mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

harmonic analysis, Harish-Chandra's ''Ξ'' function is a

special Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer Literature * ''Specia ...

spherical function on a

semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...

, studied by . Harish-Chandra used it to define Harish-Chandra's Schwartz space. 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 In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a cons ...

''G''=''NAK'' *''g'' is an element of ''G'' *''ρ'' is a

Weyl vector In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the c ...

*''a''(''g'') is the element ''a'' in the Iwasawa decomposition ''g''=''nak''

References

* * {{DEFAULTSORT:Harish-Chandra's Xi function Harmonic analysis Representation theory Special functions