标题： Olshanski 的无限维运动群的球函数固定秩 显示英文标题

标题： Olshanski spherical functions for infinite dimensional motion groups of fixed rank

摘要： 考虑与域 $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ 上的运动群相关的 Gelfand 对 $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$，其中 $p\geq q$ 固定且 $q$，以及归纳极限 $p\to\infty$，Olshanski 球面对 $(G_\infty,K_\infty)$。 我们把所有Olshanski球函数的$(G_\infty,K_\infty)$分类为在正半定$q\times q$矩阵的锥体$\Pi_q$上的函数，并证明它们作为$(G_p,K_p)$的球函数在$p\to\infty$时的（局部）一致极限出现。后者由$\Pi_q$上的贝塞尔函数给出。此外，我们确定了所有正定的Olshanski球函数，并讨论了与矩阵贝塞尔函数相关的正积分表示。我们还将这些结果扩展到与非紧格拉斯曼流形的Cartan运动群相关的对$(M_{p,q} \rtimes (U_p\times U_q),(U_p\times U_q))$。 这里 类型B的Dunkl-Bessel函数（对于有限$p$）和类型A的Dunkl-Bessel函数（对于$p\to\infty$）作为球函数出现。 显示英文摘要

摘要： Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski spherical functions of $(G_\infty,K_\infty)$ as functions on the cone $\Pi_q$ of positive semidefinite $q\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to\infty$. The latter are given by Bessel functions on $\Pi_q$. Moreover, we determine all positive definite Olshanski spherical functions and discuss related positive integral representations for matrix Bessel functions. We also extend the results to the pairs $(M_{p,q} \rtimes (U_p\times U_q),(U_p\times U_q))$ which are related to the Cartan motion groups of non-compact Grassmannians. Here Dunkl-Bessel functions of type B (for finite $p$) and of type A (for $p\to\infty$) appear as spherical functions.