  Extending work of Klyachko and Perling, we develop a combinatorial
description of pure equivariant sheaves of any dimension on an arbitrary
nonsingular toric variety $X$. Using geometric invariant theory (GIT), this
allows us to construct explicit moduli spaces of pure equivariant sheaves on
$X$ corepresenting natural moduli functors (similar to work of Payne in the
case of equivariant vector bundles). The action of the algebraic torus on $X$
lifts to the moduli space of all Gieseker stable sheaves on $X$ and we express
its fixed point locus explicitly in terms of moduli spaces of pure equivariant
sheaves on $X$. One of the problems arising is to find an equivariant line
bundle on the side of the GIT problem, which precisely recovers Gieseker
stability. In the case of torsion free equivariant sheaves, we can always
construct such equivariant line bundles. As a by-product, we get a
combinatorial description of the fixed point locus of the moduli space of
$\mu$-stable reflexive sheaves on $X$. As an application, we show in a sequel
how these methods can be used to compute generating functions of Euler
characteristics of moduli spaces of $\mu$-stable torsion free sheaves on
nonsingular complete toric surfaces.
