Harish-Chandra's c-function

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.

Gindikin–Karpelevich formula

The c-function has a generalization c_w(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, c_s0 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:

$c_{s_{1}s_{2}}(\lambda )=c_{s_{1}}(s_{2}\lambda )c_{s_{2}}(\lambda )$

provided

$\ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2}).$

This reduces the computation of c_s to the case when s = s_α, the reflection in a (simple) root α, the so-called "rank-one reduction" of Gindikin & Karpelevich (1962). In fact the integral involves only the closed connected subgroup G^α corresponding to the Lie subalgebra generated by ${\mathfrak {g}}_{\pm \alpha }$ where α lies in Σ₀⁺. Then G^α is a real semisimple Lie group with real rank one, i.e. dim A^α = 1, and c_s is just the Harish-Chandra c-function of G^α. 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 c_s(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.

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, ISBN 978-3-540-07017-7, MR 0422509
• 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 (2): 241–310, doi:10.2307/2372786, ISSN 0002-9327, JSTOR 2372786, MR 0094407
• Harish-Chandra (1958b), "Spherical Functions on a Semisimple Lie Group II", American Journal of Mathematics, 80 (3), The Johns Hopkins University Press: 553–613, doi:10.2307/2372772, ISSN 0002-9327, JSTOR 2372772
• 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
• 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, MR 1306516, Reprinted in (Doran & Varadarajan 2000)
• Helgason, Sigurdur (2000) [1984], Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2673-7, MR 1790156
• 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
• 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 (2): 386–403, doi:10.2307/2373718, ISSN 0002-9327, JSTOR 2373718, MR 0399357