I have communicated with a professor about joining his group as a Ph.D. student. He had shown his interest and told me to apply to that school after a Skype interview, although the deadline for starting the application process for that semester had already crossed then. I had applied to that school and completed my application. However, after that, he has become reluctant about me and is not telling anything clearly even after 1.5 months since I completed my application. I went to a conference last week and send him an email asking if he was also attending the same conference and invited him to come during my talk. He replied to me two days later and said that he had another schedule at the same time, but he can meet me another time to go through my slides. We met and he told me he would contact me 1 week or two weeks later. It has been 1 week passed and I have also finished my MSc. defense and now it is very urgent for me to take a decision about what am I going to do next. In this situation, I am wondering what I should do. Should I send him an email clearly asking if he is going to take me in his group?
I had applied to that school and completed my application. However, after that, he has become reluctant about me and is not telling anything clearly even after 1.5 months since I completed my application.
In cases like this, it is not incumbent on academic staff to pre-empt administrative decisions that are made as part of the normal admission processes of the university. You have made a formal application for admission, so there is already a process in motion that will lead to an outcome. Presumably this professor will need to make a decision in that process, and either you will be accepted or rejected for that application. Sending him an email asking for an answer is essentially just asking him to pre-empt the administrative decision in that process, and he probably will not be inclined to do this.
If you decide to email this professor, a better approach would be to let him know your circumstances and ask him if he has any concerns about your application that you might be able to address with further work. If there is some reason for urgency in your application, let him know what those circumstances are, and stress that you are happy to work on any deficiencies in your application. Be careful with the level of persistence that you show here, since your professor is certainly not required to pre-empt the normal mechanism for decisions on admissions, or give any assurances outside of the formal admissions processes.
If you are concerned that this application process is taking too long, and there is some reason for urgency on your side, another option would be to email the relevant administrators at the university that deal with this process and make a preliminary inquiry asking how long it usually takes to receive an outcome for a PhD application. (You say that you have been waiting 1.5 months, which is about 6-7 weeks. That is not particularly long for an admissions process for a PhD candidate.) If the process is taking too long, you can make further inquiries with the administrators, and they will horse-whip the professor into making a decision. It is better to have this pressure coming from them than from you.