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
Ξ ( g ) = ∫ K a ( k g ) ρ d k , {\displaystyle \Xi (g)=\int _{K}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
- Harish-Chandra (1966), "Two theorems on semi-simple Lie groups", Annals of Mathematics, Second Series, 83 (1): 74–128, doi:, ISSN , JSTOR , MR
- Wallach, Nolan R (1988), , Pure and Applied Mathematics, vol. 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, MR