The complex geomety of a domain related to $\mu$-synthesis

We describe the basic complex geometry and function theory of the {\em pentablock} $\mathcal{P}$, which is the bounded domain in ${\mathbb{C}}^3$ given by $$\mathcal{P}=\{(a_{21},\mathrm{t}\mathrm{r}A,\det A):A={\left[\begin{array}{c}{a}_{ij}\end{array}\right]}_{i,j=1}^2\in \mathbb{B}\}$$ where $\mathbb{B}$ denotes the open unit ball in the space of $2\times 2$ complex matrices. We prove several characterizations of the domain. We describe its distinguished boundary and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show that $\mathcal{P}$ is intimately connected with the problem of $\mu$-synthesis for a certain cost function $\mu$ on the space of $2\times 2$ matrices defined in connection with robust stabilization by control engineers. We demonstrate connections between the function theories of $\mathcal{P}$ and $\mathbb{B}$. We show that $\mathcal{P}$ is polynomially convex and starlike.