For any triple
$(i,a,\mu )$
consisting of a vertex
$i$
in a quiver
$Q$, a positive
integer $a$, and a
dominant
${\mathrm{GL}}_{a}$weight
$\mu $, we define a quiver current
${H}_{\mu}^{(i,a)}$ acting on the tensor power
${\Lambda}^{\phantom{\rule{0.17em}{0ex}}Q}$ of symmetric functions
over the vertices of $Q$.
These provide a quiver generalization of parabolic Garsia–Jing creation
operators in the theory of Hall–Littlewood symmetric functions. For a triple
$(i=({i}_{1},\dots ,{i}_{m}),a=({a}_{1},\dots ,{a}_{m}),(\mu (1),\dots ,\mu (m)))$
of sequences of such data, we define the quiver Hall–Littlewood function
${H}_{\mu (\cdot )}^{i,a}$ as the result
of acting on
$1\in {\Lambda}^{\phantom{\rule{0.17em}{0ex}}Q}$
by the corresponding sequence of quiver currents. The quiver
Kostka–Shoji polynomials are the expansion coefficients of
${H}_{\mu (\cdot )}^{i,a}$ in the
tensor Schur basis. These polynomials include the Kostka–Foulkes polynomials and
parabolic Kostka polynomials (Jordan quiver) and the Kostka–Shoji polynomials
(cyclic quiver) as special cases.
We show that the quiver Kostka–Shoji polynomials are graded multiplicities
in the equivariant Euler characteristic of a vector bundle (determined by
$\mu (\cdot )$)
on Lusztig’s convolution diagram determined by the sequences
$i$ and $a$.
For certain compositions of currents we conjecture higher cohomology vanishing of
the associated vector bundle on Lusztig’s convolution diagram. For quivers with no
branching, we propose an explicit positive formula for the quiver Kostka–Shoji
polynomials in terms of catabolizable multitableaux.
We also relate our constructions to
$K$theoretic
Hall algebras, by realizing the quiver Kostka–Shoji polynomials as natural structure
constants and showing that the quiver currents provide a symmetric function lifting of
the corresponding shuffle product. In the case of a cyclic quiver, we explain how the
quiver currents arise in Saito’s vertex representation of the quantum toroidal algebra of
type
${\mathfrak{\U0001d530}\mathfrak{\U0001d529}}_{r}$.
