Electronic ISBN:  9781470400774 
Product Code:  MEMO/105/500.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 105; 1993; 100 ppMSC: Primary 11; 22; 26; 32;
This work studies the local theory for certain RankinSelberg 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 halfplane 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.
ReadershipMathematicians 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


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This work studies the local theory for certain RankinSelberg 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 halfplane 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.

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