${\bf 1}+O_{n}(V).$ Moreover $\omega+O_{n}(V)$ is a central element of $A_{n}(V).$

2\) For $0\leq m\leq n,$ the map $\psi_n: v+O_n(V)\mapsto o_M(v)$ from $A_n(V)$ to $\End M(m)$ makes $M(m)$ an $A_n(V)$-module.

3\) $M$ is irreducible if and only if $M(n)$ is a simple $A_n(V)$-module for all $n.$

4\) The identity map on $V$ induces an onto algebra homomorphism from $A_n(V)$ to $A_m(V)$ for $0\leq m\leq n.$

5\) Two irreducible admissible $V$-modules $M^1$ and $M^2$ with $M^1(0)\ne 0$ and $M^2(0)\ne 0$ are isomorphic if and only if $M^1(n)$ and $M^2(n)$ are isomorphic $A_n(V)$-modules for any $n\geq 0$.

Parts 1)-2) and 4)-5) are proved in \[DLM4\], and 3) is given in \[DM4\].

Dual pair I
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In the next two sections we assume that $V$ is a simple vertex operator algebra. Our main goal is to generalize the duality result obtained in \[DLM1\] on $V$ to an arbitrary irreducible $V$-module. More precisely, let $G$ be a finite automorphism group of $V$ and $M$ an irreducible $V$-module. Then $M$ is a module for the vertex operator subalgebra $V^G$ of $G$-invariants. There are three questions one can ask: (1) Is $M$ is a completely reducible $V^G$-module? (2) What are irreducible modules for $V^G$ which occur as 