Quivers, Invariants and Quotient Correspondence

This paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type. We first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized Gelfand-MacPherson correspondence. Next, we introduce parabolic quivers and extend the above from the actions of reductive groups to the actions of parabolic subgroups. Interestingly, the above geometry finds its natural counterparts in the representation theory as the branching rules and transfer principle in the context of the reciprocity algebra. The last half of the paper establishes this connection.