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Peter–Weyl Iwahori Algebras

Part of: Lie groups

Published online by Cambridge University Press:  21 June 2019

Dan Barbasch  and
Allen Moy
Dan Barbasch
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853–0099, USA Email: barbasch@math.cornell.edu
Allen Moy
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay Road, Hong Kong Email: amoy@ust.hk
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Abstract

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 $\bullet$.

Keywords

convolution algebraIwahori–Hecke algebraidempotentMorita equivalenceparahoric subgroupPeter–Weyl idempotent⋆-algebra

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields
Secondary: 22E35: Analysis on $p$-adic Lie groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 5 , October 2020 , pp. 1304 - 1323
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author D.B. is partly supported by NSA grant H98230-16-1-0006. Author A.M. is partly supported by Hong Kong Research Grants Council grant CERG #16301915.

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