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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
  • H**P(x) = log a where x = kman is the decomposition of x in G = KMAN.

Notes

References

References

  1. {{harvtxt. Harish-Chandra. 1970; {{harvtxt. Harish-Chandra. 1972
  2. {{harvtxt. Harish-Chandra. 1975; {{harvtxt. Harish-Chandra. 1976a; {{harvtxt. Harish-Chandra. 1976b
  3. {{harvtxt. Trombi. 1989
  4. {{harvtxt. Harish-Chandra. 1970
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