Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard action of $\mathbb G_m$ on $\mathbb A^n$, then the $\mathbb G_m$-equivariant K-theory of $\mathbb A^n$ can be computed using a localization sequence corresponding to the inclusion of the origin inside $\mathbb A^n$. (In the localization sequence in equivariant K-theory, one uses that the quotient of $\mathbb A^n \backslash \{0\}$ is $\mathbb P^{n-1}$ and uses the projective bundle theorem, for example, to proceed.)

How does one compute $K_i^G(\mathbb A^n)$ in other cases? Is there a general procedure for doing so? For example, take $G = \mathbb G_m \times \cdots \times \mathbb G_m$ (product of $n$ copies) and consider the *diagonal action* on $\mathbb A^n$. How does one compute $K_i^G(\mathbb A^n)$ in this case? Is it easier to write down the description of $K_0^G(\mathbb A^n)$?