Journal of the Mathematical Society of Japan

On Witten multiple zeta-functions associated with semisimple Lie algebras II

Yasushi KOMORI, Kohji MATSUMOTO, and Hirofumi TSUMURA

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Abstract

This is a continuation of our previous result, in which properties of multiple zeta-functions associated with simple Lie algebras of Ar type have been studied. In the present paper we consider more general situation, and discuss the Lie theoretic background structure of our theory. We show a recursive structure in the family of zeta-functions of sets of roots, which can be explained by the order relation among roots. We also point out that the recursive structure can be described in terms of Dynkin diagrams. Then we prove several analytic properties of zeta-functions associated with simple Lie algebras of Br, Cr, and Dr types.

Article information

Source
J. Math. Soc. Japan Volume 62, Number 2 (2010), 355-394.

Dates
First available in Project Euclid: 7 May 2010

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1273236709

Digital Object Identifier
doi:10.2969/jmsj/06220355

Zentralblatt MATH identifier
05725848

Mathematical Reviews number (MathSciNet)
MR2662849

Subjects
Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 17B20: Simple, semisimple, reductive (super)algebras 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)

Keywords
Witten zeta-functions root systems Lie algebras Bernoulli polynomials

Citation

KOMORI, Yasushi; MATSUMOTO, Kohji; TSUMURA, Hirofumi. On Witten multiple zeta-functions associated with semisimple Lie algebras II. Journal of the Mathematical Society of Japan 62 (2010), no. 2, 355--394. doi:10.2969/jmsj/06220355. http://projecteuclid.org/euclid.jmsj/1273236709.


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References

  • S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith., 98 (2001), 107–116.
  • N. Bourbaki, Groupes et Algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968.
  • D. Essouabri, Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications à la théorie analytique des nombres, Thèse, Univ. Henri Poincaré – Nancy I, 1995.
  • D. Essouabri, Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres, Ann. Inst. Fourier, 47 (1997), 429–483.
  • P. E. Gunnells and R. Sczech, Evaluation of Dedekind sums, Eisenstein cocycles, and special values of $L$-functions, Duke Math. J., 118 (2003), 229–260.
  • J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972.
  • Y. Komori, K. Matsumoto and H. Tsumura, Zeta-functions of root systems, In: The Conference on $L$-functions, Fukuoka, 2006, (eds. L. Weng and M. Kaneko), World Scientific, 2007, pp.,115–140.
  • Y. Komori, K. Matsumoto and H. Tsumura, Zeta and $L$-functions and Bernoulli polynomials of root systems, Proc. Japan Acad., Series A, 84 (2008), 57–62.
  • Y. Komori, K. Matsumoto and H. Tsumura, Functional relations for zeta-functions of root systems, In: Number Theory: Dreaming in Dreams – Proceedings of the 5th China-Japan Seminar, (eds. T. Aoki, S. Kanemitsu and J.-Y. Liu), World Scientific, 2010, pp.,135–183.
  • Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras III, preprint, arXiv:0907.0955
  • Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras IV, Glasgow Math. J., to appear.
  • K. Matsumoto, On the analytic continuation of various multiple zeta-functions, In: Number Theory for the Millennium II, (eds. M. A. Bennett et al.), A K Peters, 2002, pp.,417–440.
  • K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J., 172 (2003), 59–102.
  • K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory, 101 (2003), 223–243.
  • K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, In: Proc. Session in Analytic Number Theory and Diophantine Equations, (eds. D. R. Heath-Brown and B. Z. Moroz), Bonner Math. Schriften, 360, Bonn, 2003, 17 pp.
  • K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions III, Comment. Math. Univ. St. Pauli, 54 (2005), 163–186.
  • K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier, 56 (2006), 1457–1504.
  • L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc., 33 (1958), 368–371.
  • H. Samelson, Notes on Lie Algebras, Universitext, Springer-Verlag, 1990.
  • L. Tornheim, Harmonic double series, Amer. J. Math., 72 (1950), 303–314.
  • H. Tsumura, On Witten's type of zeta values attached to $SO(5)$, Arch. Math., 82 (2004), 147–152.
  • E. Witten, On quantum gauge theories in two dimensions, Commun. Math. Phys., 141 (1991), 153–209.
  • D. Zagier, Values of zeta functions and their applications, In: First European Congress of Mathematics, Vol.,II, (eds. A. Joseph et al.), Progr. Math., 120, Birkhäuser, 1994, pp.,497–512.