text: Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime.
  @cite_4: I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- Combinatorial Descriptions.
  @cite_6: We report on progress and problems concerning the analytical behaviour of the zeta functions of groups and rings. We also describe how these generating functions are special cases of adelic cone integrals for which our results hold.
  @cite_8: We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end we introduce Igusa-type rational functions encoding these polynomials and prove that they satisfy certain functional equations. Some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on Coxeter groups of type @math . While our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements, we formulate a precise conjecture how in this situation, too, the polynomials may be described in terms of parabolic length functions.
summary: More generally, it is of interest to compute the (normal) subgroup zeta functions of other finitely generated nilpotent groups, and their behavior under base extension. We refer the reader to @cite_6 for a comprehensive list of examples. In his MSc thesis @cite_4 , Bauer has generalized many of our results to the normal zeta functions of the higher Heisenberg groups @math for all @math , where @math is a centrally amalgamated product of @math Heisenberg groups. In other words, if @math is a ring and we view elements of @math as row vectors, and if @math denotes the @math identity matrix, then ) a , b R^m, , c R . The paper @cite_8 arose from the (uncompleted) project to compute the subgroup zeta functions @math .