On the distribution of eigenspaces in classical groups over finite rings

Abstract

Jeffrey D. Achter establishes in [2] a connection between the distribution of class groups of function fields and the distribution of eigenspaces in symplectic groups. Gunter Malle relates this idea in [10] to the number field case. Motivated by these texts we compute here for a finite p-torsion R-module H, where R is a Dedekind ring with finite quotients and q=|R/p|, the limitPG,∞,q,f(H):=limn→∞PG,n,q,f(H):=limn→∞|{g∈Gn(R/pf)|ker(g−1)≅H}||Gn(R/pf)| for certain classical groups G of increasing dimension n. In doing so we extend the results of Jason Fulman ([4] and [5]) concerning distributions of eigenspaces over finite fields. Furthermore we give a reasonable backup for the conjecture of G. Malle (2.1 in [10]).