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# Multivariate Rankin-Selberg Integrals on $GL_4$ and $GU(2,2)$

Published:2018-03-14
Printed: Dec 2018
• Aaron Pollack,
School of Mathematics, Institute for Advanced Study , Princeton, NJ 08540, USA
• Shrenik Shah,
Department of Mathematics, Columbia University , New York, NY 10027 , USA
 Format: LaTeX MathJax PDF

## Abstract

Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\operatorname{GSp}_4$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\operatorname{PGL}_4$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\operatorname{SL}_4(\mathbf{C})$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\operatorname{PGU}(2,2)$ representing the product of the degree 8 standard $L$-function and the degree 6 exterior square $L$-function.
 Keywords: automorphic form, L-function, Rankin-Selberg method, unitary group, exterior square, Langlands program
 MSC Classifications: 11F66 - Langlands $L$-functions; one variable Dirichlet series and functional equations 11F55 - Other groups and their modular and automorphic forms (several variables)

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