"They support you."
Translation:Αυτοί σε υποστηρίζουν.
Can we say "Αυτοί σου υποστηρίζουν"?
No. In order to know in which case the weak pronoun form appears before the verb, you need to know how the (Subject +) Verb + Object sentence is structured.
When the object follows the verb on its own (Verb + object), the object appears as a weak pronoun form before the verb in the accusative case:
Αυτοί υποστηρίζουν εσένα - Αυτοί σε υποστηρίζουν and
Αυτοί θέλουν εσένα στην ομάδα τους - Αυτοί σε θέλουν στην ομάδα τους (They want you in their team)
For verbs whose syntax is Verb + σε + object, the object, direct or indirect, appears as a weak pronoun form before the verb in the genitive case:
Αυτή μοιάζει στον μπαμπά της - Αυτή του μοιάζει (στον = σε + τον) (She takes after her father)
Αυτοί δίνουν το δώρο σε εσένα - Αυτοί σου δίνουν το δώρο (They give you the present)
Αυτοί λένε κάτι σε εσένα - Αυτοί σου λένε κάτι (They tell you something)
In what kind of construction does a direct object require σε ? Doesn't the very presence of σε indicate that it is an indirect object?
That's a good point, however, in the example "Αυτή μοιάζει στον μπαμπά της - Αυτή του μοιάζει (στον = σε + τον)" there is only one object, therefore direct. "Αυτή μοιάζει του μπαμπά της" would be another way of writing that but it's unusual to put the object in the genitive so σε+article is favoured.
Just because there's only one object doesn't mean it's a direct object (although it might be in the English translation.)
Different languages consider objects differently as direct or indirect.
German uses the Dative (which is basically the same as Greek genitive used as an object — I expect it was genitive in ancient Greek) with the verb for help, (Er hilft mir), which in English considered a direct object.
Likewise with the verb that means μοιάζει (Sie gleicht der Mutter.), where in English we use resembles, or looks like with a different preposition.
English uses either to or for to indicate indirect object.
He gave it to her.
She made it for him.
It's not. Σε υποστηρίζουν. and Σε στηρίζουν are also correct. If yours was rejected please let us know where it was.