Orthogonal linear group-subgroup pairs with the same invariants (Q2497426)

Let \(G\) be an algebraic group acting on an irreducible variety \(X\) over a field \(k\). Then \(G\) acts on the field of rational functions \(k(X)\) of \(X\). Let \(k(X)^G\) be the subfield be the \(G\)-invariants. The triple \((G,H,X)\) is called exceptional if \(H\) is a subgroup of \(G\) and \(k(X)^H=k(X)^G\). In this article, the author classifies exceptional triples for the case that \(X=V\) is a complex finite-dimensional vector space, and \(H\) and \(G\) are both connected semisimple orthogonal groups with \(H\subset G\subset \text{GL}(V)\). This is a continuation of the author's work in [\textit{S. Solomon}, J. Lie Theory 15, 105--123 (2005; Zbl 1098.14035)], in which he classifies exceptional triples \((G,H,V)\) where \(G\) and \(H\) are both connected irreducible linear groups with \(H\subset G\subset \text{GL}(V)\). As a motivation, it is explained how the classification of exceptional triples is a first step in the classification of Gel'fand pairs.