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 [math]\displaystyle{ \mathfrak{a}_+^* }[/math] onto [math]\displaystyle{ -\mathfrak{a}_+^* }[/math]. By Harish-Chandra's integral formula, c_s0 is Harish-Chandra's c-function:

[math]\displaystyle{ c(\lambda)=c_{s_0}(\lambda). }[/math]

The c-functions are in general defined by the equation

[math]\displaystyle{ \displaystyle A(s,\lambda)\xi_0 =c_s(\lambda)\xi_0, }[/math]

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:

[math]\displaystyle{ c_{s_1s_2}(\lambda) =c_{s_1}(s_2 \lambda)c_{s_2}(\lambda) }[/math]

provided

[math]\displaystyle{ \ell(s_1s_2)=\ell(s_1)+\ell(s_2). }[/math]

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). In fact the integral involves only the closed connected subgroup G^α corresponding to the Lie subalgebra generated by [math]\displaystyle{ \mathfrak{g}_{\pm \alpha} }[/math] 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

[math]\displaystyle{ 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))}, }[/math]

where

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

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:

[math]\displaystyle{ 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))}, }[/math]

where the constant c₀ is chosen so that c(–iρ)=1 (Helgason 2000).

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, ISBN 978-3-540-07017-7 
• 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, https://books.google.com/books?id=mk-4pl9IftMC 
• 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 
• Gindikin, S. G.; Karpelevich, F. I. (1969), "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, https://www.ams.org/bookstore?fn=20&arg1=trans2series&ikey=TRANS2-85 
• 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 
• 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 
• 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 
• 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, Reprinted in (Doran Varadarajan), ISBN 978-0-7923-2787-5, https://books.google.com/books?id=mk-4pl9IftMC&pg=273 
• Helgason, Sigurdur (2000), Groups and geometric analysis, Mathematical Surveys and Monographs, 83, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2673-7, https://books.google.com/books?id=exqJ3RtPMYYC 
• 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, https://www.ams.org/bookstore-getitem/item=TRANS2-210 
• Langlands, Robert P. (1971), Euler products, Yale University Press, ISBN 978-0-300-01395-5, http://publications.ias.edu/rpl/paper/37 
• 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 
• 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 
• 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