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:

${\displaystyle c(\lambda )=c_{s_{0}}(\lambda ).}$

The c-functions are in general defined by the equation

${\displaystyle \displaystyle A(s,\lambda )\xi _{0}=c_{s}(\lambda )\xi _{0},}$

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:

${\displaystyle c_{s_{1}s_{2}}(\lambda )=c_{s_{1}}(s_{2}\lambda )c_{s_{2}}(\lambda )}$

provided

${\displaystyle \ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2}).}$

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 Gindikin & Karpelevich (1962). In fact the integral involves only the closed connected subgroup G^α corresponding to the Lie subalgebra generated by ${\displaystyle {\mathfrak {g}}_{\pm \alpha }}$ 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

${\displaystyle c_{s_{\alpha }}(\lambda )=c_{0}{2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0})) \over \Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+1+i(\lambda ,\alpha _{0}))\Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0}))},}$

where

${\displaystyle c_{0}=2^{m_{\alpha }/2+m_{2\alpha }}\Gamma \left({1 \over 2}(m_{\alpha }+m_{2\alpha }+1)\right)}$

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 c_s(λ), as follows:

${\displaystyle c(\lambda )=c_{0}\prod _{\alpha \in \Sigma _{0}^{+}}{2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0})) \over \Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+1+i(\lambda ,\alpha _{0}))\Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0}))},}$

where the constant c₀ is chosen so that c(–iρ)=1 (Helgason 2000, p.447).

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

There is a similar c-function for p-adic Lie groups. Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie group.

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