Gindikin–Karpelevich formula

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,.

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 ${\displaystyle {\mathfrak {a}}_{+}^{*}}$ onto ${\displaystyle -{\mathfrak {a}}_{+}^{*}}$. By Harish-Chandra's integral formula, c_s0 is Harish-Chandra's c-function:

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