Counting conjugacy classes in Sylow \(p\)-subgroups of Chevalley groups. (Q878673)

Let \(G\) be a connected split reductive group defined over a finite field \(K=\mathbb F_{q_0}\) and let \(G(q)\) denote the group of \(\mathbb F_q\)-rational points of \(G\) for a power \(q\) of \(q_0\). For closed \(K\)-subgroups \(M,N\) of \(G\) with \(N\) normalised by \(M\), we denote the number of \(M(q)\)-conjugacy classes in \(N(q)\) by \(k_{M,N}(q)\) and let us form the zeta function \[ \zeta_{M,N}(z)=\exp\left(\sum_{s = 1}^\infty\tfrac{k_{M,N}(q_0^s)}{s}z^s\right)\in\mathbb C[\![z]\!]. \] Let \(B\) denote a Borel subgroup in \(G\) defined over \(K\) and let \(U\) be the unipotent radical of \(B\). The aim of the paper under review is to prove that the zeta functions \(\zeta_{U,U}(z)\), \(\zeta_{B,U}(z)\) and \(\zeta_{B,B}(z)\) are rational functions in \(z\) whose numerators and denominators can be assumed to be in \(1+z\mathbb Z[z]\). The proof is geometric and is based on Dwork's theorem on the rationality of the zeta function associated to a \(K\)-variety.