We introduce new methods from -adic integration into the study of representation zeta functions associated to compact -adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of -adic analytic pro- groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact -adic analytic groups, we exhibit a link between the relevant -adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension.
Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups , where is a compact discrete valuation ring of characteristic , and of the groups , where is an unramified quadratic extension of . These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type . Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type defined over number fields.
"Representation zeta functions of compact -adic analytic groups and arithmetic groups." Duke Math. J. 162 (1) 111 - 197, 15 January 2013. https://doi.org/10.1215/00127094-1959198