Abstract
The Fourier transform of a $C^{\infty}$ function, $f$, with compact support on a real reductive Lie group $G$ is given by a collection of operators $\phi(P, \sigma, \lambda):=\pi^{P}(\sigma, \lambda)(f)$ for a suitable family of representations of $G$, which depends on a family, indexed by $P$ in a finite set of parabolic subgroups of $G$, of pairs of parameters $(\sigma, \lambda)$, $\sigma$ varying in a set of discrete series, $\lambda$ lying in a complex finite dimensional vector space. The $\pi^{P}(\sigma, \lambda)$ are generalized principal series, induced from $P$. It is easy to verify the holomorphy of the Fourier transform in the complex parameters. Also it satisfies some growth properties. Moreover an intertwining operator between two representations $\pi^{P}(\sigma, \lambda)$, $\pi^{P’}(\sigma’, \lambda’)$ of the family, implies an intertwining property for $\phi(P, \sigma, \lambda)$ and $\phi(P’, \sigma’, \lambda’)$. There is also a way to introduce ”successive (partial) derivatives” of the family of representations, $\pi^{P}(\sigma, \lambda)$, along the parameter $\lambda$, and intertwining operators between subquotients of these successive derivatives imply the intertwining property for the successive derivatives of the Fourier transform $\phi$. We show that these properties characterize the collections of operators $(P, \sigma, \lambda) \mapsto \phi(P, \sigma, \lambda)$ which are Fourier transforms of a $C^{\infty}$ function with compact support, for $G$ linear. The proof, which uses Harish-Chandra’s Plancherel formula, rests on a similar result for left and right $K$-finite functions, which is due to J. Arthur. We give also a proof of Arthur’s result, purely in term of representations, involving the work of A. Knapp and E. Stein on intertwining integrals and Langlands and Vogan’s classifications of irreducible representations of $G$.
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