A Decomposition of Twisted Equivariant 𝛫-Theory

Abstract

For 𝐺 a finite group, a normalized 2-cocycle α ∈ 𝛧²(𝐺, 𝕊¹) and 𝑋 a 𝐺-space on which a normal subgroup 𝐴 acts trivially, we show that the α-twisted 𝐺-equivariant 𝛫-theory of 𝑋 decomposes as a direct sum of twisted equivariant 𝛫-theories of 𝑋 parametrized by the orbits of an action of 𝐺 on the set of irreducible α-projective representations of 𝐴. This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant K-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups 𝐷₂ₙ with 𝑛 ≥ 2, an even integer.