In this paper(http://arxiv.org/pdf/1204.6131.pdf), the following statement
$V =K^n$ and its dual are isomorphic $SL(2,K)$-modules,
seems to be common sense. Here, $K$ is a field of characteristic $0$.
I don't know if $G = SL(2,K)$ acts on the dual of $V$, namely $V^*$, in this fashion:
$(gf)(x) = f(g^{-1}x)$ for any $g \in G$, $f \in V^*$ and $x \in V$.
If this is the case, let $e_1, \cdots, e_n$ be a basis of $V$, and $x_1, \cdots, x_n$ be a dual basis of $V^*$, that is, $x_i(e_j) = \delta_{ij}$, then for any $g \in G$, $g(e_i) = \sum_{j=1}^n a_{ji}e_j$ for a matrix $A = (a_{ij}) \in GL(n,K)$. Let $A^{-1} = (a_{ij}')$ be its inverse.
Suppose that $V$ and $V^*$ are isomorphic $G$-modules and $\phi: V \rightarrow V^*, e_i \mapsto \sum_{j=1}^n b_{ji}x_j$ gives such an isomorphism. Then as $(gx_i) (e_j) = x_i(g^{-1}e_j) = x_i(\sum_{k=1}^n a_{kj}'e_k) = a_{ij}'$, $g(x_i) = \sum_{j=1}^n a_{ij}'x_j$.
So $g(\phi(e_i)) = g(\sum_{j=1}^n b_{ji}x_j) = \sum_{j=1}^n b_{ji}(gx_j) = \sum_{j=1}^n b_{ji} (\sum_{k=1}^n a_{jk}' x_k) = \sum_{j,k=1}^n b_{ji} a_{jk}' x_k$, and $\phi(g(e_i)) = \phi(\sum_{j=1}^n a_{ji} e_j) = \sum_{j=1}^n a_{ji} \phi(e_j) = \sum_{j=1}^n a_{ji} (\sum_{k=1}^n b_{kj} x_k) = \sum_{j,k=1}^n a_{ji} b_{kj} x_k$.
They are equal, so if I let $B = (b_{ij})$, then $B^T A^{-1} = A^T B^T$, that is $BA = (A^{-1})^T B$.
But if I let $K = \mathbb C$, and $G$ acts on $\mathbb C^2$ in the natural way, it seems impossible to find $B \in GL(2, \mathbb C)$ such that $BA = (A^{-1})^T B$ for any $A \in G$. (For example, there is no such common $B$ for $A_1 = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $A_2 = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$.)
So, the statement in the paper does not seem to be exactly the case. But I trust the author and I believe there must be something wrong in my method or calculation. Would anyone please tell me where I am wrong?
Moreover, the author explained to be in the language of the representation theory of Lie algebras. I would be appreciate if someone is willing to explain to me the correspondence of representation of the algebraic group and the Lie algebra in case of $SL(2,K)$.
Thanks a lot.