^\times\times\R[\epsilon],\, (a-1)b=0\}.$$ Write $a=a_0+A+\epsilon(a_0'+A')$ with $a_0,a_0'\in\F_q$ and $A,A'\in(t_1,\dots,t_n)$, similarly for $b$.

The equation $(a-1)b=0$ is equivalent to the following system in $\k$

$$\begin{cases}(a_0+A)(b_0+B)=(b_0+B),\\
(a_0+A)(b_0'+B')+(a_0'+A')(b_0+B)=b_0'+B'.
\end{cases}$$ We have the following cases :

\(i) If $a_0\neq 