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_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 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
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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
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