Vertex functions of type $D$ Nakajima quiver varieties

Abstract: We study the quasimap vertex functions of type $D$ Nakajima quiver varieties. When the quiver varieties have isolated torus fixed points, we compute the coefficients of the vertex functions in the $K$-theoretic fixed point basis. We also give an explicit combinatorial description of zero-dimensional type $D$ quiver varieties and their vertex functions using the combinatorics of minuscule posets. Using Macdonald polynomials, we prove that these vertex functions can be expressed as products of $q$-binomial functions, which proves a degeneration of the conjectured 3d mirror symmetry of vertex functions. We provide an interpretation of type $D$ spin vertex functions as the partition functions of the half-space Macdonald processes of Barraquand, Borodin, and Corwin. This hints that the geometry of quiver varieties may provide new examples of integrable probabilistic models.