  For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$,
Steinberg constructed the induction and restriction functors between the
category of modules over the Steinberg algebra $A_R(\mathcal{G})$ and the
category of modules over the isotropy group algebra $R\mathcal{G}_x$. In this
paper, we prove a graded version of these functors and related results for the
graded Steinberg algebra of a graded ample groupoid. As an application, the
spectral simple and graded simple modules over the Leavitt path algebra
$L_K(E)$ are classified. In particular, we show that many of previously known
simple and graded simple $L_K(E)$-modules, including the Chen simple modules,
are induced from (graded or non-graded) simple modules over isotropy group
algebras.

Rings and Algebras