# 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

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

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