I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$.
I can easily find the above info for orthogonal groups and unitary groups (the representations coming from certain spaces of harmonic polynomials) but cannot seem to find such a description for the groups $\text{Sp}(n)$.
In particular I am aware that the irreducible representations of $\text{Sp}(2)$ are classified by Young diagram parameters $(a,b)$ with $a\geq b \geq 0$ but I would like to compute with such spaces so need an explicit description if possible.
Best Answer
Consider the Lie algebra $\frak{sp}(n)$ of the symplectic group ${\rm Sp}(n)={\rm Sp}(n; \mathbb{C})\cap{\rm SO}(4n)$.
For $n=1$, let us denote by $V$ the representation of $\frak{sp}(1)$ on $\mathbb{C}^{2}$. Then, any irreducible $\frak{sp}(1)$-representation is isomorphic to some symmetric power ${\rm Sym}^{p}V$ of $V$.
For $n>1$ let us denote by $W$ the $2n$-dimensional complex representation of $\frak{sp}(n)$. Then, the symmetric powers ${\rm Sym}^{p}W$ are also irreducible. But not the exterior powers $\Lambda^{p}W$. Let us denote the kernel of the contraction $\Lambda^{p}W\to \Lambda^{p-2}W$ induced by the invariant 2-form on $W$, by $\Lambda_{0}^{p}W$. For $p\leq n$ one can show that this submodule $\Lambda_{0}^{p}W$ is also irreducible. One can moreover define the modules ${\rm Ker}\{\Lambda^{2}({\rm Sym}^{2}W)\to {\rm Sym}^{2}W\}$ and ${\rm Ker}\{\Lambda^{2}({\Lambda}^{2}_{0}W)\to {\rm Sym}^{2}W\}$, but for $n=2$ the second $\frak{sp}(n)$-module is trivial. Some details can be found in the classical book of Simon Salamon, ''Riemannian geometry and holonomy groups''. In particular, for ${\rm Sp}(2)$ see pages 80-84. Finally notice that since $\frak{sp}(2)\cong\frak{so}(5)$, the spin representation of ${\rm SO}(5)$ is isomorphic to the standard representation $W$ of ${\rm Sp}(2)$ and the standard representation of ${\rm SO}(5)$ is isomorphic with $\Lambda_{0}^{2}W$. Of course, using the LiE program you can take a quick view of the complex irreducible representations of ${\rm Sp}(2)$, in terms of highest weights.
added The irreducible representation of ${\rm Sp}(2)$ with highest weight (2, 2) is of dimension 81. Since ${\rm Sp}(2)$ is connected and simply-connected, it coincides wth ${\rm Spin}(5)$. Now, this irreducible representation is identified with the isotropy representation of the homogeneous space ${\rm SO}(14)/{\rm SO}(5)$, which is strongly isotropy irreducible. Notice finally that also the coset ${\rm Spin}(10)/{\rm Sp}(2)$ is a strongly isotropy irreducible space (of dimension 35). Here, the isotropy representation has highest weight (2, 1).