Eisenstein integral explained

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.^[1] 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 ^{-1} _{K\psi(xk)\tau(k} )\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
• H_P(x) = log a where x = kman is the decomposition of x in G = KMAN.

Notes and References

1. ;