Abstract

For a split reductive group $G$ over a number field $k$, let $\rho$ be an $n$-dimensional complex representation of its complex dual group $G^{\vee }(ℂ)$. For any irreducible cuspidal automorphic representation $\sigma$ of $G(\mathbb{𝔸})$, where $\mathbb{𝔸}$ is the ring of adeles of $k$, in [Jiang and Luo 2021], the authors introduce the $(\sigma ,\rho )$-Schwartz space ${\mathcal{𝒮}}_{\sigma ,\rho }({\mathbb{𝔸}}^{\times })$ and $(\sigma ,\rho )$-Fourier operator ${\mathcal{ℱ}}_{\sigma ,\rho }$, and study the $(\sigma ,\rho ,\psi )$-Poisson summation formula on ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_1$, under the assumption that the local Langlands functoriality holds for the pair $(G,\rho )$ at all local places of $k$, where $\psi$ is a nontrivial additive character of $k\setminus \mathbb{𝔸}$. Such general formulas on ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_1$, as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture [Langlands 1970] on global functional equation for the automorphic $L$-functions $L(s,\sigma ,\rho )$. In order to understand such Poisson summation formulas, we continue with Jiang and Luo [2021] and develop a further local theory related to the $(\sigma ,\rho )$-Schwartz space ${\mathcal{𝒮}}_{\sigma ,\rho }({\mathbb{𝔸}}^{\times })$ and $(\sigma ,\rho )$-Fourier operator ${\mathcal{ℱ}}_{\sigma ,\rho }$. More precisely, over any local field $k_{\nu }$ of $k$, we define distribution kernel functions $k_{{\sigma }_{\nu },\rho ,{\psi }_{\nu }}(x)$ on ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_1$ that represent the $({\sigma }_{\nu },\rho )$-Fourier operators ${\mathcal{ℱ}}_{{\sigma }_{\nu },\rho ,{\psi }_{\nu }}$ as convolution integral operators, i.e., generalized Hankel transforms, and the local Langlands $\gamma$-functions $\gamma (s,{\sigma }_{\nu },\rho ,{\psi }_{\nu })$ as Mellin transform of the kernel functions. As a consequence, we show that any local Langlands $\gamma$-functions are the gamma functions in the sense of I. Gelfand, M. Graev, and I. Piatetski-Shapiro [Gelfand et al. 2016] and of A. Weil [1995].

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