Say I have some irreducible $Sp_{2n}(\mathbb{C})$ representation, such as $\Gamma_{0,1,0,1}$.
Consider the subgroup of $Sp_{2n}(\mathbb{C})$ isomorphic to $SL_n(\mathbb{C})$, consisting of matrices of the form
$$\begin{pmatrix} M&0\\ 0&(M^{t})^{-1} \end{pmatrix}$$
with $M \in SL_n(\mathbb{C})$. What steps can I take to decompose my $Sp_{2n}(\mathbb{C})$ representation as a direct sum of irreducible $SL_n(\mathbb{C})$ representations? Either by hand or by a program such as LiE. I believe $M \in SL_n(\mathbb{C})$ acts as $V \oplus V^*$, where $V$ is the standard representation of $SL_n(\mathbb{C})$ and $V^*$ is its dual.