Integral models of representations of the current groups of simple Lie groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 2, pp. 205-271
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For the class of locally compact groups $P$ that can be written as the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group $\mathbb R^*_+$ of automorphisms of $P_0$, a new model of representations of the current groups $P^X$ is constructed. The construction is applied to the maximal parabolic subgroups of all simple groups of rank 1. In the case of the groups $G=\mathrm{SO}(n,1)$ and $G=\mathrm{SU}(n,1)$, an extension is constructed of representations of the current groups of their maximal parabolic subgroups to representations of the current groups $G^X$. The key role in the construction is played by a certain $\sigma$-finite measure (the infinite-dimensional Lebesgue measure) in the space of distributions.
Bibliography: 32 titles.
Keywords:
current group, integral model, Fock representation, canonical representation, special representation, infinite-dimensional Lebesgue measure.
@article{RM_2009_64_2_a1,
author = {A. M. Vershik and M. I. Graev},
title = {Integral models of representations of the current groups of simple {Lie} groups},
journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
pages = {205--271},
publisher = {mathdoc},
volume = {64},
number = {2},
year = {2009},
language = {en},
url = {http://geodesic.mathdoc.fr/item/RM_2009_64_2_a1/}
}
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%0 Journal Article %A A. M. Vershik %A M. I. Graev %T Integral models of representations of the current groups of simple Lie groups %J Trudy Matematicheskogo Instituta imeni V.A. Steklova %D 2009 %P 205-271 %V 64 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/RM_2009_64_2_a1/ %G en %F RM_2009_64_2_a1
A. M. Vershik; M. I. Graev. Integral models of representations of the current groups of simple Lie groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 64 (2009) no. 2, pp. 205-271. http://geodesic.mathdoc.fr/item/RM_2009_64_2_a1/