高维超球上不变测度的Laplace变换的性质
The Behavior of Laplace Transform of the Invariant Measure on the Hypersphere of High Dimension
摘要：
考虑群$SL(n,\Bbb R), n=1 ...$的Cartan子群$SDiag(n,\Bbb R)$的超球面序列$M_{n,r}$，即齐次传递空间，研究了这些空间上相应的不变测度序列$m_n$的正规极限.在紧群和齐性空间的情况下，例如-对于经典对$(SO(n), S^{n-1}), n=1 ...$ -对应测度的极限是经典的无限维高斯测度-这是众所周知的麦克斯韦-庞加莱引理。同时，高斯测度是关于无限正交群$O(\infty)$作用的唯一（直到标量）不变测度。这种重合意味着对$(SO(n), S^{n-1})$的系列的大正则系综和小正则系综之间的渐近等价性。我们的主要结果表明，非紧群的情况，例如$(SDiag(n,\Bbb R),M_{n,r})$（定义见下文），是完全不同的：极限的措施$m_n $不存在字面意义上，我们表明，只有规范化的对数极限的拉普拉斯变换的这些措施确实存在。在同一时间存在的措施，这是不变的措施，就连续模拟的Cartan子群的组$GL(\infty)$ -这是所谓的无限维勒贝格（见\cite{V}）。这种差异是一个证据之间的不等价的大和小的非紧情况下的典型合奏。
英文摘要：
We consider the sequence of the hyperspheres $M_{n,r}$ i.e. the homogeneous transitive spaces - of the Cartan subgroup $SDiag(n,\Bbb R)$ of the group $SL(n,\Bbb R), n=1 ...$, and studied the normalized limit of the corresponding sequence of the invariant measures $m_n$ on those spaces. In the case of compact groups and homogeneous spaces, as example - for classical pairs $(SO(n), S^{n-1}), n=1 ...$ - 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 $O(\infty)$. This coincidences means the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs $(SO(n), S^{n-1})$. Our main result shows that situation for noncompact groups, for example for the case $(SDiag(n,\Bbb R),M_{n,r})$ (the definitions see below), is completely different: the limit of measures $m_n $ 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 $GL(\infty)$ - this is so called infinite dimensional Lebesgue (see \cite{V}). This difference is an evidence of the non-equivalence between the grand and small canonical ensembles for the noncompact case.
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