Promote eleven, briefly, at assembly (7)
I believe the answer is:
elevate
'promote' is the definition.
(I know that promote can be written as elevate)
'eleven briefly at assembly' is the wordplay.
'briefly' means to remove the last letter.
'assembly' is an anagram indicator.
'eleven' with its final letter removed is 'eleve'.
'eleve'+'at'='eleveat'
'eleveat' is an anagram of 'ELEVATE'.
(Other definitions for elevate that I've seen before include "'Raise, upgrade (7)'" , "Put on a pedestal" , "Raise, exalt (in rank)" , "Lift to higher place or position" , "Uplift" .)