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 (1966, section 16).

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

Wallach (1988, 4.5) 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

References

• Harish-Chandra (1966), "Two theorems on semi-simple Lie groups", Annals of Mathematics. Second Series 83: 74–128, ISSN 0003-486X, JSTOR 1970472, MR 0194556
• Wallach, Nolan R (1988), Real reductive groups. I, Pure and Applied Mathematics 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, MR 929683