Eisenstein integral

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
 * $$\displaystyle E(P:\psi:\nu:x) = \int_K\psi(xk)\tau(k^{-1})\exp((i\nu-\rho_P)H_P(xk)) \, dk$$

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.