Monomial representations of a nilpotent Lie group $G$ have beenstudied successfully during the last few years by several people,including Grélaud, Corwin and Greenleaf, Fujiwara and Lipsman. They are constructed by induction, starting from a unitary character of a $G$-subgroup. Starting from a subalgebra $\mathfrak {k}$ of the complexification $\mathfrak {g}^\mathbb {C} $ of the Lie algebra $\mathfrak {g}$ of $G$, and from a form $f$ of $\mathfrak {g}^*$ such that $f([\mathfrak {k}, \mathfrak {k}]) = \{0\}$ one can also construct the associated holomorphically induced representation. This gives another way to obtain unitary representations for $G$. It generalizes the previous one, as can easily be seen. This construction was used by Auslander and Kostant in 1971, assuming that $\mathfrak {k}$ is a so-called positive polarization. Their goal was to study irreducible unitary representations of general solvable groups. Since then, no attempts seems to have been made to use this method to consider non irreducible unitary representations. This work is a first attempt to fill in this gap. Benoist's study of the monomial representation associated to the trivial character of the fixed points subgroup for an involution of $G$, which was carried out in 1985, showed it was a good starting example for studying more general monomial representations. In the same way, we study here the holomorphically induced representation $(\rho ,\mathcal {H} )$ associated to the trivial functional on the fixed points for an involution of $\mathfrak {g}^\mathbb {C}$, hoping it will give some insight of what might happen in more general instances. First, we consider the space $(\mathcal {H} ^{-\infty }_\pi )^{\mathfrak k}$ of “spherical”, or $\mathfrak {k}$-annihilated, vectors for each irreducible unitary representation $\pi $ of $G$, associated to the $G$-orbit $\Omega $ in $\mathfrak {g}^*$ by the Kirillov bijection. We prove that this space is of dimension at most one, and find a suitable cone $\Theta $ such that the equivalence $\Omega \cap \Theta \not =\emptyset \ \Longleftrightarrow \ \dim {(\mathcal {H} ^{-\infty }_\pi )^{\mathfrak k}} =1$ holds. These results imply that $\rho $ is always multiplicity free. We then prove the equivalence $ \mathcal {H} \not =\{0\} \ \Longleftrightarrow \ {\overset {\circ }\Theta }\not =\emptyset $, where $\overset {\circ }\Theta $ is the interior of $\Theta $ in the subspace $(\mathfrak {k}\cap \mathfrak {g})^\perp $ of ${\mathfrak g}^*$.