I'm not a researcher yet, I'm defending my thesis next year and would like to get acquainted with recent results and branches of representation theory Mainly groups and C*-algebras Can I ask this in this?
You might have a look at a list of analogous questions (questions asking for open problem lists), to see which ones were well-received and why:
If you keep it focussed, then a request for an open problem list makes a decent big-list question, I think.
You can ask such questions; the community will not receive them well. The major reason is that the MathOverflow forum is more for answers to specific technical questions than for philosophical discussions or broad overviews as might be found in other resources such as mathematical texts.
Here is a strategy that might work to achieve your goal. Get a list of three or four topics that are of potential interest. Ask questions on those specific topics and see who answers. Invite them into a chat room and pump them as best as you can. There are policies for chat rooms, but I imagine there will be less or no community objections to such questions asked to well placed persons in chat.