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 & Karpelevič (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, 429, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0064335, 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, 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, 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, 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, The Johns Hopkins University Press, 80 (3): 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: 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, 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, MR 1306516, 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, MR 1790156
• 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, 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 MR0222089
• 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, MR 0435301
• Wallach, Nolan R (1975), "On Harish-Chandra's generalized C-functions", American Journal of Mathematics, 97: 386–403, doi:10.2307/2373718, ISSN 0002-9327, JSTOR 2373718, MR 0399357