The complex geomety of a domain related to $μ$-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 $\mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix} a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\}$ 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 $μ$-synthesis for a certain cost function $μ$ 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.

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