  Let $G$ be a compact connected Lie group, or more generally a path connected
topological group of the homotopy type of a finite CW-complex, and let $X$ be a
rational nilpotent $G$-space. In this paper we analyze the homotopy type of the
homotopy fixed point set $X^{hG}$, and the natural injection $k\colon
X^G\hookrightarrow X^{hG}$. We show that if $X$ is elliptic, that is, it has
finite dimensional rational homotopy and cohomology, then each path component
of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the
inclusion $k$ based on which we can prove, for instance, that for $G$ a torus,
$\pi_*(k)$ is injective in rational homotopy but, often, far from being a
rational homotopy equivalence.
