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_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 onto . By Harish-Chandra's integral formula, c_s₀ is Harish-Chandra's c-function:

The c-functions are in general defined by the equation

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:

provided

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 . In fact the integral involves only the closed connected subgroup G^α corresponding to the Lie subalgebra generated by where α lies in Σ₀⁺. 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

where

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 c_s(λ), as follows:

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.

References