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 "c"(λ) depending on an element "w" of the Weyl group . The unique element of greatest length "s", is the unique element that carries the Weyl chamber formula_1 onto formula_2. By Harish-Chandra's integral formula, "c" is Harish-Chandra's c-function: The c-functions are in general defined by the equation where ξ is the constant function 1 in L("K"/"M"). The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions: provided This reduces the computation of c to the case when "s" = "s", the reflection in a (simple) root α, the so-called "rank-one reduction" of . In fact the integral involves  only the closed connected subgroup "G" corresponding to the Lie subalgebra generated by formula_7 where α lies in Σ. Then "G" is a real semisimple Lie group with real rank one, i.e. dim "A" = 1, and c is just the Harish-Chandra c-function of "G". In this case the c-function can be computed directly and is given by where and α=α/〈α,α〉. The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of c(λ), as follows: where the constant "c" 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/"c" times Lebesgue measure. p-adic Lie groups. There is a similar "c"-function for "p"-adic Lie groups.