标题： 格拉斯曼流形上的不变函数 显示英文标题

标题： Invariant Functions on Grassmannians

摘要： 已知，在$\bbr^n$中的单位球面上的每个函数，如果关于某个坐标轴的旋转是不变的，则完全由一个变量的函数确定。 当函数的不变性降低其实际自变量的维度时，类似的结果适用于每个紧致对称空间，并可以在李理论的框架内得到。 在本文中，这一现象对于在$G_{n,i}$上的函数给出了精确的意义，该函数是$\bbr^n$中$i$维子空间的格拉斯曼流形，并且在保持任意固定维数的补坐标子空间的正交变换下是不变的。 得到了相应的积分公式。 我们的方法依赖于双施蒂费尔分解，而不涉及李理论。 显示英文摘要

摘要： It is known, that every function on the unit sphere in $\bbr^n$, which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces dimension of its actual argument, hold for every compact symmetric space and can be obtained in the framework of Lie-theoretic consideration. In the present article, this phenomenon is given precise meaning for functions on the Grassmann manifold $G_{n,i}$ of $i$-dimensional subspaces of $\bbr^n$, which are invariant under orthogonal transformations preserving complementary coordinate subspaces of arbitrary fixed dimension. The corresponding integral formulas are obtained. Our method relies on bi-Stiefel decomposition and does not invoke Lie theory.