We give basic properties of the parabolic induction and coinduction functors associated to
-algebras modelled
on the pro- Iwahori
Hecke
-algebras
and
of a reductive
-adic group
and of a Levi
subgroup
when
is a
commutative ring. We show that the parabolic induction and coinduction functors are
faithful, have left and right adjoints that we determine, respect finitely generated
-modules,
and that the induction is a twisted coinduction.
I dedicate this work to the memory of
Robert Steinberg, having in mind both a nice encounter in
Los Angeles and the representations named after him,
which play such a fundamental role in the representation
theory of reductive p-adic groups.
Keywords
parabolic induction, pro-$p$ Iwahori Hecke algebra, alcove
walk basis
