  Consider an action of a connected compact Lie group on a compact complex
manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$
of rank 1. The purpose of this paper is to establish holomorphic Morse
inequalities \`{a} la Demailly for the invariant part of the Dolbeault
cohomology of tensor powers of $L$ twisted by $E$. To do so, we define a moment
map $\mu$ by the Kostant formula and we define the reduction of $M$ under a
natural hypothesis on $\mu^{-1}(0)$. Our inequalities are given in term of the
curvature of the bundle induced by $L$ on this reduction.
