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Univariate and bivariate zeta functions of unipotent group schemes of type $G$

Michele Zordan

Abstract

We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class $2$. These groups arise from a generalisation of one the families of unipotent groups schemes treated by Stasinski and Voll, and Lins. The univariate zeta functions are obtained by specialising the respective bivariate zeta functions defined by Lins. These are also used to deduce a formula for a joint distribution on Weyl groups of type $B$.

Univariate and bivariate zeta functions of unipotent group schemes of type $G$

Abstract

We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class . These groups arise from a generalisation of one the families of unipotent groups schemes treated by Stasinski and Voll, and Lins. The univariate zeta functions are obtained by specialising the respective bivariate zeta functions defined by Lins. These are also used to deduce a formula for a joint distribution on Weyl groups of type .

Paper Structure

This paper contains 27 sections, 16 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. A family of group schemes
  3. Main results
  4. Twist representation zeta functions
  5. Local representation zeta function
  6. Topological zeta functions
  7. Bivariate zeta functions
  8. Class counting zeta function
  9. Statistics on Weyl groups
  10. Acknowledgement
  11. Preliminaries on bivariate zeta functions
  12. Igusa-type integrals
  13. Commutator matrices
  14. Bivariate zeta functions as integrals
  15. Computing the bivariate representation zeta function
  16. ...and 12 more sections

Key Result

Theorem A

Let $\mathcal{O}$ be the ring of integers of a number field $K$, and let $\zeta_K$ be the Dedekind zeta function of $K$. Then

Theorems & Definitions (37)

  • Definition 1
  • Theorem A
  • Definition 2
  • Theorem B
  • Corollary 1
  • Definition 3: lin2019bivFGH
  • Theorem C
  • Corollary 2
  • Remark 1
  • Corollary 3
  • ...and 27 more