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Peter–Weyl Iwahori Algebras
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 5 / October 2020
- Published online by Cambridge University Press:
- 21 June 2019, pp. 1304-1323
- Print publication:
- October 2020
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- Article
- Export citation
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The Peter–Weyl idempotent
$e_{\mathscr{P}}$ of a parahoric subgroup 
$\mathscr{P}$ is the sum of the idempotents of irreducible representations of 
$\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with 
$e_{\mathscr{P}}$ 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 
$\bullet$.
