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Let $G$ be a topological group, say a compact Lie group, and $e^*_G$ a multiplicative $\mathbb Z$-graded $G$-equivariant cohomology theory defined on $G$–CW complexes. Is there some analogue result to Brown representability in this context saying such a theory is represented as

$$e^n_G(X) = [X,\Sigma^n E]_G$$

for some naive $G$-equivariant ring spectrum $E$?

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