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Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the setof equivalence classes of irreducible admissible representations of$\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes ofn-dimensional Frobenius semisimple Weil-Deligne representations of $W'_F$. Thelocal Langlands correspondence(LLC) establishes the reciprocity maps$\textrm{Rec}_{n,F}: \mathcal{A}_n(F)\longrightarrow \mathcal{G}_n(F)$ ,satisfying some nice properties. An important invariant under thiscorrespondence is the L- and $\epsilon$-factors. This is also expected to betrue under parallel compositions with a complex analytic representations of$\textrm{GL}_n(\mathbb{C})$. J.W. Cogdell, F. Shahidi, and T.-L. Tsai provedthe equality of the symmetric and exterior square L- and $\epsilon$-factors [7]in 2017. But the twisted symmetric and exterior square L- and $\epsilon$-factorare new and very different from the untwisted case. In this paper we willdefine the twisted symmetric square L- and $\gamma$-factors using$\textrm{GSpin}_{2n+1}$, and establish the equality of the corresponding L- and$\epsilon$-factors. We will first reduce the problem to the analytic stabilityof their $\gamma$-factors for supercuspidal representations, then prove thesupercuspidal stability by establishing general asymptotic expansions ofpartial Bessel function following the ideas in [7].

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