at.algebraic topology – How much vanishing of odd K-groups implies the vanishing of odd equivariant K-groups?

The main quetion is

For a compact Lie group $G$, and a $G$-space $X$ with $K^1(X)=0$.
How much can we say about the vanishing of $K_G^1(X)$? Moreover, how much $K^0_G(X)=K^0(X)times R(G)$?

Here $K_G^1(X)$ stands the Atiyah–Segal equivariant K-theory which is known to be periodic.
However, the similar questions can be asked for algebraic K–theory which is not peroidic.

For a reductive group $G$, and a $G$-variety $X$ with $K^{text{odd}}(X)=0$.
How much can we say about the vanishing of $K_G^{text{odd}}(X)$? Moreover, how much $K^0_G(X)=K^0(X)times R(G)$?

Actually, this assertion is used under wierd condition in Kazhdan and Lusztig’s paper Proof of the Deligne-Langlands conjecture for Hecke algebras Page 160 1.3(l). It used K-homology, which is in principle in analogy.
I also feel curious why it is necessary to take completion. Since as far as I believe taking localization is enough, since taking completion is faithfully flat.

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Note that the parallel question in equivariant cohomology is more or less known. Since $H^*(BG;mathbb{Q})$ has only even dimensiona, the Serre–Leray sepectral sequence implies that
$$H^{text{odd}}(X;mathbb{Q})=0Longrightarrow begin{cases}
H^{text{odd}}_G(X;mathbb{Q})=0\
H^*_G(X)=H^*(X)otimes H_G^*(pt).
end{cases}$$

However, for K-groups, $K_G(X)$ and $K(EGtimes_GX)$ are equal only after completion (by Atiyah completion theorem). Some “torsion part” would be cancelled in the process of completion. And the Atiyah–Hirzebruch spectral sequence lies in first and fourth quadratic, there is also no direct comparison result (still after completion).

So it would be plausible to consider different levels of vanishing of $K_G^1(X)$.

  • $K_G^1(X)$ vanishes directly. It semmes to be too ideal, however, it holds for a point.

  • $K_G^1(X)$ vanishes after tensoring with $mathbb{Q}$.

We can view $R(G)otimes mathbb{C}$ as the space of class functions, so for each conjugation class of $G$, the kernel of evaluation is a maximal ideal. Let me denote $R(G)_c$ the localization of $R(G)otimesmathbb{C}$ with respect to this maximal ideal corresponding to $c$.

  • $K_G^1(X)$ vanishes after tensoring with $R(G)_c$, for any conjugation class $c$. By a classification theorem of prime ideals of $R(G)$ due to Segal, this is equivalent to the second condition.

  • $K_G^1(X)$ vanishes after tensoring with $R(G)_1$, where $1$ is the conjugation class of unit of $G$.
    This is ture and can be proved by spectral sequence and completion assetion, since for noetherian local ring, taking completion is a faithfully flat functor.

And the same time, the different level of “equivariant $K$-formal”.

  • $K_G(X)=K(X)otimes R(G)$. This is too ideal, since maybe there would not have a natural map.

  • $K_G(X)$ is a projective $R(G)$-module.

  • $K_G(X)otimes mathbb{C}$ is a projective $R(G)otimes mathbb{C}$-module.

  • $K_G(X)otimes_{R(G)} R(G)_c$ is a projective $R(G)_c$-module for all conjugation classes $c$.

  • $K_G(X)otimes_{R(G)} R(G)_1$ is a projective $R(G)_1$-module for conjugation class $1$.