A new formalism for describing equivariant quasicoherent sheaves on toric varieties is developed. Given an equivariant quasicoherent sheaf \(\mathcal{E}\) over the toric variety \(X_{\Delta}\) associated with a fan \(\Delta\), the author considers the eigenspace decomposition \(E^{\sigma}=\bigoplus_{m}E^{\sigma}_m\) of the space of global sections \(E^{\sigma}=H^0(U_{\sigma},\mathcal{E})\), where \(U_{\sigma}\subset X\) is the affine chart associated with \(\sigma\in\Delta\). The eigenspaces \(E^{\sigma}_m\) form a directed family with respect to multiplication maps by characters \(\chi(m)\), \(m\in\sigma^{\vee}\). Conversely, any collection of directed families \(\{E^{\sigma}_m\}\) satisfying some natural compatibility conditions determines an equivariant quasicoherent sheaf \(\mathcal{E}\) over \(X_{\Delta}\). Such collections are called \(\Delta\)-families. Some finiteness conditions on the \(\Delta\)-family imply that \(\mathcal{E}\) is coherent.
If \(\mathcal{E}\) is torsion free, then the above multiplication maps are injective and induce a multifiltration of the limit space \(\mathbb{E}^{\sigma}=\varinjlim_{m\in\sigma^{\vee}}E^{\sigma}_m\). The author shows that \(\mathbb{E}^{\sigma}=\mathbb{E}\) does not depend on \(\sigma\). Thus torsion free equivariant coherent sheaves are described in terms of families of multifiltrations of finite-dimensional vector spaces. If \(\mathcal{E}\) is a vector bundle, it is determined by its restriction to the union of orbits of codimension \(\leq1\). Therefore it suffices to consider only rays \(\rho\in\Delta\). As the multiplication by \(\chi(m)\) is isomorphic on \(E^{\rho}\) for \(m\in\rho^{\perp}\), the multifiltration on \(\mathbb{E}=\mathbb{E}^{\rho}\) gives rise to a \(\mathbb{Z}\)-filtration. Thus equivariant vector bundles over \(X_{\Delta}\) are in bijection with families of filtrations of finite-dimensional vector spaces indexed by rays of \(\Delta\) and satisfying certain compatibility conditions. This language is similar to that used by A. A. Klyachko [Math. USSR, Izv. 35, No. 2, 337–375 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001–1039 (1989; Zbl 0706.14010)].
The author also describes equivariant sheaves in terms of multigraded modules over the homogeneous coordinate ring of \(X_{\Delta}\), generalizing slightly the result of V. V. Batyrev and D. A. Cox [Duke Math. J. 75, 293–338 (1994; Zbl 0851.14021)]. The above theory is applied to constructing minimal resolutions for general equivariant vector bundles of rank 2 on toric surfaces.