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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:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields 2.2e+36 Analysis on $$p$$-adic Lie groups
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