Old flame blushing, holding Greek letter, died (7)
I believe the answer is:
expired
'died' is the definition.
('expire' can be a synonym of 'die')
'old flame blushing holding greek letter' is the wordplay.
'old flame' becomes 'ex' (a former partner).
'blushing' becomes 'red' (blushing makes someone's face go red).
'holding' means one lot of letters goes inside another.
'greek letter' becomes 'pi' (letter from the Greek alphabet).
'ex'+'red'='exred'
'exred' going around 'pi' is 'EXPIRED'.
(Other definitions for expired that I've seen before include "Run out" , "brought to an end" , "Died; elapsed" , "Lapsed" , "Became invalid" .)