  We develop a version of Hodge theory for a large class of smooth formally
proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective
schemes. We show that the noncommutative Hodge-de Rham sequence for the
category of equivariant coherent sheaves degenerates. This spectral sequence
converges to the periodic cyclic homology, which we canonically identify with
the topological equivariant K-theory of $X$ with respect to a maximal compact
subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We
also establish Hodge-de Rham degeneration for categories of matrix
factorizations for a large class of equivariant Landau-Ginzburg models.
