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.

The c-function has a generalization cw(λ) depending on an element w of the Weyl group.

The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions: provided 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 Gindikin & Karpelevich (1962).

In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by

Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie group.