Tangent spaces of orbit closures for representations of Dynkin quivers of type D

Abstract

Let $\Bbbk$ be an algebraically closed field, $Q$ a finite quiver, and denote by $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$ the affine $\Bbbk$-scheme of representations of $Q$ with a fixed dimension vector ${\mathbf{d}}$. Given a representation $M$ of $Q$ with dimension vector ${\mathbf{d}}$, the set ${\mathcal{O}}_M$ of points in $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ isomorphic as representations to $M$ is an orbit under an action on $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ of a product of general linear groups. The orbit ${\mathcal{O}}_M$ and its Zariski closure $\overline{{\mathcal{O}}}_M$, considered as reduced subschemes of $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$, are contained in an affine scheme ${\mathcal{C}}_M$ defined by rank conditions on suitable matrices associated to $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$. For all Dynkin and extended Dynkin quivers, the sets of points of $\overline{{\mathcal{O}}}_M$ and ${\mathcal{C}}_M$ coincide, or equivalently, $\overline{{\mathcal{O}}}_M$ is the reduced scheme associated to ${\mathcal{C}}_M$. Moreover, $\overline{{\mathcal{O}}}_M={\mathcal{C}}_M$ provided $Q$ is a Dynkin quiver of type ${\mathbb{A}}$, and this equality is a conjecture for the remaining Dynkin quivers (of type ${\mathbb{D}}$ and ${\mathbb{E}}$). Let $Q$ be a Dynkin quiver of type ${\mathbb{D}}$ and $M$ a finite dimensional representation of $Q$. We show that the equality $T_N\overline{{\mathcal{O}}}_M=T_N{\mathcal{C}}_M$ of Zariski tangent spaces holds for any closed point $N$ of $\overline{{\mathcal{O}}}_M$. As a consequence, we describe the tangent spaces to $\overline{{\mathcal{O}}}_M$ in representation theoretic terms.