Let V be a simple vertex operator algebra and G a finite automorphism
group of V. A major problem in orbifold conformal field theory is to
determine the module category of V^G. Here V^G is the G-invariant
sub-vertex operator algebra of V. Let S be a finite set of
inequivalent irreducible V-modules. Then there is a finite dimensional
semisimple associative algebra A_{\lapha}(G,S) such that A_{\alpha}(G,S)
and V^G form a dual pair on the sum of V_modules in S in the sense
of
Howe. In particular, every irreducible V-module is completely reducible
V^G-module.