Suppose we have a linear representation of the group $SL_d$ over $\mathbb{C}$. i.e. a finite dimensional vector space $V$ with a linear action of $SL_d$ on it. Let $v\in V$ be some vector and let $H\subseteq SL_d$ be the stabilizer of $v$. Is it possible that $H\cong \mathbb{C}^{\times}$?
Remark: I consider the groups $SL_d$ and $\mathbb{C}^{\times}=\mathbb{G}_m$ (and the representation) as algebraic, but I think that it is equivalent to take them as Lie groups (and the representation smooth/continuous).
My best: In the adjoint representation of $SL_2$, the stabilizer of
$$ v=\left(\begin{array}{cc}1 & 0\\0 & -1\end{array}\right)\in \mathfrak{sl}_2 $$
Is the subgroup of monomial matrices in $SL_2$, which has two connected components and the one containing the identity is the group of diagonal matrices in $SL_2$ which is isomorphic to $\mathbb{G}_m$. This is the closest I could get.
I know there is a vast theory of representations of $SL_d$, so perhaps there are some known constraints on the possible isomorphism types of stabilizers. A reference to the relevant literature would be great.