In mathematical representation theory, the Eisenstein integral is an integral introduced by Harish-Chandra; in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms. Harish-Chandra used Eisenstein integrals to decompose the regular representation of a semisimple Lie group into representations induced from parabolic subgroups.; ; Trombi gave a survey of Harish-Chandra's work on this.
Definition
Harish-Chandra defined the Eisenstein integral by
where: x is an element of a semisimple group G P = MAN is a cuspidal parabolic subgroup of G ν is an element of the complexification of a a is the Lie algebra of A in the Langlands decomposition P = MAN. K is a maximal compact subgroup of G, with G = KP. ψ is a cuspidal function on M, satisfying some extra conditions τ is a finite-dimensional unitary double representation of K HP(x) = log a where x = kman is the decomposition of x in G = KMAN.
Notes
References