The Galois Correspondence for Branched Covering Spaces and Its Relationship to Hecke Algebras

There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F ). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of nonGalois covers S/P1. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of the induced character 1H , and that the character of the homology representation of a Galois group generalizes to one for Hecke algebras, the decomposition of which depends on certain double cosets in the group corresponding to the Galois closure of the cover S/P1.