The Fourier Transform on Symmetric Spaces and Applications 1. the Fourier Transform

The symmetric spaces the title refers to are the spaces X = G=K where G is a connected semisimple Lie group with nite center and K is a maximal compact subgroup. The Fourier transform on X is deened by means of the Iwasawa decomposition G = NAK of G where N is nilpotent and A abelian. Let g; n; a; k denote the corresponding Lie algebras. We also need the group M = K A ; the centralizer of A in K: To deene the Fourier transform we write for g 2 G g = n exp A(g)k; n 2 N; A(g) 2 a; k 2 K and deene A : G=K K=M ?! a by A(gK; kM) = A(k ?1 g): If f is a function on X its Fourier transform ~ f is deened in 3a] by (1) ~ f(; b) = Z X f(x)e (?ii+)(A(x;b)) dx; b 2 B; for all (; b) 2 a c B for which the integral exists. Here is half the sum of the restricted roots. Deenition (1) is the analog of the Euclidean Fourier transform (2)