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 $K\_0^G(\mathbb A^n)$?



Equivariant algebraic K-theory of affine space
In fact, just like in the case of non-equivariant algebraic $K$-theory, there is a homotopy-invariance for equivariant $K$-theory (of nonsingular varieties). Addressing your question specifically, for any action of an algebraic group $G$ on affine space, the answer is:
$$ K^G\_i({\Bbb A}^n) = R(G) \otimes K\_i(k), $$
where $k$ is the ground field and $R(G) = K^G\_0(k)$ is the representation ring. If $G$ is a (split) torus, the representation ring is just the group ring of the character group; choosing a basis $x\_1,\ldots,x\_n$ of characters identifies $R(G)$ with the Laurent polynomial ring ${\Bbb Z}[x\_1^\pm,\ldots,x\_n^\pm]$.


(The general statement is that when $f\colon X \to Y$ is a $G$-equivariant affine bundle, there is an isomorphism $$f^\*\colon G^G\_\*(Y) \to G^G\_\*(X),$$ where $G^G\_\*$ denotes equivariant $K$-theory of coherent sheaves. For nonsingular varieties $X$ one has $G^G\_\*(X) = K^G\_\*(X)$. See Theorem 4.1 of Thomason's foundational paper "Algebraic K-theory of group scheme actions".)


