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In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. Harish-Chandra (1958a, 1958b) introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and Harish-Chandra (1970) introduced a more general c-function called Harish-Chandra's (generalized) C-function. Gindikin and Karpelevich (1962, 1969) introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function,.

Harish-Chandra's c-function
Gindikin–Karpelevich formula

The c-function has a generalization cw(λ) depending on an element w of the Weyl group. The unique element of greatest length s0, is the unique element that carries the Weyl chamber \( \mathfrak{a}_+^* onto -\mathfrak{a}_+^* \). By Harish-Chandra's integral formula, cs0 is Harish-Chandra's c-function:

\( c(\lambda)=c_{s_0}(\lambda). \)

The c-functions are in general defined by the equation

\( \displaystyle A(s,\lambda)\xi_0 =c_s(\lambda)\xi_0, \)

where ξ0 is the constant function 1 in L2(K/M). The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions:

\( c_{s_1s_2}(\lambda) =c_{s_1}(s_2 \lambda)c_{s_2}(\lambda) \)

provided

\( \ell(s_1s_2)=\ell(s_1)+\ell(s_2). \)

This reduces the computation of cs to the case when s = sα, the reflection in a (simple) root α, the so-called "rank-one reduction" of Gindikin & Karpelevič (1962). In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by \( \mathfrak{g}_{\pm \alpha} \) where α lies in Σ0+. Then Gα is a real semisimple Lie group with real rank one, i.e. dim Aα = 1, and cs is just the Harish-Chandra c-function of Gα. In this case the c-function can be computed directly and is given by

\( c_{s_\alpha}(\lambda)=c_0{2^{-i(\lambda,\alpha_0)}\Gamma(i(\lambda,\alpha_0))\over\Gamma({1\over 2} ({1\over 2}m_\alpha + 1+ i(\lambda,\alpha_0)) \Gamma({1\over 2} ({1\over 2}m_\alpha + m_{2\alpha} + i(\lambda,\alpha_0))}, \)

where

\( c_0=2^{m_\alpha/2 + m_{2\alpha}}\Gamma\left({1\over 2} (m_\alpha+m_{2\alpha} +1)\right) \)

and α0=α/〈α,α〉.

The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of cs(λ), as follows:

\( c(\lambda)=c_0\prod_{\alpha\in\Sigma_0^+}{2^{-i(\lambda,\alpha_0)}\Gamma(i(\lambda,\alpha_0))\over\Gamma({1\over 2} ({1\over 2}m_\alpha + 1+ i(\lambda,\alpha_0)) \Gamma({1\over 2} ({1\over 2}m_\alpha + m_{2\alpha} + i(\lambda,\alpha_0))}, \)

where the constant c0 is chosen so that c(–iρ)=1 (Helgason 2000, p.447).
Plancherel measure

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


Generalized C-function
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, 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, 68, Providence, R.I.: American Mathematical Society, pp. xii+551, ISBN 978-0-8218-1197-9, MR1767886
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, MR0150239
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, 85, pp. 249–258, ISBN 978-0-8218-1785-8, MR0222219
Harish-Chandra (1958a), "Spherical functions on a semisimple Lie group. I", American Journal of Mathematics 80: 241–310, ISSN 0002-9327, JSTOR 2372786, MR0094407
Harish-Chandra (1958b), "Spherical Functions on a Semisimple Lie Group II", American Journal of Mathematics (The Johns Hopkins University Press) 80 (3): 553–613, ISSN 0002-9327, JSTOR 2372772
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, Noncompact Lie groups and some of their applications (San Antonio, TX, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 429, Dordrecht: Kluwer Acad. Publ., pp. 55–67, ISBN 978-0-7923-2787-5, MR1306516, Reprinted in (Doran & Varadarajan 2000)
Helgason, Sigurdur (2000) [1984], Groups and geometric analysis, Mathematical Surveys and Monographs, 83, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2673-7, MR0754767MR1790156
Knapp, Anthony W. (2003), "The Gindikin-Karpelevič formula and intertwining operators", in Gindikin, S. G., Lie groups and symmetric spaces. In memory of F. I. Karpelevich, Amer. Math. Soc. Transl. Ser. 2, 210, Providence, R.I.: American Mathematical Society, pp. 145–159, ISBN 978-0-8218-3472-5, MR2018359
Langlands, Robert P. (1971) [1967], Euler products, Yale University Press, ISBN 978-0-300-01395-5, MR0419366
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, MRMR0222089
Macdonald, I. G. (1971), Spherical functions on a group of p-adic type, Ramanujan Institute lecture notes, 2, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, MR0435301
Wallach, Nolan R (1975), "On Harish-Chandra's generalized C-functions", American Journal of Mathematics 97: 386–403, ISSN 0002-9327, JSTOR 2373718, MR0399357

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