Harish-Chandra's Ξ function

In mathematical harmonic analysis, Harish-Chandra's Ξ function is a special spherical function on a semisimple Lie group, studied by Harish-Chandra (1966, section 16).

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

Wallach (1988, 4.5) gives a detailed description of the properties of Ξ.

Definition

\Xi (g)=\int _{K}a(kg)^{\rho }dk,

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

gollark: ```c#define let char*#define var char#define auto int*#define fn int#define new malloc#include <stdio.h>#include <stdlib.h>#include <string.h>fn main() { let s = "abcdefghijklmnqoprstuvwxyz Lyric Ly Make Macro N"; let j_ = new(1024); strcpy(j_, s); for (var i = 0; i < 33; i++) strcat(j_, s); auto q = j_; fn x = 0x6F5D5F5F; q[0] = x; printf("%s", j_);}```

gollark: No.

gollark: string operat™.

gollark: ```c#define let char*#define var char#define auto int*#define fn int#define new malloc#include <stdio.h>#include <stdlib.h>#include <string.h>fn main() { let s = "abcdefghijklmnqoprastjasdhasdua"; let j_ = new(1024); strcpy(j_, s); for (var i = 0; i < 33; i++) strcat(j_, s); auto q = j_; fn x = 0x6F5D5F5F; q[0] = x; printf("%s", j_);}```

gollark: Yes, UTF-8 is in fact backward-compatible with non-extended ASCII, apio.

References

Harish-Chandra (1966), "Two theorems on semi-simple Lie groups", Annals of Mathematics, Second Series, 83: 74–128, doi:10.2307/1970472, ISSN 0003-486X, JSTOR 1970472, MR 0194556
Wallach, Nolan R (1988), Real reductive groups. I, Pure and Applied Mathematics, 132, Boston, MA: Academic Press, ISBN 978-0-12-732960-4, MR 0929683

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.