A Decomposition of Twisted Equivariant K-Theory

( )For G a finite group, a normalized 2-cocycle alpha is an element of Z(2) (G, 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 in [Gomez 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 D-2n with n >= 2 an even integer.