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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 f 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.

References

• Harish-Chandra (1958), "Spherical Functions on a Semisimple Lie Group II", American Journal of Mathematics, 80 (3), The Johns Hopkins University Press: 553–613, doi:10.2307/2372772, ISSN 0002-9327, JSTOR 2372772
• Wallach, Nolan R (1988), Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, MR 0929683

• Representation theory of Lie groups