Peter-Weyl Iwahori Algebras

The Peter-Weyl idempotent of a parahoric subgroup is the sum of the idempotents of irreducible representations of that have a nonzero Iwahori fixed vector. The convolution algebra associated with 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, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by, and the Morita equivalence preserves irreducible and unitary modules for.