Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T22:16:51.446Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALIZED ZETA INTEGRALS ON CERTAIN REAL PREHOMOGENEOUS VECTOR SPACES

Part of: Algebraic number theory: local and $p$-adic fields Abstract harmonic analysis

Published online by Cambridge University Press:  05 September 2022

WEN-WEI LI Open the ORCID record for WEN-WEI LI [Opens in a new window]
WEN-WEI LI*
Affiliation:
Beijing International Center of Mathematical Research, Peking University No. 5, Yiheyuan Road, Beijing 100871, People’s Republic of China wwli@bicmr.pku.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X be a real prehomogeneous vector space under a reductive group G, such that X is an absolutely spherical G-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz–Bruhat functions on X against generalized matrix coefficients of admissible representations of $G(\mathbb {R})$ , twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation, as well as a functional equation in terms of abstract $\gamma $ -factors. This subsumes the archimedean zeta integrals of Godement–Jacquet, those of Sato–Shintani (in the spherical case), and the previous works of Bopp–Rubenthaler. The proof of functional equations is based on Knop’s results on Capelli operators.

MSC classification

Primary: 11S40: Zeta functions and $L$-functions
Secondary: 11S90: Prehomogeneous vector spaces 43A85: Analysis on homogeneous spaces
Type
Article
Information
Nagoya Mathematical Journal , Volume 249 , March 2023 , pp. 50 - 87
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work is supported by the National Natural Science Foundation of China (Grant No. 11922101).

References

Aizenbud, A., Gourevitch, D., and Minchenko, A., Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs , Sel. Math. New Ser. 22 (2016), 2325–2345.CrossRefGoogle Scholar
Beilinson, A. and Bernstein, J., “A proof of Jantzen conjectures” in I. M. Gelfand Seminar, Adv. Soviet Math. 16, Amer. Math. Soc., Providence, RI, 1993, 1–50.Google Scholar
Benson, C. and Ratcliff, G., A classification of multiplicity free actions , J. Algebra 181 (1996), 152–186.CrossRefGoogle Scholar
Bernstein, J. and Krötz, B., Smooth Fréchet globalizations of Harish-Chandra modules , Israel J. Math. 199 (2014), 45–111.CrossRefGoogle Scholar
Bochnak, J., Coste, M., and Roy, M.-F., Real Algebraic Geometry, Ergeb. Math. Grenzgeb. (3) [Results Math. Relat. Areas (3)] 36, Springer, Berlin, 1998, Translated from the 1987 French original, Revised by the authors.CrossRefGoogle Scholar
Bopp, N. and Rubenthaler, H., Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces, Mem. Amer. Math. Soc. 174(821), Amer. Math. Soc., Providence, RI, 2005, viii + 233.CrossRefGoogle Scholar
Bruns, W. and Gubeladze, J., Polytopes, Rings, and K-Theory, Springer Monogr. Math., Springer, Dordrecht, 2009.Google Scholar
Brylinski, J.-L. and Delorme, P., Vecteurs distributions H-invariants pour les séries principales généralisées d’espaces symétriques réductifs et prolongement méromorphe d’intégrales d’Eisenstein , Invent. Math. 109 (1992), 619–664.CrossRefGoogle Scholar
Casselman, W., Canonical extensions of Harish-Chandra modules to representations of G , Canad. J. Math. 41 (1989), 385–438.CrossRefGoogle Scholar
Chriss, N. and Ginzburg, V., Representation Theory and Complex Geometry, Mod. Birkhäuser Class., Birkhäuser Boston, Boston, MA, 2010, Reprint of the 1997 edition.CrossRefGoogle Scholar
Goldfeld, D. and Hundley, J., Automorphic Representations and L-Functions for the General Linear Group, II, Cambridge Stud. Adv. Math. 130, Cambridge Univ. Press, Cambridge, 2011, With exercises and a preface by Xander Faber.Google Scholar
Howe, R. and Umeda, T., The Capelli identity, the double commutant theorem, and multiplicity-free actions , Math. Ann. 290 (1991), 565–619.CrossRefGoogle Scholar
Kac, V. G., Some remarks on nilpotent orbits , J. Algebra 64 (1980), 190–213.CrossRefGoogle Scholar
Kimura, T., Introduction to Prehomogeneous Vector Spaces, Transl. Math. Monogr. 215, Amer. Math. Soc., Providence, RI, 2003, Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author.Google Scholar
Knop, F., A Harish-Chandra homomorphism for reductive group actions , Ann. Math. 140 (1994), 253–288.CrossRefGoogle Scholar
Knop, F., “Some remarks on multiplicity free spaces” in Representation Theories and Algebraic Geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer Acad., Dordrecht, 1998, 301–317.CrossRefGoogle Scholar
Knop, F., Krötz, B., and Schlichtkrull, H., The tempered spectrum of a real spherical space , Acta Math. 218 (2017), 319–383.CrossRefGoogle Scholar
Kobayashi, T. and Oshima, T., Finite multiplicity theorems for induction and restriction , Adv. Math. 248 (2013), 921–944.CrossRefGoogle Scholar
Krötz, B. and Schlichtkrull, H., Multiplicity bounds and the subrepresentation theorem for real spherical spaces , Trans. Amer. Math. Soc. 368 (2016), no. 4, 2749–2762.CrossRefGoogle Scholar
Leahy, A. S., A classification of multiplicity free representations , J. Lie Theory 8 (1998), 367–391.Google Scholar
Li, W.-W., “Towards generalized prehomogeneous zeta integrals” in Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms, Lecture Notes in Math. 2221, Springer, Cham, 2018, 287–318.CrossRefGoogle Scholar
Li, W.-W., Zeta Integrals, Schwartz Spaces and Local Functional Equations, Lecture Notes in Math. 2228, Springer, Cham, 2018.CrossRefGoogle Scholar
Li, W.-W., On the regularity of D-modules generated by relative characters , Transform. Groups 27 (2022), 525–562.CrossRefGoogle Scholar
Saito, H., Convergence of the zeta functions of prehomogeneous vector spaces , Nagoya Math. J. 170 (2003), 1–31.CrossRefGoogle Scholar
Sakellaridis, Y., Spherical varieties and integral representations of L-functions , Algebra Number Theory 6 (2012), 611–667.CrossRefGoogle Scholar
Sato, F., Zeta functions in several variables associated with prehomogeneous vector spaces. I. Functional equations , Tohoku Math. J. (2) 34 (1982), 437–483.CrossRefGoogle Scholar
Sato, F., “On functional equations of zeta distributions” in Automorphic Forms and Geometry of Arithmetic Varieties, Adv. Stud. Pure Math. 15, Academic Press, Boston, MA, 1989, 465–508.Google Scholar
Sato, F., Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms . Proc. Math. Sci. 104 (1994), 99–135, K. G. Ramanathan memorial issue.CrossRefGoogle Scholar
Sato, F., Zeta functions of $\left({SL}_2\times {SL}_2\times {GL}_2,{\boldsymbol{M}}_2\oplus {\boldsymbol{M}}_2\right)$ associated with a pair of Maass cusp forms , Comment. Math. Univ. St. Pauli 55 (2006), 77–95.Google Scholar
Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector spaces , Ann. of Math. 2 (1974), 131–170.CrossRefGoogle Scholar
Tate, J., “Number theoretic background” in Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, RI, 1979, 3–26.Google Scholar
Trèves, F., Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.Google Scholar