Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

Abstract

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. REPRESENTATION ZETA FUNCTIONS OF NILPOTENT GROUPS AND GENERATING FUNCTIONS FOR WEYL GROUPS OF TYPE BBy A. STASINSKI and C. VOLL Abstract. 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 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 p-adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.

Odd length for even hyperoctahedral groups and signed generating functions

“…In the next section we recall some definitions, notation, and results that are used in the sequel. In §3 we define a new statistic on the even hyperoctahedral group which is a natural analogue of the odd length statistics that have already been defined in types A and B in [9] and [18], and study some general properties of the corresponding signed generating functions. These include a complementation property, the identification of subsets of the quotients over which the corresponding signed generating function always vanishes, and operations on a quotient that leave the corresponding signed generating function unchanged.…”

The average size of the kernel of a matrix and orbits of linear groups

“…We note that permutation statistics have previously featured in explicit formulae for representation zeta functions ; see also .…”

“…For further positive results establishing such functional equations, see, in particular, work of du Sautoy and Lubotzky , Voll , Avni et al . [, § 4] and Stasinski and Voll [, Theorem A]. Using the formalism developed above, we may deduce the following; recall the notation from § 3.3.…”

“…[, § 3.2]; cf. . The average sizes of the kernels of all these spaces of matrices are known or can be deduced as by‐products or our investigations in § 5.3.…”

“…1) for the numbers of traceless, antisymmetric and symmetric d × d matrices over F q , respectively, due to Carnevale et al [13, § 3.2]; cf [82]…”