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We give a general formula for the equivariant complex $K$-theory $K_G^*(V)$of a finite dimensional real linear space $V$ equipped with a linear action ofa compact group $G$ in terms of the representation theory of a certain doublecover of $G$. Using this general formula, we give explicit computations invarious interesting special cases. In particular, as an application we obtainexplicit formulas for the $K$-theory of $C_r^*(GL(n,\RR))$, the reduced groupC*-algebra of $GL(n,\RR)$.

Equivariant K-theory of finite dimensional real vector spaces