Peter--Weyl Iwahori algebras

The Peter-Weyl idempotent e(p) of a parahoric subgroup P is the sum of the idempotents of irreducible representations of P that have a nonzero Iwahori fixed vector. The convolution algebra associated with e(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 *, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by 4., and the Morita equivalence preserves irreducible and unitary modules for *.