INVERSION OF HOROSPHERICAL INTEGRAL TRANSFORM ON REAL SEMISIMPLE LIE GROUPS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 1047-1071 ◽  
Author(s):  
Anton ZORICH
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Inversion Formula ◽  
Integral Transform ◽  
Regular Representation ◽  
Integral Transforms ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Complex Semisimple ◽  
Local Inversion

There exists the wonderful integral transform on complex semisimple Lie groups, which assigns to a function on the group the set of its integrals over "generalized horospheres" — some specific submanifolds of the Lie group. The local inversion formula for this integral transform, discovered in 50's for [Formula: see text] by Gel'fand and Graev, made it possible to decompose the regular representation on [Formula: see text] into irreducible ones. In case of real semisimple Lie group the situation becomes more complicated, and usually there is no reasonable analogous integral transform at all. Nevertheless, in the present paper we succeed to define the integral transforms on the Lorentz group and some other real semisimple Lie groups, which are in a sense analogous to the integration over "horospheres". We obtain the inversion formulas for these integral transforms.

On Some q-Analogs of a Theorem of Kostant-Rallis

2000 ◽  
Vol 52 (2) ◽  
pp. 438-448 ◽  
Author(s):  
N. R. Wallach ◽  
J. Willenbring
Keyword(s):  
Conjugacy Class ◽  
Quantum Computation ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Cartan Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Real Group ◽  
Qubit States

AbstractIn the first part of this paper generalizationsof Hesselink’s q-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, ) (that is SO(4) acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

scholarly journals Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 17-29 ◽  
Author(s):  
Kenneth D. Johnson
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Nilpotent Subgroup ◽  
Iwasawa Decomposition ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Simply Connected

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

scholarly journals Representations of complex semisimple Lie groups and Lie algebras

1966 ◽  
Vol 72 (3) ◽  
pp. 522-526 ◽  
Author(s):  
K. R. Parthasarathy ◽  
R. Ranga Rao ◽  
V. S. Varadarajan
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Semisimple Lie Groups ◽  
Complex Semisimple
Sign in / Sign up

Export Citation Format

Share Document