# Rankin-Selberg Convolutions for \(\mathrm{SO}_{2ℓ+1}×\mathrm{GL}_{n}\) : Local Theory

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*David Soudry*

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.

#### Table of Contents

# Table of Contents

## Rankin-Selberg Convolutions for $\mathrm{SO}_{2+1}x\mathrm{GL}_{n}$ : Local Theory

- Contents v6 free
- 0. Introduction and Preliminaries 18 free
- 1. The Integrals to be Studied 1118 free
- 2. Estimates for Whittaker Functions on G[sub(l)](Nonarchimedean Case) 1522
- 3. Estimates for Whittaker Functions on G[sub(l)] (Archimedean Case) 1724
- 4. Convergence of the Integrals (Nonarchimedean Case) 2128
- 5. Convergence of the Integrals (Archimedean Case) 2532
- 6. A(W,ξ[sub(r,s)]) Can Be Made Constant (Nonarchimedean Case) 2936
- 7. An Analog in the Archimedean Case 3340
- 8. Uniqueness Theorems 4451
- 9. Application of the Intertwining Operator 5360
- 10. Definition of Local Factors 6168
- 11. Multiplicativity of the γ…Factor (Case I < n, First Variable) 6572
- 12. The Unramified Computation 96103