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 \mathfrak{a}+^* onto -\mathfrak{a}+^*. By Harish-Chandra's integral formula, c**s0 is Harish-Chandra's c-function:

: c(\lambda)=c_{s_0}(\lambda).

The c-functions are in general defined by the equation

: \displaystyle A(s,\lambda)\xi_0 =c_s(\lambda)\xi_0,

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:

: c_{s_1s_2}(\lambda) =c_{s_1}(s_2 \lambda)c_{s_2}(\lambda)

provided

:\ell(s_1s_2)=\ell(s_1)+\ell(s_2).

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 . In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by \mathfrak{g}_{\pm \alpha} 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

:c_{s_\alpha}(\lambda)=c_0{2^{-i(\lambda,\alpha_0)}\Gamma(i(\lambda,\alpha_0))\over\Gamma({1\over 2} ({1\over 2}m_\alpha + 1+ i(\lambda,\alpha_0)) \Gamma({1\over 2} ({1\over 2}m_\alpha + m_{2\alpha} + i(\lambda,\alpha_0))},

where

:c_0=2^{m_\alpha/2 + m_{2\alpha}}\Gamma\left({1\over 2} (m_\alpha+m_{2\alpha} +1)\right) 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 cs(λ), as follows: :c(\lambda)=c_0\prod_{\alpha\in\Sigma_0^+}{2^{-i(\lambda,\alpha_0)}\Gamma(i(\lambda,\alpha_0))\over\Gamma({1\over 2} ({1\over 2}m_\alpha + 1+ i(\lambda,\alpha_0)) \Gamma({1\over 2} ({1\over 2}m_\alpha + m_{2\alpha} + i(\lambda,\alpha_0))}, 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