Peter–Weyl Iwahori Algebras

Barbasch, Dan ; Moy, Allen

Canadian journal of mathematics, Tome 72 (2020) no. 5, pp. 1304-1323

Résumé

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 $\bullet$.

Détail

DOI : 10.4153/S0008414X19000324

Mots-clés : convolution algebra, Iwahori–Hecke algebra, idempotent, Morita equivalence, parahoric subgroup, Peter–Weyl idempotent, ⋆-algebra