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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A decomposition for certain real semisimple Lie groups


Author: H. Lee Michelson
Journal: Trans. Amer. Math. Soc. 213 (1975), 177-193
MSC: Primary 22E30; Secondary 43A80
MathSciNet review: 0385002
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Abstract: For a class of real semisimple Lie groups, including those for which G and K have the same rank, Kostant introduced the decomposition $ G = K{N_0}K$, where $ {N_0}$ is a certain abelian subgroup of N, and conjectured that the Jacobian of the decomposition with respect to Haar measure, as well as the spherical functions, would be polynomial in the canonical coordinates of $ {N_0}$. We compute here the Jacobian, which turns out to be polynomial precisely when the equality of ranks is satisfied. We also compute those spherical functions which restrict to polynomials on $ {N_0}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0385002-6
PII: S 0002-9947(1975)0385002-6
Article copyright: © Copyright 1975 American Mathematical Society