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Towards Generalized Prehomogeneous Zeta Integrals

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Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2221))

Abstract

Let X be a prehomogeneous vector space under a connected reductive group G over \(\ensuremath {\mathbb {R}}\). Assume that the open G-orbit X + admits a finite covering by a symmetric space. We study certain zeta integrals involving (1) Schwartz functions on X, and (2) generalized matrix coefficients on \(X^+(\ensuremath {\mathbb {R}})\) of Casselman–Wallach representations of \(G(\ensuremath {\mathbb {R}})\), upon a twist by complex powers of relative invariants. This merges representation theory with prehomogeneous zeta integrals of Igusa et al. We show their convergence in some shifted cone, and prove their meromorphic continuation via the machinery of b-function together with V. Ginzburg’s results on admissible D-modules. This provides some evidence for a broader theory of zeta integrals associated to affine spherical embeddings.

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Acknowledgements

The author is grateful to Jeffrey Adams, Wee Teck Gan, Dihua Jiang, Eitan Sayag and Jun Yu for inspiring conversations. The author also appreciates the referee’s pertinent comments.

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Authors and Affiliations

  1. Beijing International Center for Mathematical Research, School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China

    Wen-Wei Li

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  1. Wen-Wei Li
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Correspondence to Wen-Wei Li .

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  1. Aix-Marseille Université CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France

    Volker Heiermann

  2. School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India

    Dipendra Prasad

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Li, WW. (2018). Towards Generalized Prehomogeneous Zeta Integrals. In: Heiermann, V., Prasad, D. (eds) Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms. Lecture Notes in Mathematics, vol 2221. Springer, Cham. https://doi.org/10.1007/978-3-319-95231-4_6

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