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Rankin-Selberg Convolutions for $$\mathrm{SO}_{2\ell +1}\times \mathrm{GL}_n$$ : Local Theory
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Memoirs of the American Mathematical Society
1993; 100 pp; softcover
Volume: 105
ISBN-10: 0-8218-2568-2
ISBN-13: 978-0-8218-2568-6
List Price: US$34 Individual Members: US$20.40
Institutional Members: US\$27.20
Order Code: MEMO/105/500

This work studies the local theory for certain Rankin-Selberg convolutions for the standard $$L$$-function of degree $$2\ell n$$ of generic representations of $$\mathrm{ SO}_{2\ell +1}(F)\times \mathrm{GL}_n(F)$$ over a local field $$F$$. The local integrals converge in a half-plane and continue meromorphically to the whole plane. One main result is the existence of local gamma and $$L$$-factors. The gamma factor is obtained as a proportionality factor of a functional equation satisfied by the local integrals. In addition, Soudry establishes the multiplicativity of the gamma factor ($$\ell < n$$, first variable). A special case of this result yields the unramified computation and involves a new idea not presented before. This presentation, which contains detailed proofs of the results, is useful to specialists in automorphic forms, representation theory, and $$L$$-functions, as well as to those in other areas who wish to apply these results or use them in other cases.

Mathematicians working in automorphic forms, representation theory of reductive groups over local fields, $$L$$-functions and $$\epsilon$$ functions.

• Introduction and preliminaries
• The integrals to be studied
• Estimates for Whittaker functions on $$G_\ell$$ (nonarchimedean case)
• Estimates for Whittaker functions on $$G_\ell$$ (archimedean case)
• Convergence of the integrals (nonarchimedean case)
• Convergence of the integrals (archimedean case)
• $$A(W,\xi _{\tau ,s})$$ can be made constant (nonarchimedean case)
• An analog in the archimedean case
• Uniqueness theorems
• Application of the intertwining operator
• Definition of local factors
• Multiplicativity of the $$\gamma$$-factor (case $$\ell < n$$, first variable)
• The unramified computation
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