Minimal polynomials of elements of order \(p\) in irreducible representations of Chevalley groups over fields of characteristic \(p\)
Let \(K\) be an algebraically closed field of nonzero characteristic, let \(G\) be a semisimple algebraic group over \(K\), and let \(G_\mathbb{C}\) be the algebraic group of the same type over the field \(\mathbb{C}\) of complex numbers. The minimal polynomial has the form \((x-1)^k\) with some natural \(k\) for any unipotent element \(t\) and an irreducible representation \(\rho\) of the Chevalley group \(G\). So, it is defined by the number \(k=k_\rho(t)\). This number is constant on conjugacy classes. The author constructs a natural bijection \(f\) from the set of unipotent conjugacy classes of \(G\) onto the same classes of \(G_\mathbb{C}\) such that, for every unipotent conjugacy class \(c\), the equality \(k_\varphi(c)=\min\{p,k_{\varphi_\mathbb{C}}(f(c))\}\) holds.
scientific article; zbMATH DE number 1431906