Harish-Chandra's c-function explained

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 lengths₀, is the unique element that carries the Weyl chamber

ak{a} ^{*} _{+} onto - ak{a} ^{*} _{+} . By Harish-Chandra's integral formula, c_s0 is Harish-Chandra's c-function:c( λ ) = c _{s _{0}} ( λ ).

The c-functions are in general defined by the equation

\displaystyle A(s, λ )\xi _{0} = c _{s(λ)\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}} ( λ ) = c _{s _{1}} (s _{2} λ )c _{s _{2}} ( λ )

provided

\ell(s _{1s} _{2)=\ell(s} _{1)+\ell(s} _{2).}

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

ak{g} _{\pm} 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 byc _{s _{\alpha}} ( λ ) = c ^{- i( λ ,\alpha _{0)}} _{0 { 2} \Gamma(i( λ ,\alpha _{0))\over\Gamma({1\over} 2 } ({ 1 \over 2 }m _{\alpha} + 1+ i( λ ,\alpha _{0))} \Gamma({ 1 \over 2 } ({ 1 \over 2 }m _{\alpha} + m _{2\alpha} + i( λ ,\alpha _{0))},}

where

c ^{m _{\alpha/2} + m _{2\alpha}} _{0=2} \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 c_s(λ), as follows:

c( λ ) = c _{0\prod} ^{\alpha\in\Sigma ^{+} _{0}} { 2 ^{- i( λ ,\alpha _{0)}} \Gamma(i( λ ,\alpha _{0))\over\Gamma({1\over} 2 } ({ 1 \over 2 }m _{\alpha} + 1+ i( λ ,\alpha _{0))} \Gamma({ 1 \over 2 } ({ 1 \over 2 }m _{\alpha} + m _{2\alpha} + i( λ ,\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.