In $K$-Theory, actually also in equivariant cohomology theory, there exists a useful theorem as known Borel-Hsiang-Atiyah-Segal Localization theorem. For $K$-Theory
Theorem: Let $G$ be a compact Lie group and $X$ a locally compact $G$-space. Let $\gamma $ be a conjugacy class in $G$, and $i:X^{\gamma }\longrightarrow X$ the inclusion. Then the homomorphism $i^{\ast }:K_{G}\left( X\right) \longrightarrow K_{G}\left( X^{\gamma }\right) $ becomes an isomorphism $ \left( i^{\ast }\right) _{\gamma }:K_{G}\left( X\right) _{\gamma }\longrightarrow K_{G}\left( X^{\gamma }\right) _{\gamma } $ when localized at the prime ideal of $R\left( G\right) $ determined by $ \gamma $.
I generalized this theorem to finite dimensional compact group (non-Lie) actions for equivariant cohomology theory.
https://journals.tubitak.gov.tr/cgi/viewcontent.cgi?article=1684&context=math
Now, I'm wondering if the above theorem is true for finite dimensional compact group actions.
But, I don't know how there is a relationship between the prime ideals of the representation ring of finite dimensional compact groups and the prime ideals of the representation rings of compact Lie groups, I have no idea how difficult this question is.
Segal determined prime ideals of representation ring of a compact lie group. He said that on page one (in him article ''The representation ring of a compact Lie group'') that I confine myself to the case of a compact Lie group G: in general $R\left( G\right) =\underrightarrow{\lim }R\left( G/N\right) $, where $N$ runs through the compact normal subgroups of $G$ such that $G/N$ is a compact Lie group. But he doesn't do anything other than that.
Can you give an idea about this problem? Is it a problem worth working on? Where to start?
