# Rankin-Selberg convolutions for ${\rm SO}_{2l+1}\times {\rm GL}_n$: local
theory

### About this Title

**David Soudry**

Publication: Memoirs of the American Mathematical Society

Publication Year
1993: Volume 105, Number 500

ISBNs: 978-0-8218-2568-6 (print); 978-1-4704-0077-4 (online)

DOI: http://dx.doi.org/10.1090/memo/0500

MathSciNet review: 1169228

MSC: Primary 11F70; Secondary 11F67, 22E50

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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.

Readership

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

### Table of Contents

**Chapters**

- 0. Introduction and preliminaries
- 1. The integrals to be studied
- 2. Estimates for Whittaker functions on $G_l$ (nonarchimedean case)
- 3. Estimates for Whittaker functions on $G_l$ (archimedean case)
- 4. Convergence of the integrals (nonarchimedean case)
- 5. Convergence of the integrals (archimedean case)
- 6. $A(W, \xi _{r,s})$ can be made constant (nonarchimedean case)
- 7. An analog in the archimedean case
- 8. Uniqueness theorems
- 9. Application of the intertwining operator
- 10. Definition of local factors
- 11. Multiplicativity of the $\gamma $-factor (case $l < n$, first variable)
- 12. The unramified computation