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In mathematical [[abstract harmonic analysis|harmonic analysis]], '''Harish-Chandra's ''Ξ'' function''' is a special spherical function on a [[semisimple Lie group]], studied by {{harvs|txt|last=Harish-Chandra|authorlink=Harish-Chandra|year=1966|loc=section 16}}.


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Harish-Chandra used it to define [[Harish-Chandra's Schwartz space]].


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{{harvtxt|Wallach|1988|loc=4.5}} gives a detailed description of the properties of ''Ξ''.
 
==Definition==
 
:<math>\Xi(g)=\int_Ka(kg)^\rho dk,</math>
 
where
 
*''K'' is a maximal compact subgroup of a semisimple Lie group with [[Iwasawa decomposition]] ''G''=''NAK''
*''g'' is an element of ''G''
*''ρ'' is a [[Weyl vector]]
*''a''(''g'') is the element ''a'' in the Iwasawa decomposition ''g''=''nak''
 
==References==
 
*{{Citation | last1=Harish-Chandra | title=Two theorems on semi-simple Lie groups | jstor=1970472 | mr=0194556 | year=1966 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=83 | pages=74–128}}
*{{Citation | last1=Wallach | first1=Nolan R | title=Real reductive groups. I | publisher=[[Academic Press]] | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-732960-4 | mr=929683  | year=1988 | volume=132}}
 
[[Category:Harmonic analysis]]
[[Category:Representation theory]]

Revision as of 10:11, 22 March 2013

In mathematical harmonic analysis, Harish-Chandra's Ξ function is a special spherical function on a semisimple Lie group, studied by Template:Harvs.

Harish-Chandra used it to define Harish-Chandra's Schwartz space.

Template:Harvtxt gives a detailed description of the properties of Ξ.

Definition

where

  • K is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition G=NAK
  • g is an element of G
  • ρ is a Weyl vector
  • a(g) is the element a in the Iwasawa decomposition g=nak

References

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