a "representative" is an entity separate and distinct from the person, not a part of him/herself.
The Holy Spirit shall remain a mystery to our finite minds into eternity to come, too, I think.
Being the infinite "Spirit in the fulness of the Godhead", representing God and Christ jointly as their Spirit is an infinite task..., for which he is well suited: I don't think we need worry about separating apples and pears, here - between God and his rep, since the "Spirit of Jesus" dwells in us, bringing us the 'presence of Jesus'. Infinity is big enough for the Spirit to have its own personality: we needn't delve deeper than that regarding its nature, or what?
How we deal with Jesus' gentle Spirit is necessary to study out, though.