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. introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and introduced  a more general c-function called Harish-Chandra's (generalized) C-function. introduced the Gindikin–Karpelevich formula, a product formula for 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, cs 0 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. 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.

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. and found an analogous product formula for the c-function of a p-adic Lie group.