$\widehat{\mathcal V}$. We shall show that this can be easily done, making use of [(\[rango\])]{}.

To make simpler the solution of the problem, let us assume that an orthonormal basis has been introduced in $\real^n$ such that the first $\nu$ elements of the basis yield a basis for the vector space $V$. Then, if we denote by $V_\bot$ the orthogonal complement of $V$ in $\real^n$, the subspace $V_\bot$ is invariant under ($K$ and) $H$, owing to the orthogonality of the transformations of $G$. Since $V$ contains all the $H$-invariant vectors of $\real^n$, there is no non trivial $H$-invariant vector in $V_\bot$.

To attain our goal, we shall start from a convenient parametrization of the principal stratum $\widehat \Sigma$ of $V/K$ in terms of $l$ real parameters $\lambda$, related to a MIB $(\lambda_1(v),\dots ,\lambda_l(v))$ of the ring of polynomial $K$-invariant functions of $v \in V$. This parametrization will be global and one-to-one, if $K$ turns out to be coregular. In this case, the range of $\lambda$ has to be restricted to the positivity region of the $\widehat P$-matrix $\widehat\Lambda(\lambda)$, associated with the MIB $\{\lambda\}$:

[$${\widehat\Lambda_{\alpha\beta}(\lambda(v)) =
\Lambda_{\alpha\beta}(v) = \sum_{i=1}^\nu\,
\partial_i \lambda_\alpha(x_1,\dots ,x_\nu)\, \partial_i \lambda_\beta(x_1,\dots ,x_\nu).}\label{4}$$]{} As recalled in the previous section, 