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 Template:Harvs.

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

Template:Harvtxt gives a detailed description of the properties of Ξ.

Definition

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

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

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