The Fourier transform of a C ∞ function, f , with compact support on a real reductive Lie group G is given by a collection of operators φ(P, σ, λ) := π P (σ, λ)(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 (σ, λ), σ varying in a set of discrete series, λ lying in a complex finite dimensional vector space. The π P (σ, λ) are generalized principal series, induced from P . It is.
