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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 fP of a function f on the group G is given by

\( f^P(m) =a^{-\rho}\int_Nf(nm)\;dn \)

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

Harish-Chandra (1958), "Spherical Functions on a Semisimple Lie Group II", American Journal of Mathematics (The Johns Hopkins University Press) 80 (3): 553–613, ISSN 0002-9327, JSTOR 2372772
Wallach, Nolan R (1988), Real reductive groups. I, Pure and Applied Mathematics, 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, MR 929683

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