标题： 仿射李群上的球面函数 显示英文标题

标题： Spherical functions on affine Lie groups

摘要： 我们描述了在群K上的向量值共轭等变函数的两种情况——K是一个紧致单李群，以及K是一个仿射李群。 我们将这些函数构造为K的表示之间某些 intertwining 算子的加权迹。对于紧致群$K$，Peter-Weyl 定理表明所有等变函数都可以写成这种迹的线性组合。 接下来，我们计算了$K$上作用于共轭等变函数的拉普拉斯算子的径向部分，并得到一个完全可积的矩阵系数量子系统，在特定情况下与三角Calogero-Sutherland-Moser多粒子系统一致。 在仿射李群的情况下，我们证明了具有固定齐次度数的等变函数空间是有限维的，并由 intertwining 算子的加权迹生成。 这个空间与椭圆曲线上的Wess-Zumino-Witten共形块空间相同。 我们计算了仿射李群上作用于等变函数的二阶拉普拉斯算子的径向部分，发现它是一个特定的抛物型偏微分算子，当中心电荷趋于负的对偶Coxeter数（临界水平）时退化为椭圆Calogero-Sutherland-Moser哈密顿量。 该哈密顿量的量子积分是在临界水平下为中心的高阶Sugawara算子的径向部分。 显示英文摘要

摘要： We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.