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Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type \(B\). (English) Zbl 1286.11140
Summary: We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and \(\mathfrak{p}\)-adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class-\(2\)-nilpotent groups obtained from three infinite families of group schemes, generalizing the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type \(B\). This allows us to establish a formula for the joint distribution of three functions, or “statistics”, on such Weyl groups. Finally, we compare our explicit formulae to \(\mathfrak{p}\)-adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.

11M41 Other Dirichlet series and zeta functions
14L15 Group schemes
20G05 Representation theory for linear algebraic groups
22E50 Representations of Lie and linear algebraic groups over local fields
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