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. {{harvs|txt|last=Harish-Chandra|authorlink=Harish-Chandra|year1=1958a|year2=1958b}} introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and {{harvs|txt|last=Harish-Chandra|year=1970}} introduced a more general c-function called Harish-Chandra's (generalized) C-function. {{harvs|txt|last=Gindikin|authorlink=Simon Gindikin|last2=Karpelevich|author2-link=Fridrikh Israilevich Karpelevich|year1=1962|year2=1969}} 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 $\mathfrak{a}_+^*$ 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 c_s to the case when s = s_α, the reflection in a (simple) root α, the so-called

"rank-one reduction" of {{harvtxt|Gindikin|Karpelevich|1962}}. In fact the integral involves only the closed connected subgroup G^α corresponding to the Lie subalgebra generated by $\mathfrak{g}_{\pm \alpha}$ 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

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

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 c_s(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 {{harv|Helgason|2000|loc=p.447}}.

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.

{{harvs|txt|last=Macdonald|year1=1968|year2=1971}} and {{harvtxt|Langlands|1971}} found an analogous product formula for the c-function of a p-adic Lie group.

References

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Category:Lie groups