Quiver Hall–Littlewood functions and Kostka–Shoji polynomials

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in quiver $Q$, positive integer $a$, and dominant $GL_a$-weight $\mu$, we define current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ symmetric functions over vertices $Q$. These provide generalization parabolic Garsia-Jing creation operators theory Hall-Littlewood functions. $(\mathbf{i},\mathbf{a},\mu(\bullet))$ sequences such data, function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as result $1\in\Lambda^Q$ by corresponding sequence currents. The Kostka-Shoji polynomials are expansion coefficients Schur basis. include Kostka-Foulkes Kostka (Jordan quiver) (cyclic special cases. We show that graded multiplicities equivariant Euler characteristic vector bundle Lusztig's convolution diagram determined $\mathbf{i},\mathbf{a}$. certain compositions currents conjecture higher cohomology vanishing associated diagram. quivers with no branching propose an explicit formula for terms catabolizable multitableaux. also relate our constructions to $K$-theoretic Hall algebras, realizing natural structure constants showing lifting shuffle product. In case cyclic quiver, explain how arise Saito's representation quantum toroidal algebra type $\mathfrak{sl}_r$.