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Archimedean zeta integrals for $GL(3)\times GL(2)$
About this Title
Miki Hirano, Taku Ishii and Tadashi Miyazaki
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 278, Number 1366
ISBNs: 978-1-4704-5277-3 (print); 978-1-4704-7166-8 (online)
DOI: https://doi.org/10.1090/memo/1366
Published electronically: May 23, 2022
Keywords: Whittaker functions,
automorphic forms,
zeta integrals
Table of Contents
Chapters
- Introduction
1. Whittaker functions
- 1. Basic objects
- 2. Preliminaries for $GL(n,\mathbb {R})$
- 3. Whittaker functions on $GL(2,\mathbb {R})$
- 4. Whittaker functions on $GL(3,\mathbb {R})$
- 5. Preliminaries for $GL(n,\mathbb {C})$
- 6. Whittaker functions on $GL(2,\mathbb {C})$
- 7. Whittaker functions on $GL(3,\mathbb {C})$
2. Archimedean zeta integrals for $GL(3)\times GL(2)$
- 8. Preliminaries
- 9. The local zeta integrals for $GL(3,\mathbb {R})\times GL(2,\mathbb {R})$
- 10. The local zeta integrals for $GL(3,\mathbb {C})\times GL(2,\mathbb {C})$
- A. Archimedean zeta integrals for $GL(2)\times GL(m)$ ($m=1,2$)
Abstract
In this article, we give explicit formulas of archimedean Whittaker functions on $GL(3)$ and $GL(2)$. Moreover, we apply those to the calculation of archimedean zeta integrals for $GL(3)\times GL(2)$, and show that the zeta integral for appropriate Whittaker functions is equal to the associated $L$-factors.- Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1995. Translated from the German manuscript; Corrected reprint of the 1985 translation. MR 1410059
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