A Decomposition of Twisted Equivariant K-Theory

For $G$ a finite group, a normalized 2-cocycle $\alpha\in Z^{2}(G,\mathbb{S}^{1})$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $\alpha$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $\alpha$-projective representations of $A$. This generalizes the decomposition obtained by Gomez and Uribe for equivariant $K$-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups $D_{2n}$ with $n\ge 1$ an even integer.