One of the glitterati jilts a famous surgeon (6)
I believe the answer is:
lister
'famous surgeon' is the definition.
(Joseph Lister, father of modern surgery)
'one of the glitterati jilts a' is the wordplay.
'one of the glitterati' becomes 'A-lister' (glitterati and A-list can both mean famous people).
'jilts' is a deletion indicator (some letters are abandoned or jilted).
'alister' with 'a' taken away is 'LISTER'.
(Other definitions for lister that I've seen before include "Man introducing antiseptics" , "Plough" , "surgeon?" , "Pioneer of antiseptic surgery" , "One with leanings" .)