On dimension data and local vs. global conjugacy (Q2900297)

For a complex, reductive algebraic group \(G\), and a closed algebraic subgroup \(H\), consider the set \(\{ \dim V^H \}\) where \(V\) runs over finite-dimensional representations of \(G\). These dimension data may or may not determine \(H\) up to isomorphism.NEWLINENEWLINEFor instance, if \(H,H'\) are connected and semi-simple and have the same dimension data, then they are isomorphic as proved by \textit{M. Larsen} and \textit{R. Pink} [Invent. Math. 102, No. 2, 377--398 (1990; Zbl 0687.22004)]. In contrast, in \(GL(n)\), there exist families of non-isomorphic reductive subgroups \(H,H'\) with the same dimension data. When \(G = GL(n, \mathbb{C}), O(n, \mathbb{C}), Sp(2n, \mathbb{C})\) or \(SO(2n+1)\) and \(H,H'\) are connected, reductive and irreducible when composed with the standard representation of \(G\), and having the same dimension data, then not only are \(H,H'\) isomorphic, they are conjugate subgroups in \(G\). However, if \(H,H'\) are finite with the same dimension data in a reductive group \(G\), it turns out that the elements of \(H\) and \(H'\) are individually conjugate in \(G\) but the groups are not (globally) conjugate. For \(G=SO(2n)\), the author gives the first counterexample of connected, irreducible subgroups which have the same dimension data but are not (globally) conjugate.NEWLINENEWLINEThe author proves a more general result. The local versus global conjugacy problems play a role in multiplicity one theorems in the theory of automorphic forms. Indeed, the author proves that under the assumption of Langlands's functoriality principle in certain cases, there exist cusp forms on \(SO(2n)\) over a number field which appear with multiplicity bigger than \(1\). This follows from the author's example where local conjugacy holds but global conjugacy fails.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].