Lattices of Minimum Covolume in Chevalley Groups over Positive Characteristic Local Fields
We will show that G(IFq[t]) is a lattice of minimum covolume among the lattices in G(Tq((t-1))), if G is a simply connected classical Chevalley group and p > 7. This is analogous to the Lubotzky's result [Lu90] for the case of SIL2. Moreover, we will show that up to an automorphism of G(IFq((t-1))), lattice of minimum covolume is unique. Along the way, we would prove a quantitative version of a theorem by Kazhdan and Margulis. Namely we prove that any lattice in G = G(IFq((t-1))) can be pushed out of the 1" congruence subgroup by applying an adjoint automorphism of G, where l is the maximum coefficient appearing in the highest root of G.