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Let $G$ be a group and let $V$ be a complex projective representation of $G$, so that $G$ acts on the projectivization $\mathbb{P}(V)$. Is there any way to calculate the $G$-equivariant complex $K$-theory of $\mathbb{P}(V)$? If one has a genuine representation, then there is a projective bundle formula.

The specific example I have in mind is follows: one has a $p$-dimensional projective representation of $C_p \times C_p$ (e.g., given by the following two $p$-by-$p$ matrices: the cyclic permutation matrix, and the matrix with the powers of a primitive $p$th root of unity on the diagonal), which induces an action of $C_p \times C_p$ on $\mathbb{CP}^{p-1}$ without fixed points. Where can I find the equivariant $K$-theory of this space? I suspect this is written down somewhere, and I would appreciate either a reference or argument.

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    $\begingroup$ A projective representation of $G$ corresponds to a linear representation of a central extension of $G$ by a circle. Let's denote it $\tilde G$. The $G$-equivariant complex $K$-theory of ${\mathbb P}(V)$ is the same as the $\tilde G$ equivariant $K$-theory of $S(V)$ (the unit sphere of $V$). If I am not confused, this is the representation ring of $\tilde G$, quotiened by the ideal generated by $V$. In your example, $\tilde G$ is the subgroup of $U(p)$ generated by the center and your $C_p\times C_p$. So now we have to calculate the representation ring of this group. Shouldn't be too hard... $\endgroup$
    – Gregory Arone
    Aug 30, 2016 at 6:13
  • $\begingroup$ @GregoryArone: that sounds like a good idea. Thanks! $\endgroup$
    – Akhil Mathew
    Aug 30, 2016 at 22:43

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