Check old air-conditioned halls (9)
I believe the answer is:
auditoria
'halls' is the definition.
(I've seen this before)
'check old air-conditioned' is the wordplay.
I cannot really understand how this works, but
'check' could be 'audit' (an audit is an examination or check) and 'audit' is present in the answer.
'old' could be 'o' (common abbreviation eg in OE for Old English) and 'o' is found within the answer.
an anagram of 'air' is 'ria' which is present in the answer.
No letters remain.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for auditoria that I've seen before include "Locations for houses" , "lecture theatres" , "Theatres, concert halls" , "Where concertgoers sit" , "music halls" .)