Unitriangular actions on quadratic forms and character degrees. (Q2568364)

Let \(G\) be a classical group defined over a finite field \(F\) of characteristic \(p\), and let \(U\) be a Sylow \(p\)-subgroup of \(G\). Much interest has recently been shown in such questions as determining the degrees of the irreducible complex characters of \(U\) and finding generating functions for the number of conjugacy classes of \(U\). In the case that \(G\) is the general linear group, Isaacs showed in 1995 that the degrees of the irreducible characters of \(U\) are powers of \(|F|\). There is also a conjecture that, for each positive integer \(n\), there is a polynomial \(f_n(x)\) with integer coefficients such that, if \(U=U_n(F)\) is the Sylow \(p\)-subgroup of the general linear group of degree \(n\) over \(F\), then the number of conjugacy classes in \(U\) is \(f_n(|F|)\), but as far as we know, this has not been proved. Building on the work of Isaacs, Previtali (the author of the work under review), Szegedy, and Sangroniz have generalized Isaacs's result to all finite classical groups \(G\) and corresponding Sylow \(p\)-subgroups \(U\), except for the symplectic and orthogonal groups in characteristic 2. For the Sylow 2-subgroups of these latter two types of groups, the analogue of Isaacs's theorem is false. Investigation of the characters and conjugacy classes of \(U\) when \(G\) is a symplectic or orthogonal group of characteristic 2 leads one to study the action of the Sylow 2-subgroup of the general linear group of degree \(n\) over \(F\) on the set of quadratic forms defined on a vector space of dimension \(n\) over \(F\). The paper under review contains a detailed analysis of this action and provides where possible canonical representatives of orbits and generating functions for orbit sizes. The information gained is used to deduce properties of the irreducible character degrees of the Sylow 2-subgroups of these classical groups in characteristic 2. Using a similar analysis of the action of the Sylow \(p\)-subgroup of the general linear group on sesquilinear forms, the author shows that when \(G\) is the unitary group \(\text{U}(2n,q^2)\), the degrees of the irreducible characters of a Sylow \(p\)-subgroup of \(G\) are the powers \(q^j\), where \(0\leq j\leq n^2-n\).