Republican is greeting Hindu sage (5)
I believe the answer is:
rishi
'hindu sage' is the definition.
(I've seen this before)
'republican is greeting' is the wordplay.
'republican' becomes 'r' (abbreviation).
'greeting' becomes 'hi' (hi is a kind of greeting).
'r'+'is'+'hi'='RISHI'
(Other definitions for rishi that I've seen before include "Poet" , "Spiritual leader" , "wise man" , "holy man" , "Hindu sage or saint" .)