Gindikin–Karpelevich Formula

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 length
''s''₀, is the unique element that carries the Weyl chamber $\mathfrak_+^*$ onto $-\mathfrak_+^*$. By Harish-Chandra's integral formula, ''c''_''s''0 is Harish-Chandra's c-function:

:$c(\lambda)=c_(\lambda).$

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:

:$c_(\lambda) =c_(s_2 \lambda)c_(\lambda)$

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 . In fact the integral involves only the closed connected subgroup ''G''^α corresponding to the Lie subalgebra generated by $\mathfrak_$ 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

:$c_(\lambda)=c_0,$

where

:$c_0=2^\Gamma\left( (m_\alpha+m_ +1)\right)$
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:
:$c(\lambda)=c_0\prod_,$
where the constant ''c''₀ 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/''c''² 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

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*{{Citation , last1=Wallach , first1=Nolan R , title=On Harish-Chandra's generalized C-functions , jstor=2373718 , mr=0399357 , year=1975 , journal=American Journal of Mathematics , issn=0002-9327 , volume=97 , issue=2 , pages=386–403 , doi=10.2307/2373718

Lie groups