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) gives a detailed description of the properties of Ξ.

Definition

[math]\displaystyle{ \Xi(g)=\int_Ka(kg)^\rho dk, }[/math]

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 (1): 74–128, doi:10.2307/1970472, ISSN 0003-486X 
• Wallach, Nolan R (1988), Real reductive groups. I, Pure and Applied Mathematics, 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, https://archive.org/details/realreductivegro0000wall