# 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 s$_0$, 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 ξ$_0$ is the constant function 1 in L$^2$(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 Σ$_0$$^+$. Then G$^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 α$_0$=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 c$_0$ 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/c$^2$ 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.