Abstract In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case SU ( n + 1 ) meets all the conditions that we impose in Part 1. For any k ∈ N 0 we obtain families of orthogonal polynomials in n variables with values in the N × N -matrices, where N = ( n + k k ) . The case k = 0 leads to the classical Heckman-Opdam polynomials of type A n with geometric parameter. For k = 1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n ≥ 2 . We also give explicit expressions of the spherical functions that determine the matrix weight for k = 1 . These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n = 1 . The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for ( n , k ) equal to ( 2 , 1 ) and ( 3 , 1 ) .

中文翻译：

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SU(n + 1)×SU(n + 1) 和多变量矩阵值正交多项式的不可约表示的矩阵元素

摘要 在第 1 部分中，我们研究了在适当的重数自由度假设和分支规则的附加条件下的任意秩紧对称对上的球函数。球函数在子群的有限维表示的线性算子的空间中取值，因此球函数是矩阵值的。在这些假设下，这些函数可以用几个变量中的矩阵值正交多项式来描述，其中变量的数量是紧对称对的秩。此外，这些多项式被唯一确定为微分算子的可交换代数的同时特征函数。在第 2 部分中，我们验证组案例 SU ( n + 1 ) 满足我们在第 1 部分中强加的所有条件。对于任何 k ∈ N 0，我们获得 n 个变量中的正交多项式族，其值在 N × N 矩阵中，其中 N = ( n + kk ) 。k = 0 的情况导致具有几何参数的 A n 类型的经典 Heckman-Opdam 多项式。对于 k = 1，我们获得了最完整的结果。在这种情况下，我们给出了矩阵权重的显式表达式，当 n ≥ 2 时，我们证明它是不可约的。我们还给出了确定 k = 1 矩阵权重的球函数的显式表达式。这些表达式用于计算确定一般 k 到可逆上三角矩阵的矩阵权重的球函数。这概括并给出了 Koornwinder 最初为 n = 1 情况获得的公式的新证明。具有矩阵值多项式作为同时特征函数的微分算子的交换族包含一个一阶元素。我们给出了 (n, k) 等于 (2, 1) 和 (3, 1) 的一阶和二阶微分算子的显式公式。