Nikolaus Conference 2023

Let \(k\) be any field and let \(G\) be a connected reductive algebraic \(k\)-group. Associated to \(G\) is an invariant called the index of \(G\) (a Dynkin diagram along with some additional combinatorial information). Tits showed that the \(k\)-isogeny class of \(G\) is uniquely determined by its index and the \(k\)-isogeny class of its anisotropic kernel. Let \(H\) be a connected reductive \(k\)-subgroup of maximal rank in \(G\). One can define an invariant of the \(G(k)\)-conjugacy class of \(H\) in \(G\) called the embedding of indices of \(H \subset G\). Using this, one can begin to classify the maximal connected subgroups of maximal rank in \(G\) up to an invariant called "index-conjugacy" for any arbitrary field \(k\) and \(G\) absolutely simple. This has been done for absolutely simple groups of exceptional type by D. Sercombe. We will continue this work for absolutely simple groups of classical type and I will present partial results for types \(D_4\) and \(A_n\). This is a joint work with Damian Sercombe and Vanthana Ganeshalingam.