Intersection of conjugacy classes with Bruhat cells in Chevalley groups: the cases \(\text{SL}_n(K)\), \(\text{GL}_n(K)\). (Q872178)

Let \(K\) be a field and let \(G\) be either \(\text{SL}_n(K)\) or \(\text{GL}_n(K)\). Let \(C\) be a conjugacy class of \(G\). The aim is to give conditions under which \(C\) intersects a given Bruhat cell \(B\dot w B\) of \(G\). Two cases are treated: The conjugacy class of a semisimple element \(g\) and the conjugacy class of an element with eigenvalues in \(K\). First one reduces to the case that \(w\) is a product of simple reflections with each simple reflection occurring at most once. For the semisimple case a combinatorial criterion is given in terms of the degrees and multiplicities of irreducible factors of the minimal polynomial of \(g\), and the cycle structure of \(w\). In the unipotent case a similar criterion is given involving the sizes of the Jordan blocks of \(g\).