The Behavior of Laplace Transform of the Invariant Measure on the Hypersphere of High Dimension

We consider the sequence of the hyperspheres Mn,r i.e. the homogeneous transitive spaces of the Cartan subgroup SDiag(n, R) of the group SL(n, R), n = 1 . . . , and studied the normalized limit of the corresponding sequence of the invariant measures mn on those spaces. In the case of compact groups and homogeneous spaces, as example for classical pairs (SO(n), Sn−1), n = 1 . . . the limit of corresponding measures is the classical infinite dimensional gaussian measure this is well-known Maxwell-Poincare lemma. Simultaneously that Gaussian measure is a unique (up to scalar) invariant measure with respect to the action of infinite orthogonal group O(∞). This coincidences means the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), Sn−1). Our main result shows that situation for noncompact groups, for example for the case (SDiag(n, R),Mn,r) (the definitions see below), is completely different: the limit of measures mn does not exist in literal sense, and we show that only normalized logarithmic limit of the Laplace transform of those measures does exist. In the same time there exists the measure which is invariant measure with respect to continuous analogue of Cartan subgroup of the group GL(∞) this is so called infinite dimensional Lebesgue (see [1]). This difference is an evidence of the non-equivalence between the grand and small canonical ensembles for the noncompact case. 1 Setting of the problem. Consider the series of classical Lie groupsGn and corresponding homogeneous space Mn, equipped with Gn-invariant measures mn. Does there exist the natural (weak) limit of such sequences of measures as a measure defined in some infinite-dimensional space? 1 There are at least two specifications of the question: we can try to find a measure which is invariant under the action of an infinite dimensional analog of our groups; and the second approach is to find a direct limit of the finite dimensional measures. These two approaches coincide for the classical case of orthogonal group SO(n) and spheres S as homogeneous spaceswe obtain a standard Gaussian measure in both cases in the first approach as unique (up to the scalar) ergodic measure which is invariant under infinite dimensional orthogonal group SO(∞) (Schoenberg theorem see [1, 2]) and in the second approach -as Maxwell-Poincare lemma (MP-lemma) about the weak limit of the sequence of normalized Lebesgue measures on the n − 1dimensional spheres of the radius c √ n (see detail discussion of this case in [1]). We consider the generalization of this scheme for noncompact group. The first nontrivial example is given by the case of the diagonal (Cartan) subgroup SDiag(n,R) of the group SL(nR). More exactly, in the paper [1] we considered the group of positive diagonal matrices with determinant 1,the group SDiag+(n,R), and its homogeneous space the manifold, which sometimes called ”hypersphere of radius r”: