Harish-Chandra's Ξ function

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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

$\Xi (g)=\int _{K}a(kg)^{\rho }dk,$

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

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