Title: Graded Steinberg algebras and their representations,
Abstract: We study the category of left unital graded modules over the Steinberg
algebra of a graded ample Hausdorff groupoid. In the first part of the paper,
we show that this category is isomorphic to the category of unital left modules
over the Steinberg algebra of the skew-product groupoid arising from the
grading. To do this, we show that the Steinberg algebra of the skew product is
graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash
product of the Steinberg algebra of the underlying groupoid with the grading
group. In the second part of the paper, we study the minimal (that is,
irreducible) representations in the category of graded modules of a Steinberg
algebra, and establish a connection between the annihilator ideals of these
minimal representations, and effectiveness of the groupoid.
Specialising our results, we produce a representation of the monoid of graded
finitely generated projective modules over a Leavitt path algebra. We deduce
that the lattice of order-ideals in the $K_0$-group of the Leavitt path algebra
is isomorphic to the lattice of graded ideals of the algebra. We also
investigate the graded monoid for Kumjian--Pask algebras of row-finite
$k$-graphs with no sources. We prove that these algebras are graded von Neumann
regular rings, and record some structural consequences of this.