Keeping Greek character in, eating (9)

'keeping' is the definition.
The definition and answer can be both to do with social activities as well as being present participle verbs.
Perhaps there's an association between them I don't understand?

'greek character in eating' is the wordplay.
'greek character' becomes 'eta' (letter from the Greek alphabet).
'in' means one lot of letters goes inside another.
'eating' becomes 'dining' (dining is a kind of eating).
'eta' inserted within 'dining' is 'DETAINING'.

Can you help me to learn more?