$\mathbb{C}^{2l}$ as a representation of $\mathfrak{sp}(2l)$ I read somewhere that $\mathbb{C}^{2l}$ can be considered as a representation of $\mathfrak{sp}(2l)$, but I don’t understand how the action could be defined. Is it an irreducible representation? If so, what would its highest weight be? I thought of computing the eigenvalues of elements $h$ in the Cartan subalgebra $\mathfrak{h} = span\left\{E_{ii} - E_{l+i, l+i} \right\}_{i=1..l}$, acting on the standard basis of $\mathbb{C}^{2l}$. Does this make any sense?
 A: Taking the standard basis of $\Bbb C^{2l}$ lets us regard elements of $\mathfrak{sp}(2l) := \mathfrak{sp}(\Bbb C^{2l})$ as $(2l) \times (2l)$ matrices in the usual way.
With this matrix realization in hand, the irreducible $(2l)$-dimensional representation of $\mathfrak{sp}(2l)$ is just the set $\Bbb C^{2l}$ of column vectors, where $X \in \mathfrak{sp}(2l)$ acts by multiplication $\mathfrak{sp}(2l) \times \Bbb C^{2l} \to \Bbb C^{2l}$ on the left. This is the standard or defining representation of $\mathfrak{sp}(2l)$.
Remark This representation is isomorphic to its dual via the canonical identification $\Bbb C^{2l} \leftrightarrow (\Bbb C^{2l})^*$, $x \leftrightarrow \omega(x, \,\cdot\,)$, where $\omega : \Bbb C^{2l} \times \Bbb C^{2l} \to \Bbb C$ is the symplectic form on $\Bbb C^{2l}$.
Picking a particular $\omega$ lets us talk about weights more concretely. A typical convention is
$$\omega(x, y) := {}^\top \!x J y, \qquad J := \pmatrix{\cdot&I_l\\-I_l&\cdot} .$$
Like you say, a natural choice for a Cartan subalgebra $\mathfrak h$ with this convention is $$\mathfrak{h} := (H_i), \qquad H_i := E_{ii} - E_{l + i, l + i} .$$

If we denote by $(L_i)$ the basis of $\mathfrak{h}^*$ dual to the basis $(H_i)$ of $\mathfrak{h}$ and by $(e_i, e_{l+i})$ the standard basis of $\Bbb C^{2l}$, we can verify directly that each $e_i$ is an eigenvector of the $h$-action of eigenvalue $L_i$ and each $e_{l+i}$ is an eigenvector of eigenvalue $-L_i$. Explicitly, $H_j \cdot e_i = L_i(H_j) e_i$ and similarly for the elements $e_{l+i}$.

In the usual convention the highest weight is $L_1$, and when we want to emphasize the role of roots or weights we sometimes denote this representation by $\Gamma_{1,0,0}$.
The weights, all of which have multiplicity $1$, form a single orbit of the action of the Weyl group, so the general theory of representations of simple Lie algebras implies that representation is irreducible.
For much more, see Fulton & Harris' Representation Theory: A First Course; $\S\S$16–17 treat the representation theory of the Lie algebras $\mathfrak{sp}(2l)$.
