For a split reductive group
$G$
over a number field
$k$,
let
$\rho $ be an
$n$dimensional
complex representation of its complex dual group
${G}^{\vee}(\u2102)$.
For any irreducible cuspidal automorphic representation
$\sigma $ of
$G(\mathbb{\mathbb{A}})$, where
$\mathbb{\mathbb{A}}$ is the ring
of adeles of
$k$,
in [Jiang and Luo 2021], the authors introduce the
$(\sigma ,\rho )$Schwartz
space
${\mathcal{\mathcal{S}}}_{\sigma ,\rho}({\mathbb{\mathbb{A}}}^{\times})$ and
$(\sigma ,\rho )$Fourier operator
${\mathcal{\mathcal{F}}}_{\sigma ,\rho}$, and study the
$(\sigma ,\rho ,\psi )$Poisson summation
formula on
${\mathrm{GL}}_{1}$,
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
$k\setminus \mathbb{\mathbb{A}}$. Such
general formulas on ${\mathrm{GL}}_{1}$,
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
${\mathcal{\mathcal{S}}}_{\sigma ,\rho}({\mathbb{\mathbb{A}}}^{\times})$ and
$(\sigma ,\rho )$Fourier operator
${\mathcal{\mathcal{F}}}_{\sigma ,\rho}$. More precisely, over
any local field
${k}_{\nu}$ of
$k$, we define distribution
kernel functions
${k}_{{\sigma}_{\nu},\rho ,{\psi}_{\nu}}(x)$
on
${\mathrm{GL}}_{1}$ that represent
the
$({\sigma}_{\nu},\rho )$Fourier
operators
${\mathcal{\mathcal{F}}}_{{\sigma}_{\nu},\rho ,{\psi}_{\nu}}$ 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. PiatetskiShapiro
[Gelfand et al. 2016] and of A. Weil [1995].
