Hall polynomials for symplectic groups. I (Q1895547)

Let \(G\) be the symplectic group \(\text{Sp}(2m,F)\) where \(F=\overline{\mathbf F}_q\), and \(P\) a parabolic subgroup of \(G\) with Levi subgroup \(L\) isomorphic to \(\text{GL}(m,F)\). Let \(p:P\to L\) be the natural projection. Let \(u\), \(w\) be fixed unipotent elements of \(G\), \(L\) respectively. The author studies the subvariety \(X_{u,w}\) of \(G/P\) defined by \(X_{u,w}=\{xP\mid uxP=xP\) and \(p(x^{-1}ux)\) is conjugate to \(w\) in \(L\}\). Let \(\lambda\), \(\mu\) be the partitions of \(2m\) and \(m\) determined by the Jordan forms of \(u\), \(w\) respectively. The polynomial \(g^\lambda_\mu(q)\) defined as the number of \({\mathbf F}_q\)- rational points of \(X_{u,w}\) is a symplectic analogue of a Hall polynomial for \(\text{GL}(n,q)\), and is important in the character theory of \(G\). Using the symplectic geometry afforded by the natural representation of \(G\) on \(F^m\), the author computes the polynomials \(g^\lambda_\mu(q)\) and gives a closed formula for them when \(\lambda=r^d\) for some \(r\) and \(\mu\) is arbitrary.