Cooling capacity

2013-05-03T23:45:02Z

155.98.6.146: Please reference http://en.wikipedia.org/wiki/Kelvin; Kelvin is an absolute scale, not a degree. One does not say degrees kelvin and therefor the degrees symbol is not used.

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 group]]s. {{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,.

==Harish-Chandra's ''c''-function==

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==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 <math>\mathfrak{a}_+^*</math> onto <math>-\mathfrak{a}_+^*</math>. By Harish-Chandra's integral formula, ''c''''s''0 is Harish-Chandra's '''c'''-function:

:<math> c(\lambda)=c_{s_0}(\lambda).</math>

The '''c'''-functions are in general defined by the equation

:<math> \displaystyle A(s,\lambda)\xi_0 =c_s(\lambda)\xi_0,</math>

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:

:<math> c_{s_1s_2}(\lambda) =c_{s_1}(s_2 \lambda)c_{s_2}(\lambda)</math>

provided

:<math>\ell(s_1s_2)=\ell(s_1)+\ell(s_2).</math>

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|Karpelevič|1962}}. In fact the integral involves only the closed connected subgroup ''G''α corresponding to the Lie subalgebra generated by <math>\mathfrak{g}_{\pm \alpha}</math> 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

:<math>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))},</math>

where

:<math>c_0=2^{m_\alpha/2 + m_{2\alpha}}\Gamma\left({1\over 2} (m_\alpha+m_{2\alpha} +1)\right)</math>
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:
:<math>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))},</math>
where the constant ''c''0 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/''c''2 times Lebesgue measure.

==Generalized C-function==

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==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==

*{{Citation | last1=Cohn | first1=Leslie | title=Analytic theory of the Harish-Chandra C-function | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0064335 | mr=0422509 | year=1974 | volume=429}}
*{{Citation | editor1-last=Doran | editor1-first=Robert S. | editor2-last=Varadarajan | editor2-first=V. S. | title=Proceedings of the AMS Special Session on Representation Theory and Noncommutative Harmonic Analysis, held in memory of Harish-Chandra on the occasion of the 75th anniversary of his birth, in Baltimore, MD, January 9–10, 1998 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Proceedings of Symposia in Pure Mathematics | isbn=978-0-8218-1197-9 | mr=1767886 | year=2000 | volume=68 | chapter=The mathematical legacy of Harish-Chandra | pages=xii+551|url=http://books.google.com/books?id=mk-4pl9IftMC}}
*{{Citation | last1=Gindikin | first1=S. G. | last2=Karpelevich | first2=F. I. | title=Plancherel measure for symmetric Riemannian spaces of non-positive curvature | mr=0150239 | year=1962 | journal=Soviet Math. Dokl. | issn=0002-3264 | volume=3 | pages=962–965}}
*{{Citation | last1=Gindikin | first1=S. G. | last2=Karpelevich | first2=F. I. | title=Twelve Papers on Functional Analysis and Geometry | origyear=1966 | url=http://www.ams.org/bookstore?fn=20&arg1=trans2series&ikey=TRANS2-85 | series=American Mathematical Society translations | isbn=978-0-8218-1785-8 | mr=0222219 | year=1969 | volume=85 | chapter=On an integral associated with Riemannian symmetric spaces of non-positive curvature | pages=249–258}}
*{{Citation | last1=Harish-Chandra | title=Spherical functions on a semisimple Lie group. I | jstor=2372786 | mr=0094407 | year=1958a | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=80 | pages=241–310}}
*{{Citation | last1=Harish-Chandra | title=Spherical Functions on a Semisimple Lie Group II | jstor=2372772 | publisher=The Johns Hopkins University Press | year=1958b | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=80 | issue=3 | pages=553–613}}
*{{Citation | last1=Harish-Chandra | title=Harmonic analysis on semisimple Lie groups | doi=10.1090/S0002-9904-1970-12442-9 | mr=0257282 | year=1970 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=76 | pages=529–551}}
*{{Citation | last1=Helgason | first1=Sigurdur | editor1-last=Tanner | editor1-first=Elizabeth A. | editor2-last=Wilson. | editor2-first=Raj | title=Noncompact Lie groups and some of their applications (San Antonio, TX, 1993) | url=http://books.google.com/books?id=mk-4pl9IftMC&pg=273 | publisher=Kluwer Acad. Publ. | location=Dordrecht | series=NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. | isbn=978-0-7923-2787-5 | mr=1306516 |id= Reprinted in {{harv|Doran|Varadarajan|2000}} | year=1994 | volume=429 | chapter=Harish-Chandra's c-function. A mathematical jewel | pages=55–67}}
*{{Citation | last1=Helgason | first1=Sigurdur | title=Groups and geometric analysis | origyear=1984 | url=http://books.google.com/books?id=exqJ3RtPMYYC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-2673-7 | id={{MR|0754767}}{{MR|1790156}} | year=2000 | volume=83}}
*{{Citation | last1=Knapp | first1=Anthony W. | editor1-last=Gindikin | editor1-first=S. G. | title=Lie groups and symmetric spaces. In memory of F. I. Karpelevich | url=http://www.ams.org/bookstore-getitem/item=TRANS2-210 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Amer. Math. Soc. Transl. Ser. 2 | isbn=978-0-8218-3472-5 | mr=2018359 | year=2003 | volume=210 | chapter=The Gindikin-Karpelevič formula and intertwining operators | pages=145–159}}
*{{Citation | last1=Langlands | first1=Robert P. | title=Euler products | origyear=1967 | url=http://publications.ias.edu/rpl/paper/37 | publisher=Yale University Press | isbn=978-0-300-01395-5 | mr=0419366 | year=1971}}
*{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Spherical functions on a p-adic Chevalley group | doi=10.1090/S0002-9904-1968-11989-5 | mr=MR0222089 | year=1968 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=74 | issue=3 | pages=520–525}}
*{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Spherical functions on a group of p-adic type | publisher=Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras | series=Ramanujan Institute lecture notes | mr=0435301 | year=1971 | volume=2}}
*{{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 | pages=386–403}}

[[Category:Lie groups]]

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Cooling capacity