Relation between the dual module and the dual of a vector space.

Question

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite orthogonal group acting on V and U is G-invariant and self dual

Answer 1

Indeed one has to be careful with the notion of dual module:

In the context of representation theory of groups (more generally, when studying modules over Hopf algebras), the dual of a $G$-module $V$ is usually defined as having the vector space dual $V^{\ast}$ as its underlying vector space, and with the $G$-action defined by $(g.\varphi)(v) := \varphi(g^{-1}.v)$.

In contrast, when looking at modules over an arbitrary ring $R$, one might mimick the definition of dual vector space in associating to an $R$-module $M$ the module $\text{Hom}_R(M,R)$. However, this will usually not be an involution, not even for finitely generated $R$-modules. For example, if $R$ is a domain and $M$ is a torsion module (e.g. $R={\mathbb Z}$ and $M={\mathbb Z}/2{\mathbb Z}$), then $\text{Hom}_R(M,R)=0$.