The auxiliary verb 져요 turns a descriptive verb in infinitive form into something that means get or become. Aside from the plain infinitive form, this auxilliary verb can be attached to verbs in other forms as shown below using 나빠요 (is worse or is bad):
- 나빠졌어요 (nappajyeosseoyo) – got worse or got bad. Yes, the past indicator is added after the auxiliary form.
- 나빠질 거에요 (nappajil keo.eyo) – will get hot. Note that it’s not 져 but rather 지 because as mentioned in previous post future from is normally added on plain base so in this case 져 becomes 지 since its just an abbreviation of 지어요.
- 나빠지지만 (nappajijiman) – gets hot, but.
- 나빠질 까요? (nappajil kkayo?) – do you suppose it will get hot?
From this, I am thinking that when the verbs taking the 져 form should be taken as the one compound verb itself such that any rules in changing its form whether it be past, future or other applicable verb endings will be followed with consideration of compound verb ending as either 지 or 져.
So even for the auxiliary verb 고 싶어, 져 can be used. The meaning becomes ‘gets to want to do <verb>’. Altogether the form becomes plain base+고 싶어져요.