Are these good reasons for starting a new PhD project?
There are dozens of parameters that seem to be influencing this decision. I will try to touch on a few points as best as I am able:


Coming up with a PhD topic is hard for a lot of PhD students. This is a common feeling; it does not mean you cannot succeed as a PhD student. 
How does representation theory relate to your current subject? Are there papers connecting the two subjects? If so, what do they say? If not, can you discover a link between the two subjects (representation theory and applied?) Your advisor seems to think there is a link. What is that link? (These are questions I would be asking myself as a way to try and discover a dissertation direction).
Graduate students are sometimes distant with one another. They all have their own projects to be doing. That's the reality. I would look for social connections outside of your department if necessary. 
It may be easier to change advisors internally, if possible, than switching universities. Being from the US, I have no idea how feasible this is in your case. 
In my opinion, passion for your subject is overrated as a PhD student. Graduate students I have advised are rarely deeply passionate about our research. But they realize that it is 4-5 years of learning and thinking about a subject in order to develop the ability to perform independent research.
I work at a national lab in the US now. No one cares about the topic of my PhD (probability theory). My master's degree was in representation theory. I never use these topics in my current work. But I've proven that I can think about math and statistics, and I use these skills in my job now as a researcher. I do not know your goals (academia or industry), but industry usually only cares if you have a PhD in a generally relevant field and if you can program. It is that binary.  
Being in your early 30s when you finish a PhD is somewhat normal. There is nothing magical about being 27 and having a PhD. It just means that you have to start going to a full time job earlier in life. If your finances are in okay shape, finishing a PhD at 32 is not the end of the world. 
We speak of a PhD being able to do "independent research." This does not mean you never need help or collaboration. I have a PhD. I collaborate with other researchers. Sometimes I come up with an idea, sometimes they do. And some of their ideas blow my mind--I would never have thought of them in a millions years. 
Being honest, it sounds like your advisor may not be the best advisor on the planet. At least not for your case. Some students really like a hands off advisor, some do not. Both types of PhD students exist and both are successful. Changing universities to get to a better advisor position may not be as huge of a deal as you think. It might actually make your work more enjoyable.