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Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter XIII of May's book, Equivariant Homotopy and Equivariant Cohomology). One impetus for introducing this definition is that it allows one to properly formulate an equivariant analogue of the suspension isomorphism. Indeed, if $E_G^*$ is an $RO(G)$-graded theory with reduced counterpart $\tilde{E}_G^*$, then we have natural $\mathbb{Z}$-module isomorphisms $$\tilde{E}_G^{\alpha}(X)\cong\tilde{E}_G^{\alpha+V}(\Sigma^V(X)).$$ Here, $\alpha\in RO(G)$, $V$ is a real orthogonal $G$-module, $X$ is a $G$-space, and $\Sigma^V(X)$ is its equivariant reduced suspension with respect to $V$.

To recover an underlying $\mathbb{Z}$-grading, one identifies a non-negative integer $n$ with the trivial $G$-module $[\mathbb{R}^n]\in RO(G)$ so that $$\tilde{E}_G^n(X):=\tilde{E}_G^{[\mathbb{R}^n]}(X).$$ The usual suspension isomorphism then determines the negative grading components of $\tilde{E}_G^*$.

$\textbf{Question:}$ Suppose that $V$ is a finite-dimensional orthogonal $G$-representation. What is the relationship between $E_G^{n+V}(\text{pt})$ and $E_G^{n+\dim(V)}(\text{pt})$?

I suspect this is well-known and understood. Yet, I am having difficulty finding references that directly address this issue. I would therefore appreciate any and all advice concerning references.

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There is no relationship in general. If $E$ is a module over the $G$-equivariant $MU$ spectrum then $E_G^{n+V}(\text{pt})$ is naturally isomorphic to $E^{n+\text{dim}_{\mathbb{R}}(V)}(\text{pt})$ for all complex representations $V$. The same works for real representations if $E$ is a module over the equivariant $MO$.

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This is very helpful. Might you have a reference for this result? –  Peter Crooks Jul 24 at 19:55

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