Rationality of the invariant field for a class of representations of the real orthogonal groups

We give a sufficient condition for the invariant field of a regular representation of ${\mathrm{SO}}_n\left(R\right)$ to be rational. We define a length ${\lambda }^{\left(n\right)}$ on representations of ${\mathrm{SO}}_n\left(R\right)$, depending on their weights. This length is at most $\lfloor \frac{n}{2}\rfloor$. If the representation of ${\mathrm{SO}}_n\left(R\right)$ or $O_n\left(R\right)$ on $V$ contains the standard representation $R^n$ with multiplicity greater than this length, its invariant field is rational. To prove this, we construct a sequence of Seshadri slices. Each slice reduces the problem to a representation of a subgroup preserving the inequality between the multiplicity of the standard representation and the length.

We first treat the case of regular representations of positive length, which form the general case. We next emphasize the special case of representations of non-positive length, which correspond to the representations of ${\mathrm{SO}}_n\left(R\right)$ on matrices $M_{n,k}\left(R\right)$. For them, in addition to proving the rationality of the invariant field, we explicitly construct an algebraically independent set of generating invariants.