# 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 $$K_{S}=K_{infty}prodlimits_{v be the maximal compact subgroup. Let $$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.