If you are a new supervisor, here's a small piece of advice: Learn now how to respond to a death of an employee on the job, at home, or suddenly without warning (automobile accident, etc.) Your employees will react with fright and confusion and they glance your direction as a leader to take cues on what they should do and how they should react.
This is the natural response to death of a coworker in the workplace. You will notice that some employees handle such an incident amazingly well, and others will struggle much more, particularly if they have other personal problems in their lives, and most especially if the death or event coincides with loss in their personal lives in some other respect, no matter how unrelated. The issue of loss and grief is at the heart of response.
Don't think that you can shove everyone back on to the tread mill as if they must be on a productivity routine that is more important that processing, talking out, and helping the family members of the deceased employees. In large measure, you will have to let things play out.
Your employees have experienced a dramatizing event and now must address the crisis of sudden loss and all the ramifications that go along with it. All employees want to return to their normal routines, family, and work-life as quickly as possible, but providing assistance to help them do this may be necessary, prudent, and wise. Denial of the emotional impacts of such an event can be compounded by an employer's unwillingness (often because of their own denial) to provide an opportunity to "process" or "talk out" the event.
Some employees may be reluctant to discuss the event and others are more willing. Much depends on an individual's past coping skills. Although you shouldn't force the issue, allowing time for employees to consider the event, their role in it, and feelings will speed their emotional recovery quicker. Set an example as a supervisor by being a role model for openness and willingness to talk about what happened. And pay attention to protracted responses that might indicate one of your employees could use additional counseling or assistance. To not respond invites stress claims, time off work, absenteeism, depression, and even increased likelihood of an accident due to problems concentration.