Harish Chandra's Ξ function - Alchetron, the free social encyclopedia

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

Ξ ( g ) = ∫ K a ( k g ) ρ d k ,

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's Ξ function Wikipedia

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