Peter--Weyl Iwahori algebras

TL;DR

This paper introduces Peter-Weyl Iwahori algebras, showing they are Morita equivalent to Iwahori-Hecke algebras and preserve key module properties under this equivalence.

Contribution

It establishes Morita equivalence between Peter-Weyl Iwahori algebras and Iwahori-Hecke algebras, preserving hermitian, unitary modules, and anti-involution structures.

Findings

Morita equivalence between Peter-Weyl Iwahori and Iwahori-Hecke algebras

Preservation of hermitian and unitary modules under equivalence

Compatibility with anti-involution structures

Abstract

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.