Harish-Chandra transform

In mathematical representation theory, the Harish-Chandra transform is a linear map from functions on a reductive Lie group to functions on a parabolic subgroup. It was introduced by Harish-Chandra (1958, p.595).

The Harish-Chandra transform f^P of a function f on the group G is given by

${\displaystyle f^{P}(m)=a^{-\rho }\int _{N}f(nm)\,dn}$

where P = MAN is the Langlands decomposition of a parabolic subgroup.