In
mathematical harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
, Harish-Chandra's ''Ξ'' function is a
special spherical function on a
semisimple Lie group, studied by .
Harish-Chandra used it to define
Harish-Chandra's Schwartz space.
gives a detailed description of the properties of ''Ξ''.
Definition
:
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
*
*
{{DEFAULTSORT:Harish-Chandra's Xi function
Harmonic analysis
Representation theory
Special functions