Local Langlands Correspondence for the Twisted Exterior and Symmetric Square ε-Factors of GL(N)

Dongming She, Purdue University

Abstract

Let F be a non-Archimedean local field. Let An(F) be the set of equivalence classes of irreducible admissible representations of GLn(F), and Gn(F) be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of W'F . The local Langlands correspondence(LLC) establishes the reciprocity maps Recn,F : An(F) → Gn(F) , satisfying some nice properties. An important invariant under this correspondence is the L- and ε-factors. This is also expected to be true under parallel compositions with a complex analytic representations of GLn(C). J.W. Cogdell, F. Shahidi, and T.-L. Tsai proved the equality of the symmetric and exterior square L- and ε-factors [7] in 2017. But the twisted symmetric and exterior square L- and ε-factor are new and very different from the untwisted case. In this paper we will define the twisted symmetric square L- and γ-factors using GSpin2n+1, and establish the equality of the corresponding L- and ε-factors. We will first reduce the problem to the analytic stability of their γ-factors for supercuspidal representations, then prove the supercuspidal stability by establishing general asymptotic expansions of partial Bessel function following the ideas in [7].

Degree

Ph.D.

Advisors

Shahidi, Purdue University.

Subject Area

Economics

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