The Peter-Weyl idempotent $e_{\mathcal{P}}$ of a parahoric subgroup ${\mathcal{P}}$ is the sum of the idempotents of irreducible representations of $\mathcal{P}$ which have a nonzero Iwahori fixed vector. The convolution algebra associated to $e_{\mathcal{P}}$ is called a Peter-Weyl Iwahori algebra. We show any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algbera have a natural $\mathbb{C}^\star$-algebra structure, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution denoted as $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for the $\bullet$-involution.

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PETER-WEYL IWAHORI 代数

平行子群 ${\mathcal{P}}$ 的 Peter-Weyl 幂等 $e_{\mathcal{P}}$ 是 $\mathcal{P}$ 的不可约表示的幂等的和，这些表示具有非零的岩堀固定向量。与 $e_{\mathcal{P}}$ 相关的卷积代数称为 Peter-Weyl Iwahori 代数。我们证明任何 Peter-Weyl Iwahori 代数是 Morita 等价于 Iwahori-Hecke 代数。Iwahori-Hecke 代数和 Peter-Weyl Iwahori 代数都具有自然的 $\mathbb{C}^\star$-代数结构，并且 Morita 等价保留了不可约的厄密模和酉模。两个代数都有另一个反对合，表示为 $\bullet$，并且 Morita 等价为 $\bullet$-对合保留了不可约和酉模。