Virtual Nikolaus Conference 2020

Let \(k\) be any field. Let \(G\) be a connected reductive algebraic \(k\)-group. Associated to \(G\) is an invariant that is called the index of \(G\). Tits showed that, up to \(k\)-anisotropy, the \(k\)-isogeny class of \(G\) is uniquely determined by its index. Moreover, for the cases where \(G\) is absolutely simple, Tits classified all possibilities for the index of \(G\).

Let \(H\) be a connected reductive \(k\)-subgroup of maximal rank in \(G\). We introduce an invariant of the pair \(H < G\) called the embedding of indices of \(H < G\). This consists of the index of \(H\) and the index of \(G\) along with an embedding map that satisfies certain compatibility conditions. We show that, up to \(k\)-anisotropy, the \(G(k)\)-conjugacy class of \(H\) in \(G\) is uniquely determined by its embedding of indices. Moreover, for the cases where \(G\) is absolutely simple of exceptional type and \(H\) is maximal connected in \(G\), we classify all possibilities for the embedding of indices of \(H < G\). Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when \(k\) has cohomological dimension \(1\) (resp. \(k=R\), \(k\) is \(p\)-adic).