  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 of
a compact group $G$ in terms of the representation theory of a certain double
cover of $G$. Using this general formula, we give explicit computations in
various interesting special cases. In particular, as an application we obtain
explicit formulas for the $K$-theory of $C_r^*(GL(n,\RR))$, the reduced group
C*-algebra of $GL(n,\RR)$.
