Harish-Chandra's Ξ function

In mathematical harmonic analysis, Harish-Chandra's Ξ function is a special spherical function on a semisimple Lie group, studied by.

Harish-Chandra used it to define Harish-Chandra's Schwartz space.

gives a detailed description of the properties of Ξ.

Definition

 * $$\Xi(g)=\int_Ka(kg)^\rho dk,$$

where


 * K is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition G=NAK
 * g is an element of G
 * ρ is a Weyl vector
 * a(g) is the element a in the Iwasawa decomposition g=nak