Harish-Chandra C-function
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Harish-Chandra's ''c''-function is a function related to the intertwining operator between two principal series representations, that appears in the
Plancherel measure In mathematics, Plancherel measure is a measure defined on the set of irreducible unitary representations of a locally compact group G, that describes how the regular representation breaks up into irreducible unitary representations. In some cas ...
for
semisimple Lie group In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...
s. introduced a special case of it defined in terms of the asymptotic behavior of a
zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vect ...
of a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
, 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 In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
. 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

* * * * * * * * * * * * * *{{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 The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...
, issn=0002-9327 , volume=97 , issue=2 , pages=386–403 , doi=10.2307/2373718 Lie groups