Special Vinberg cones, invariant admissible cubics and special real manifolds

By Vinberg theory any homogeneous convex cone $\mathcal{V}$ may be realized as the cone of positive Hermitian matrices in a $T$-algebra of generalised matrices. The level hypersurfaces ${\mathcal{V}}_q\subset \mathcal{V}$ of homogeneous cubic polynomials $q$ with positive definite Hessian (symmetric) form $g_q:=-\mathrm{Hess}(\log (q)){\mathrm{\mid }}_{T{\mathcal{V}}_q}$ are the {\it special real manifolds}. Such manifolds occur as scalar manifolds of the vector multiplets in $N=2$, $D=5$ supergravity and, through the $r$-map, correspond to Kähler scalar manifolds in $N=2$ $D=4$ supergravity. We offer a simplified exposition of the Vinberg theory in terms of $\mathrm{Nil}$-algebras (= the subalgebras of upper triangular matrices in Vinberg $T$-algebras) and we use it to describe all rational functions on a special Vinberg cone that are $G_0$- or $G^{\mathrm{\prime }}$- invariant, where $G_0$ is the unimodular subgroup of the solvable group $G$ acting simply transitively on the cone, and $G^{\mathrm{\prime }}$ is the unipotent radical of $G_0$. The results are used to determine $G_0$- and $G^{\mathrm{\prime }}$-invariant cubic polynomials $q$ that are {\it admissible} (i.e. such that the hypersurface ${\mathcal{V}}_q=\{q=1\}\cap \mathcal{V}$ has positive definite Hessian form $g_q$) for rank $2$ and rank $3$ special Vinberg cones. We get in this way examples of continuous families of non-homogeneous special real manifolds of cohomogeneity less than or equal to two.