Local Langlands correspondence for the twisted exterior and symmetric square $\epsilon$-factors of ${\text{GL}}_n$

Abstract

Let $F$ be a non-Archimedean local field. Let ${\mathcal{A}}_n(F)$ be the set of equivalence classes of irreducible admissible representations of ${\text{GL}}_n(F)$, and ${\mathcal{G}}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W_F^{\prime }$. The local Langlands correspondence(LLC) establishes the reciprocity maps ${\text{Rec}}_{n,F}:{\mathcal{A}}_n(F)⟶{\mathcal{G}}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $\epsilon$-factors. This is also expected to be true under parallel compositions with a complex analytic representations of ${\text{GL}}_n(\mathbb{C})$. J.W. Cogdell, F. Shahidi, and T.-L. Tsai proved the equality of the symmetric and exterior square L- and $\epsilon$-factors [7] in 2017. But the twisted symmetric and exterior square L- and $\epsilon$-factor are new and very different from the untwisted case. In this paper we will define the twisted symmetric square L- and $\gamma$-factors using ${\text{GSpin}}_{2n+1}$, and establish the equality of the corresponding L- and $\epsilon$-factors. We will first reduce the problem to the analytic stability of their $\gamma$-factors for supercuspidal representations, then prove the supercuspidal stability by establishing general asymptotic expansions of partial Bessel function following the ideas in [7].

arXiv admin note: text overlap with arXiv:1412.1448 by other authors