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 C-function. introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function.

Harish-Chandra's ''c''-function

Gindikin–Karpelevich formula

The c-function has a generalization cw 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, cs0 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. 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 root α, the so-called
"rank-one reduction" of. In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by where α lies in Σ0+. Then Gα is a real semisimple Lie group with real rank one, i.e. dim Aα = 1,
and cs is just the Harish-Chandra c-function of Gα. 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 is an immediate consequence of this formula and the multiplicative properties of cs, as follows:
where the constant c0 is chosen so that c=1.

Plancherel measure

The c-function appears in the Plancherel theorem for spherical functions, and the Plancherel measure is 1/c2 times Lebesgue measure.

Generalized C-function

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.