Green Correspondence for Virtually Pro-p Groups∗

Let p be a prime number, G a finite group, Q a p-subgroup of G and L any subgroup of G containing the normalizer NG(Q) of Q in G. Let k be a field of positive characteristic p. In [2] J.A. Green demonstrates a fundamental correspondence between finitely generated kG-modules with vertex Q and finitely generated kL-modules with vertex Q. When L = NG(Q) the Green correspondence allows for the reduction of many questions about general modules to questions about modules with a normal vertex. Now let G be a profinite group and k a finite field of characteristic p. In [3] we took some first steps towards a modular representation theory of profinite groups. In particular we demonstrated a classification theorem for relatively projective finitely generated k[[G]]-modules, introduced vertices and sources, and showed that the expected uniqueness properties hold for these objects (under additional hypotheses in the case of sources). Here we generalize the Green correspondence (properly interpreted) to the class of virtually pro-p groups. We will reference [3] frequently, since many necessary foundational results are discussed therein. Our main result is the following: