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Peter-Weyl Iwahori algebras. (English) Zbl 1448.22003
Summary: The Peter-Weyl idempotent \(e_{\mathcal{P}}\) of a parahoric subgroup \(\mathcal{P}\) is the sum of the idempotents of irreducible representations of \(\mathcal{P}\) that have a nonzero Iwahori fixed vector. The convolution algebra associated with \(e_{\mathcal{P}}\) is called a Peter-Weyl Iwahori algebra. We show that any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algebra have a natural conjugate linear anti-involution \(\star\), and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by \(\bullet\), and the Morita equivalence preserves irreducible and unitary modules for \(\bullet\).
MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
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