A few of things:
- Know what, when and how to delegate. Delegation has such a negative connotation in industry (being seen as letting garbage flow downhill) but I believe this is because so many managers are poor at the mechanics of delegation. As a PM director, you have to learn how to delegate problems to be solved and not orders to be fulfilled (a/k/a solutions) and be comfortable with the ultimate solutions even if they weren't what you would have built as an IC in the shoes of the delegate. You must also learn to see delegation as the manner in which you are granting autonomy and growth opportunities to your direct reports and not as the way to hand off dirty work that you would rather not do to "underlings".
- Frequent contemplation of organizational design and areas of responsibility. Every week I am thinking about organizational design and at least quarterly I am having discussions with my peer leadership team in engineering and design to talk about whether we are a) staffed appropriately in the areas we need to be; b) whether we have the right skillsets and mix of seniority on our teams; c) whether we are optimizing for the strengths of PMs/eng leads/design leads while also giving them opportunities to stretch and grow in their current roles.
- Employing a variety of communication styles with a wider range of stakeholders. We all have our default, natural communication styles (for example, my management style with my direct reports defaults to Socratic) but a director has to know when to switch styles based on the recipient of the message, their level of seniority, their personality, etc. -- and be able to intuit this very quickly in the course of delivery. I have had to force myself to be more direct ("Crucial Conversations"-style) when a softer message isn't making it through, for example. A corollary here, particularly when dealing with senior stakeholders: knowing what to spend one's political capital on, and when to pull back.