Stability of γ-factors for GL( R) × GL(R)

Tung-Lin Tsai, Purdue University

Abstract

One of the successful approaches to establish the Langlands principle of functoriality is combining Converse Theorem of J.W. Cogdell and I.I. Piatetski-Shapiro and Langlands-Shahidi method. Stability of γ-factors plays a crucial role in those places where the local Langlands correspondence is lacking. It also leads to the proof of the equality of γ-factors obtained from the Langlands-Shahidi method with those obtained from other methods. We first reduce the problem to the stability of local coefficients since γ-factors are defined inductively by local coefficients in Langlands-Shahidi method. We then start with an integral representation for local coefficients established by F. Shahidi. By constructing a set of suitably chosen representatives inside the maximal torus and doing change of variables, the integral above can be written as a Mellin transform of a Bessel function up to an abelian γ-function. In order to use the philosophy suggested by J.W. Cogdell, I.I. Piatetski-Shapiro, and F. Shahidi previously to prove the stability of local coefficients under twisting a highly ramified character, we introduce the notions of orbital integrals and Shalika germs used by H. Jacquet and Y. Ye. We then prove the Bessel functions are a certain kind of orbital integrals. After using the result that orbital integrals, and hence the Bessel function, have a Shalika germ expansion established by H. Jacquet and Y. Ye, we may express the Bessel function into two parts. One part is not a smooth function and only depends on the central character of the representation. The other part is smooth in a certain sense. Finally, using the above two properties of the Bessel functions, which is the the philosophy that we pursuit, we prove the main theorem that local coefficients, and hence γ-factors, are stable under twisting by a sufficiently highly ramified character. Stability of γ-factors proved in the previous cases requires certain crucial assumptions, namely the dimension and the rank condition. This thesis reveals a new method to establish the stability of γ-factors in certain cases that previous assumptions do not hold, yet the philosophy suggested by J.W. Cogdell, I.I. Piatetski-Shapiro, and F. Shahidi is still valid. We hope to generalize this method to other cases that we are interested in eventually.

Degree

Ph.D.

Advisors

Shadidi, Purdue University.

Subject Area

Applied Mathematics|Mathematics

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