In mathematical representation theory, the Eisenstein integral is an integral introduced by Harish-Chandra^[1] 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.^[2] Trombi gave a survey of Harish-Chandra's work on this.^[3]

Definition

Harish-Chandra^[4] defined the Eisenstein integral by

${\displaystyle \displaystyle E(P:\psi$ :\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

• Harish-Chandra (1970), "Harmonic analysis on semisimple Lie groups", Bulletin of the American Mathematical Society, 76 (3): 529–551, doi:10.1090/S0002-9904-1970-12442-9, ISSN 0002-9904, MR 0257282
• Harish-Chandra (1972), "On the theory of the Eisenstein integral", in Gulick, Denny; Lipsman, Ronald L. (eds.), Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Lecture Notes in Mathematics, vol. 266, Berlin, New York: Springer-Verlag, pp. 123–149, doi:10.1007/BFb0059640, ISBN 978-3-540-05856-4, MR 0399355
• Harish-Chandra (1975), "Harmonic analysis on real reductive groups. I. The theory of the constant term", Journal of Functional Analysis, 19: 104–204, doi:10.1016/0022-1236(75)90034-8, MR 0399356
• Harish-Chandra (1976a), "Harmonic analysis on real reductive groups. II. Wavepackets in the Schwartz space", Inventiones Mathematicae, 36: 1–55, doi:10.1007/BF01390004, ISSN 0020-9910, MR 0439993, S2CID 73664990
• Harish-Chandra (1976b), "Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula", Annals of Mathematics, Second Series, 104 (1): 117–201, doi:10.2307/1971058, ISSN 0003-486X, JSTOR 1971058, MR 0439994
• Trombi, P. C. (1989), "On Harish-Chandra's theory of the Eisenstein integral for real semisimple Lie groups", in Sally, Paul J.; Vogan, David A. (eds.), Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Providence, R.I.: American Mathematical Society, pp. 287–350, ISBN 978-0-8218-1526-7, MR 1011900