---
_id: '2675638'
abstract:
- lang: eng
text: 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.
article_type: original
author:
- first_name: A.
full_name: Stasinski, A.
last_name: Stasinski
- first_name: Christopher
full_name: Voll, Christopher
id: '27296433'
last_name: Voll
citation:
ama: Stasinski A, Voll C. Representation zeta functions of nilpotent groups and
generating functions for Weyl groups of type B. *American Journal of Mathematics*.
2014;136(2):501-550.
angewandte-chemie: A. Stasinski, and C. Voll, “Representation zeta functions of
nilpotent groups and generating functions for Weyl groups of type B”, *American
Journal of Mathematics*, **2014**, *136*, 501-550.
apa: Stasinski, A., & Voll, C. (2014). Representation zeta functions of nilpotent
groups and generating functions for Weyl groups of type B. *American Journal
of Mathematics*, *136*(2), 501-550. doi:10.1353/ajm.2014.0010
apa_indent: Stasinski,
A., & Voll, C. (2014). Representation zeta functions of nilpotent groups and
generating functions for Weyl groups of type B. *American Journal of Mathematics*,
*136*(2), 501-550. doi:10.1353/ajm.2014.0010

bio1: 'Stasinski A, Voll C (2014)

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

American Journal
of Mathematics 136(2): 501-550.'
chicago: 'Stasinski,
A., and Voll, Christopher. 2014. “Representation zeta functions of nilpotent groups
and generating functions for Weyl groups of type B”. *American Journal of Mathematics*
136 (2): 501-550.

'
default: 'Stasinski A, Voll C (2014)

*American Journal of Mathematics*
136(2): 501-550.'
dgps: Stasinski,
A. & Voll, C. (2014). Representation zeta functions of nilpotent groups and
generating functions for Weyl groups of type B. *American Journal of Mathematics*,
*136*(2), 501-550. Muse - Johns Hopkins University Press. doi:10.1353/ajm.2014.0010.

frontiers: Stasinski, A., and Voll, C. (2014). Representation zeta functions of
nilpotent groups and generating functions for Weyl groups of type B. *American
Journal of Mathematics* 136, 501-550.
harvard1: Stasinski, A., & Voll, C., 2014. Representation zeta functions of
nilpotent groups and generating functions for Weyl groups of type B. *American
Journal of Mathematics*, 136(2), p 501-550.
ieee: ' A. Stasinski and C. Voll, “Representation zeta functions of nilpotent groups
and generating functions for Weyl groups of type B”, *American Journal of Mathematics*, vol.
136, 2014, pp. 501-550.'
lncs: ' Stasinski, A., Voll, C.: Representation zeta functions of nilpotent groups
and generating functions for Weyl groups of type B. American Journal of Mathematics.
136, 501-550 (2014).'
mla: 'Stasinski, A., and Voll, Christopher. “Representation zeta functions of nilpotent
groups and generating functions for Weyl groups of type B”. *American Journal
of Mathematics* 136.2 (2014): 501-550.'
wels: 'Stasinski, A.; Voll, C. (2014): Representation zeta functions of nilpotent
groups and generating functions for Weyl groups of type B *American Journal
of Mathematics*,136:(2): 501-550.'
date_created: 2014-05-12T12:20:57Z
date_updated: 2019-06-14T13:01:51Z
department:
- _id: '10020'
doi: 10.1353/ajm.2014.0010
external_id:
isi:
- '000333657000007'
intvolume: ' 136'
isi: 1
issue: '2'
language:
- iso: eng
page: 501-550
publication: American Journal of Mathematics
publication_identifier:
issn:
- 0002-9327
publication_status: published
publisher: Muse - Johns Hopkins University Press
quality_controlled: '1'
status: public
title: Representation zeta functions of nilpotent groups and generating functions
for Weyl groups of type B
type: journal_article
user_id: '89573'
volume: 136
year: '2014'
...