nt.number theory – What is the Fourier Transform of a compactly supported smooth function on Lie groups over mathbb{Q}_{S}, S contains finitely many primes and infty

Let $G$ be a semisimple Lie group defined over global Field $mathbb{Q}$. Let $S$ be a set of finitely many non-Archimedean places including Archimedean places. Let $P_{0}=M_{0}A_{0}N_{0}$ be the minimal Parabolic subgroup with Langlands decompostion defined over $mathbb{Q}$. We consider the $mathbb{Q}_{S}$ points of $P_{0}$ and name those subgroups as $M_{0,S}$, $A_{0,S}$, and $N_{0,S}$. Suppose $K_{S}=K_{infty}prodlimits_{v<infty} K_{v} $ be the maximal compact subgroup. Let $C_{c}^{infty}(G(mathbb{Q}_{S}),K_{infty})$ be space of compactly Supported smooth functions, which are bi-$K_{infty}$-finite at Archimedean place. Let $mathfrak{a}_{0,S}=text{Lie}(A_{0,S}/A_{0,S}cap K_{S})$ and $nu in mathfrak{a}_{0,S ,mathbb{C}}^*$, the complexified dual of $mathfrak{a}_{0,S}$.
Spherical Case: When we consider the smooth compactly supported bi-$K_{S}$ -invariant functions f, we can define the Fourier Transform as composition of Satake Isomorphism and Fourier transform on Abelian group:
$$hat{f}(nu)=intlimits_{A_{0,S}}intlimits_{N_{0,S}}f(an)e^{(-nu+rho)(ln(a))}dnda.$$
Bi-$K_{infty}$-case: Let us only consider function on $G(mathbb{R})$. Let $(tau,V_{tau})$ be an irreducible representation of $K_{infty}$. We consider the functions f which are bi-$K_{infty}$-finite with respect to $tau$. Let us denote this space of functions as $C_{c}^{infty}(G(mathbb{R}),K_{infty})$. Let $sigma_{infty}$ be an irreducible admissible representation of $M_{0,infty}$, such that $sigma_{infty}subset tau|_{M_{0,infty}}$. Let $lambda_{infty} in mathfrak{a}_{0,infty,mathbb{C}}^*$. Let $mathcal{P}(M_{0,infty})$ be the set of parabolic subgroups whose Levi part is $M_{0,infty}A_{0,infty}$. Let us denote the normalized principle series representation for $B in mathcal{P}(M_{0,infty})$ as
$$text{Ind}_{B}^{G(mathbb{R})}(sigma_{infty}otimes e^{-lambda_{infty}+rho})=pi_{B}(sigma_{infty},lambda_{infty}).$$
Then by Arthur’s Paley-Wiener theorem at Archimedean place we can define the operator valued Fourier Transform as the following family:
$${pi_{B}(sigma_{infty},lambda_{infty})(f): B in mathcal{P}(M_{0,infty}),sigma_{infty},lambda_{infty}}$$
Non-Archimedean case: Let $f$ be a locally constant compactly supported function on $G(mathbb{Q}_{v})$, for $vne infty$. Let $P_{v}=M_{v}N_{v}$ be the levi decompsition of a standard Parabolic containing the minimal parabolic $P_{0,v}$. Let $sigma_{v}$ be the irreducible representation of the Levi subgroup $M_{v}$. Let $X(M_{v})$ be the group of $mathbb{Q}_{v}$ rational characters of $M_{v}$. Let $mathfrak{a}_{M,v,mathbb{C}}^*=X(M_{v})otimes mathbb{C}$. We can define the following induced representation with parameters $sigma_{v},lambda_{v}$:
$$text{Ind}_{P_{v}}^{G(mathbb{Q}_{v})}(sigma_{v}otimes e^{lambda_{v}})=pi_{P_{v}}(sigma_{v},lambda_{v}).$$
Let $mathcal{F}(M_{0,v})$ be the set of parabolic subgroups whose Levi component contains $M_{0,v}$. Hence we can define the Fourier transform of $f$ via the trace Paley-Wiener theorem by Bernstein, Deligne and Kazdhan as:
$${text{Tr}pi_{P_{v}}(sigma_{v},lambda_{v})(f): P_{v}in mathcal{F}(M_{0,v}),sigma_{v},lambda_{v}}$$
Question: How can I combine these two definition of Fourier transform to get the Fourier transform of $f in C_{c}^{infty}(G(mathbb{Q}_{S}),K_{infty})$? Is it going to be same as the global induced operator?
Thank you for your patience in reading the question. I apologize if this is long read in setting up the question.