my question is the following one:
Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy representation of some symmetric space $H/G$. Then we can associate with $V$ the so-called commuting scheme $\mathcal{C}(V)$; it is a generalization of the "usual" commuting scheme associated with the adjoint representation $\mathfrak{g}$ of $G$. For the adjoint representation $\mathfrak{g}$, it is not known whether $\mathcal{C}(\mathfrak{g})$ is always reduced. But what about $\mathcal{C}(V)$, i.e., are there any known examples where $\mathcal{C}(V)$ is not reduced? I already know that $\mathcal{C}(V)$ may be reducible (contrary to $\mathcal{C}(\mathfrak{g})$), but I found nothing concerning the possible non-reduceness.
Do you know some related literature?
Thank you in advance!
