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[edit]

\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

References[edit]