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Generating functions on covering groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 February 2018

David Ginzburg
David Ginzburg*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 6997801, Israel email ginzburg@post.tau.ac.il
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Abstract

In this paper we prove a conjecture relating the Whittaker function of a certain generating function with the Whittaker function of the theta representation $\unicode[STIX]{x1D6E9}_{n}^{(n)}$ . This enables us to establish that a certain global integral is factorizable and hence deduce the meromorphic continuation of the standard partial $L$ function $L^{S}(s,\unicode[STIX]{x1D70B}^{(n)})$ . In fact we prove that this partial $L$ function has at most a simple pole at $s=1$ . Here, $\unicode[STIX]{x1D70B}^{(n)}$ is a genuine irreducible cuspidal representation of the group $\text{GL}_{r}^{(n)}(\mathbf{A})$ .

Keywords

L functionsmetaplectic covering groupsgenerating functions

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F55: Other groups and their modular and automorphic forms (several variables) 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 4 , April 2018 , pp. 671 - 684
Copyright
© The Author 2018 

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