Nikolaus Conference 2015     

Let \(q\) be a power of a prime \(p\). We denote by \(G\) a finite group of Lie type which is split over \(F_q\), and by \(U = U(G)\) a Sylow \(p\)-subgroup of \(G\). We are interested in a parametrization of the irreducible characters of \(U\), which is "uniform" over good primes \(p\). This is motivated by the determination of the generic character tables for \(U\), with a view towards the determination of decomposition numbers of \(G\). I will first present some recent results of Himstedt, Le and Magaard, that hold when the prime \(p\) is not very bad for \(G\), about the parametrization of the irreducible characters of \(U(D_4(q))\). I will then show how the ideas in this work lead to a parametrization of the irreducible characters of any \(U\), via a reduction procedure by successive subquotients of \(U\). This is obtained in a joint work with Goodwin, Le and Magaard when \(U\) is a Sylow \(p\)-subgroup of any group of Lie type of rank at most 4. In particular, for \(U(F_4(q))\), the parameterization is "uniform" over all primes \(p > 3\), where all character degrees are of the form \(q^d\) for some integer \(d\), but this does not happen for the bad prime \(p = 3\).