  We extend a comparison theorem of Anandavardhanan-Borisagar between the
quotient of the induction of a mod $p$ character by the image of an
Iwahori-Hecke operator and compact induction of a weight to the case of the
trivial character. This involves studying the corresponding non-commutative
Iwahori-Hecke algebra. We use this to give an Iwahori theoretic reformulation
of the (semi-simple) mod $p$ Local Langlands Correspondence discovered by
Breuil and reformulated functorially by Colmez. This version of the
correspondence is expected to have applications to computing the mod $p$
reductions of semi-stable Galois representations.

Number Theory;Representation Theory