Equivariant algebraic K-theory of affine space 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 $K_0^G(\mathbb A^n)$?
 A: 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".)
