A commutative model of a representation of the group \(O(n, 1)^{X}\) and a generalized Lebesgue measure in the space of distributions
Let \(X\) be a smooth manifold and \(m\) a finite measure with a continuous density and \(G\) a Lie group. The paper under review continues a series of publications devoted to the representation theory of the corresponding current group \(G^X\) of bounded locally continuous mappings from \(X\) to \(G\) under pointwise multiplication. Here \(G=O(n,1)\) and the representation is what was called the basic representation in their earlier work. The authors describe a commutative model with respect to a natural maximal unipotent subgroup \(Z\) of \(G\). Then \(Z\) and the current group \(Z^X\) are both abelian, and the spectral theorem provides an \(L^2\)-space \(L^2(\nu)\) on which \(Z^X\) acts by multiplication group. If one realizes the representation of \(G^X\) on \(L^2(\nu)\), then one speaks about a commutative model. In this paper the representation on \(L^2(\nu)\) is a construction from scratch, and it is proved that the representation constructed is indeed the basic representation by relating it to the spherical functions coming from the basic representation.
scientific article; zbMATH DE number 5161600