Harish-Chandra's c-function

In mathematics, Harish-Chandra's c-function is a function introduced by Harish-Chandra (1958a, 1958b) describing the asymptotic behavior of a zonal spherical function of a Lie group. It has a generalization called Harish-Chandra's (generalized) C-function. The Gindikin–Karpelevich formula is a product formula for Harish-Chandra's c-function, introduced by Gindikin and Karpelevich (1962, 1969).

Gindikin–Karpelevich formula

The c-function has a generalization c_w(λ) depending on an element w of the Weyl group.

Plancherel measure

The c-function appears in the Plancherel theorem for spherical functions, and the Plancherel measure is 1/c² times Lebesgue measure.

p-adic Lie groups

There is a similar c-function for p-adic Lie groups. Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie group.

References

• Cohn, Leslie (1974), Analytic theory of the Harish-Chandra C-function, Lecture Notes in Mathematics, vol. 429, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0064335, MR0422509
• Doran, Robert S.; Varadarajan, V. S., eds. (2000), "The mathematical legacy of Harish-Chandra", Proceedings of the AMS Special Session on Representation Theory and Noncommutative Harmonic Analysis, held in memory of Harish-Chandra on the occasion of the 75th anniversary of his birth, in Baltimore, MD, January 9–10, 1998, Proceedings of Symposia in Pure Mathematics, vol. 68, Providence, R.I.: American Mathematical Society, pp. xii+551, ISBN 978-0-8218-1197-9, MR 1767886
• Gindikin, S. G.; Karpelevich, F. I. (1962), "Plancherel measure for symmetric Riemannian spaces of non-positive curvature", Soviet Math. Dokl., 3: 962–965, ISSN 0002-3264, MR 0150239
• Gindikin, S. G.; Karpelevich, F. I. (1969) [1966], "On an integral associated with Riemannian symmetric spaces of non-positive curvature", Twelve Papers on Functional Analysis and Geometry, American Mathematical Society translations, vol. 85, pp. 249–258, ISBN 978-0-8218-1785-8, MR 0222219
• Harish-Chandra (1958a), "Spherical functions on a semisimple Lie group. I", American Journal of Mathematics, 80: 241–310, ISSN 0002-9327, MR0094407
• Harish-Chandra (1958b), "Spherical Functions on a Semisimple Lie Group II", American Journal of Mathematics, 80 (3), The Johns Hopkins University Press: 553–613, ISSN 0002-9327
• Harish-Chandra (1970), "Harmonic analysis on semisimple Lie groups", Bulletin of the American Mathematical Society, 76: 529–551, doi:10.1090/S0002-9904-1970-12442-9, ISSN 0002-9904, MR0257282
• Helgason, Sigurdur (1994), "Harish-Chandra's c-function. A mathematical jewel", in Tanner, Elizabeth A.; Wilson., Raj (eds.), Noncompact Lie groups and some of their applications (San Antonio, TX, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 429, Dordrecht: Kluwer Acad. Publ., pp. 55–67, ISBN 978-0-7923-2787-5, MR1306516 Reprinted in (Doran & Varadarajan 2000)
• Knapp, Anthony W. (2003), "The Gindikin-Karpelevič formula and intertwining operators", in Gindikin, S. G. (ed.), Lie groups and symmetric spaces. In memory of F. I. Karpelevich, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Providence, R.I.: American Mathematical Society, pp. 145–159, ISBN 978-0-8218-3472-5, MR 2018359
• Langlands, Robert P. (1971) [1967], Euler products, Yale University Press, ISBN 978-0-300-01395-5, MR 0419366
• Macdonald, I. G. (1968), "Spherical functions on a p-adic Chevalley group", Bulletin of the American Mathematical Society, 74 (3): 520–525, doi:10.1090/S0002-9904-1968-11989-5, ISSN 0002-9904, MR 0222089{{citation}}: CS1 maint: MR format (link)
• Macdonald, I. G. (1971), Spherical functions on a group of p-adic type, Ramanujan Institute lecture notes, vol. 2, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, MR 0435301
• Wallach, Nolan R (1975), "On Harish-Chandra's generalized C-functions", American Journal of Mathematics, 97: 386–403, ISSN 0002-9327, MR0399357