The usual answer to this question is a list of important things: strategy, vision, culture, the big decisions. That answer is wrong, and it is wrong in a way that causes real damage, because importance is not the criterion and never was.
Enormously important work is delegated correctly every day. A hospital delegates surgery. A construction firm delegates the structural calculation. In each case the person doing the work is better at it than the person accountable for it, which is the ordinary and correct arrangement. If importance decided the question, leaders would end up doing everything that mattered, badly.
The actual test
Something cannot be delegated when the act of delegating it destroys the thing itself.
This is a narrow test and it produces a short list. Most of what a leader does fails it and should be handed to someone more capable. But where the test bites, it bites completely: the delegated version is not a diminished form of the original. It is a different object, and everyone receiving it knows the difference immediately.
The list
The first enforcement of a standard. A standard becomes real at the moment it is first applied to someone who violated it. If a deputy handles that moment, the organization has learned that the standard belongs to the deputy. You may enforce it personally forever after and never recover the original signal. Later enforcement can be delegated freely, because by then the standard has an owner and everyone knows who it is.
Delivering a consequence to a person. Termination, demotion, the denial of something they wanted and expected. The content of the message can be prepared by others and often should be. The delivery cannot be handed over. A person who learns from a third party that you ended their employment has been told two things, and the second one, that they did not merit twenty minutes of your time after eleven years, is the one they will repeat for the rest of their life. Everyone still employed hears it too.
Owning a failure. A delegated apology is not an apology. It is a communications exercise, and it is received as one. The moment the words come from someone other than the person accountable, the thing being sought, which is evidence that the leader will absorb cost, has been withheld by the very act of speaking.
Your own commitment. When you tell someone you will personally see that something happens, you have created an asset that exists only because you said it. Passing the follow-through to an assistant does not reduce the commitment by some percentage. It converts your word into an ordinary work item, and the next commitment you make will be priced accordingly.
Judging the people who report directly to you. Input can and should be gathered widely. The assessment itself is the substance of the relationship between you and that person, and a leader who outsources it has ended the relationship while continuing to occupy the position within it.
Why leaders violate the list
Almost never laziness. Look at what the five items have in common: each requires the leader to be personally present for something unpleasant, and in four of the five, someone will be upset with them by name.
Consider a rural hospital district where the CEO decides to remove a nurse manager who has been there nineteen years and is widely liked. The HR director is asked to handle the conversation. The stated reasons are good ones. She is trained for it, she knows the documentation, the process will be cleaner, and there is genuine legal exposure in an unscripted discussion.
All of that is true and none of it is why. The CEO does not want to sit in a room with someone she has known for nine years and tell her it is over. That is an entirely human thing not to want. But the decision gets described internally as appropriate process, and the description is what makes it corrosive. Delegation from this list is nearly always conflict avoidance in the language of empowerment, and the organization can read the difference with complete accuracy.
The staff will not conclude that the CEO followed protocol. They will conclude that when it gets hard she is not in the room, and they will carry that conclusion into every future situation where being in the room matters. Nineteen years of service ended by a person the nurse manager barely knew.
The mirror error
The opposite failure is more common and less severe. Leaders treat far more as non-delegable than the test supports, usually work they are good at and enjoy, and defend the hoarding by describing it as accountability.
The tell is straightforward. If handing the thing to a competent person would produce a slightly worse result and no change in what the organization believes about you, it is delegable and you are keeping it for yourself. Discomfort is not the criterion either. Plenty of delegation is uncomfortable because the other person will do it differently and you will have to watch.
What the list is really measuring
Every item on it is a moment where the organization finds out something about the person in charge that cannot be established any other way.
That is the whole reason the delegated version fails. These are not tasks with outputs. They are occasions of evidence, and evidence about you specifically cannot be produced by somebody else. The work of the organization is delegable almost without limit. The demonstration of who you are under cost is not, and a leader who quietly hands off all five will hold the position for years while the thing the position was supposed to contain drains out of it.
Edited by Patrick J. Wolf, PhD