Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
-
Alexander Stasinski
und
Christopher Voll
Abstract
We compute the representation zeta functions of some finitely
generated nilpotent groups associated to unipotent group schemes
over rings of integers in number fields. These group schemes are
defined by Lie lattices whose presentations are modelled on certain
prehomogeneous vector spaces. Our method is based on evaluating
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: Sonderforschungsbereich 701 at Bielefeld University
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/F044194/1
Funding statement: Voll acknowledges support by the DFG through Sonderforschungsbereich 701 at Bielefeld University and helpful conversations with Jan Schepers. This research was supported by Engineering and Physical Sciences Research Council grant EP/F044194/1.
Acknowledgements
The remarks of two anonymous referees helped to improve this paperâs exposition.
References
[1] Avni N., Klopsch B., Onn U. and Voll C., Representation zeta functions of compact p-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), no. 1, 111â197. 10.1215/00127094-1959198Suche in Google Scholar
[2] Björner A. and Brenti F., Combinatorics of Coxeter groups, Grad. Texts in Math. 231, Springer, New York, 2005. Suche in Google Scholar
[3] Denef J., Report on Igusaâs local zeta function, SĂ©minaire Bourbaki, Vol. 1990/91, ExposĂ©s 730â744, AstĂ©risque 201â203, SociĂ©tĂ© MathĂ©matique de France, Paris (1991), 359â386. Suche in Google Scholar
[4] du Sautoy M. P. F. and Woodward L., Zeta Functions of Groups and Rings, Lecture Notes in Math. 1925, Springer, Berlin, 2008. 10.1007/978-3-540-74776-5Suche in Google Scholar
[5] Dung D. H. and Voll C., Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups, preprint 2015, http://arxiv.org/abs/1503.06947; to appear in Trans. Amer. Math. Soc. 10.1090/tran/6879Suche in Google Scholar
[6] Igusa J.-I., An Introduction to the Theory of Local Zeta Functions, AMS/IP Stud. Adv. Math. 14, American Mathematical Society, Providence, 2000. Suche in Google Scholar
[7] Kimura T., Introduction to Prehomogeneous Vector Spaces, Transl. Math. Monogr. 215, American Mathematical Society, Providence, 2003. Suche in Google Scholar
[8] Laksov D. and Thorup A., Counting matrices with coordinates in finite fields and of fixed rank, Math. Scand. 74 (1994), no. 1, 19â33. 10.7146/math.scand.a-12477Suche in Google Scholar
[9] Rossmann T., Stability results for local zeta functions of groups and related structures, preprint 2015, http://arxiv.org/abs/1504.04164. Suche in Google Scholar
[10] Rossmann T., Topological representation zeta functions of unipotent groups, J. Algebra 448 (2016), 210â237. 10.1016/j.jalgebra.2015.09.050Suche in Google Scholar
[11] Stasinski A. and Voll C., Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B, Amer. J. Math. 136 (2014), no. 2, 501â550. 10.1353/ajm.2014.0010Suche in Google Scholar
© 2017 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Probabilistic trace and Poisson summation formulae on locally compact abelian groups
- The Selberg trace formula as a Dirichlet series
- On the cohomology and their torsion of real toric objects
- Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
- Non-abelian tensor and exterior products of multiplicative Lie rings
- On the infimum of the spectrum of a relativistic Schrödinger operator
- Character correspondences for symmetric groups and wreath products
- Univalence in locally cartesian closed â-categories
- On subordinate random walks
- Tame combings and easy groups
- On Belkâs classifying space for Thompsonâs group F
- Subspace confinement for switched linear systems
- Exceptional bundles of homological dimension ${k}$
- Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
- Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
Artikel in diesem Heft
- Frontmatter
- Probabilistic trace and Poisson summation formulae on locally compact abelian groups
- The Selberg trace formula as a Dirichlet series
- On the cohomology and their torsion of real toric objects
- Outer automorphism groups of simple Lie algebras and symmetries of painted diagrams
- Non-abelian tensor and exterior products of multiplicative Lie rings
- On the infimum of the spectrum of a relativistic Schrödinger operator
- Character correspondences for symmetric groups and wreath products
- Univalence in locally cartesian closed â-categories
- On subordinate random walks
- Tame combings and easy groups
- On Belkâs classifying space for Thompsonâs group F
- Subspace confinement for switched linear systems
- Exceptional bundles of homological dimension ${k}$
- Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces
- Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups
