On Semisimple Representations of the Framed $g$-loop Quiver
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Let $Q$ be the frame $g$-loop quiver, i.e.\ a generalized ADHM quiver obtained by replacing the two loops into $g$ loops. The vector space $\bM$ of representations of $Q$ admits an involution $*$ if orthogonal and symplectic structures on the representation spaces are endowed. We prove equivalence between semisimplicity of representations of the $*$-invariant subspace $\bN$ of $\bM$ and the orbit-closedness with respect to the natural adjoint action on $\bN$. We also explain this equivalence in terms of King's stability \cite{King} and orthogonal decomposition of representations.
Abstract
Let $Q$ be the frame $g$-loop quiver, i.e.\ a generalized ADHM quiver obtained by replacing the two loops into $g$ loops. The vector space $\bM$ of representations of $Q$ admits an involution $*$ if orthogonal and symplectic structures on the representation spaces are endowed. We prove equivalence between semisimplicity of representations of the $*$-invariant subspace $\bN$ of $\bM$ and the orbit-closedness with respect to the natural adjoint action on $\bN$. We also explain this equivalence in terms of King's stability \cite{King} and orthogonal decomposition of representations.
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