write the elements of $\GL(\V)(\R[\epsilon])$ uniquely in the form $u+\epsilon v$ where $u\in\GL(\V)(\R)$ and $v\in {\rm End}_{\R-{\rm mod}}(\V(\R))$, where for $x+\epsilon y\in\V(\R[\epsilon])=\V(\R)\oplus\epsilon\V(\R)$, we have

$$(u+\epsilon v)(x+\epsilon y)=u(x)+\epsilon (u(y)+v(x)).
\label{eq1}$$

\(ii) The Lie algebra $\gl(\V)(\R)$ is then identified with $\{1+\epsilon v\,|\, v\in{\rm End}_\R(\V(\R))\}\simeq {\rm End}_\R(\V(\R))$ and the adjoint action is the conjugation action of $\GL(\V)(\R)$ on ${\rm End}_\R(\V(\R))$.

\(iii) An element $u+\epsilon v\in \GL(\V)(\R[\epsilon])$ corresponds to the element $(vu^{-1},u)\in \gl(\V)(\R)\rtimes\GL(\V)(\R)$. For $(w,g)\in \gl(\V)(\R)\rtimes\GL(\V)(\R)$ and $(x+\epsilon y)\in\V(\R[\epsilon])$, the equation $$(w,g)(x+\epsilon y)=g(x)+\epsilon \left(w(g(x))+g(y)\right),$$corresponds to (\[eq1\]) with $(w,g)=(vu^{-1},u)$.

\[GL\]

Consider the $\k$-group functor $\V^\vee$ associated with the $\k$-vector space $\V^\vee={\rm Hom}_\k(\V,\k)$. Namely, for any $\k$-algebra $\R$

$$\V^\vee(\R):= \V^\vee\otimes_\k\R\simeq\operatorname{Hom}_\k(\V,\R)\simeq {\rm Hom}_\R(\V(\R),\R)=\V(\R)^\vee.$$ The first isomorphism is [@Bour §4, Proposition 2] and the second is [@Bour §4].

Consider a homomorphism of $\k$-group schemes $\rho:\G\to\GL(\V)$ with corresponding Lie algebra homomorphism $\varrho:={\rm Lie}(
\rho):\g\to \gl(\V)$ (c.f. [@DG §4.2]).

We have the dual representation $\rho^\vee$ of $\G$ on $\V^\vee$ defined as

$$\rho^\vee_\R(g)(f)(v)=f(\rho_\R(g^{-1})(v)),$$ for all $g\in\G(\R)$, $v\in \V(\R)$ and $f\in\V^\vee(\R)$. It is equivalent to ask that $$\left\langle \rho_\R(g)(v), \rho_\R^\vee(g)(f)\right\rangle=\langle v,f\rangle,$$ for all $g,v$ and $f$ as above, where $\langle,\rangle$ denotes the natural pairing $\V(\R)\times \V^\vee(\R)\to \R$.

For a $\k$-algebra $\R$ we define the usual moment map

$$\mu_\R:\V(\R)\times \V(\R)^\vee\to \g(\R)^\vee$$

by

$$\mu_\R(v,f)=(\delta_v)^\vee(f)$$ 