Galatians 4:6 says:
And because you are sons, God has sent the Spirit of his Son into our hearts, crying, "Abba! Father!"
Is Paul referring to the Holy Spirit here? Given that the ESV capitalises the word 'Spirit', it seems likely to me, but I'm interested in knowing what evidence there is either way.
Yes, Paul is referring to the Holy Spirit.
The Holy Spirit is the Spirit of Jesus. In John 14, when Jesus said to his disciples during passion week
"'I will not leave you as orphans; I will come to you'" (v.18),
he was promising that although he would not in the very near future be with them physically, he would be with them by his Spirit. He called his post-ascension presence with them the helper and the comforter (in Greek, the paraklete), one who not only comes alongside us and puts his arm around our shoulders, but one who abides and dwells in us (v.17).
The disciples would discover not many days after Jesus made this promise to them that the Spirit of Jesus in them would also be their empowerer:
"'. . . you will receive power when the Holy Spirit has come upon you; and you shall be my witnesses both in Jerusalem, and in all Judea and Samaria, and even to the remotest part of the earth'" (Acts 1:8).
The important point to realize is this: out of obedience to his Father and his love for us, Jesus not only became one of us in his humanity through the virgin birth, but he chose to retain his humanity forever.
" Wherefore when he cometh into the world, he saith , 'Sacrifice and offering thou wouldest not, but a body hast thou prepared me: In burnt offerings and sacrifices for sin thou hast had no pleasure. Then said I, 'Lo , I come (in the volume of the book it is written of me), to do thy will, O God'" (Hebrews 10:5-6 KJV, cf. Psalm 40:6).
Not only did Jesus atone for our sins "in his own body on the tree" at Calvary (1 Peter 2:24), but he rose from the dead in bodily form and 40 days later ascended into heaven in bodily form. We Christians anticipate with great joy seeing our Lord and Savior face to face! Were it not for Jesus,
"who, although he existed in the form of God, [and] did not regard equality with God something to be grasped, but emptied himself, taking the form of a bond-servant and [being] made in the likeness of men" (Philippians 2:6),
we would still be lost in sin and condemned to hell for all eternity. We thank God, therefore, that Jesus became the God-Man for us, that he might redeem us from the curse of the Law, according to which we deserved only condemnation, since we all have sinned.
In conclusion, your question cannot be answered fully without the all-important assumption that
the Father is fully God
the Son is fully God
the Holy Spirit is fully God
Jesus did not cease being God when he became a human being, nor will he ever cease being fully God. To indwell every believer, however, only his Spirit--the Holy Spirit--became the person through whom--and the induement by which--we are assured of Christ's presence with us until we see him face to face in glory.
In Galatians chapter four what Paul is trying to convey to the Galatians that God sent not just a physical Jesus to save the world, but he sent the Spirit of Jesus.
The Trinity or what we know as God is not of the physical realm, but of the Spiritual realm. That is to say that they are from a realm existing before the Physical things were made.
So what Paul is really saying to them is that that Spiritual part of Jesus live on, and when Jesus said:
John 14:16 KJV And I will pray the Father, and he shall give you another Comforter, that he may abide with you for ever;
He was talking about the Holy Spirit coming to live in the hearts of Jesus' disciples:
John 14:26 KJV But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
That promise is just as true today as it was at Pentecost:
Acts 2:1 through 4 KJV And when the day of Pentecost was fully come, they were all with one accord in one place. And suddenly there came a sound from heaven as of a rushing mighty wind, and it filled all the house where they were sitting. And there appeared unto them cloven tongues like as of fire, and it sat upon each of them. And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance.
Jesus also told his disciples that the Holy Spirit could not come to them until after Jesus had been sacrificed on the cross:
John 14:26 KJV But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
And this is the part to which Paul was referring when writing to the Galatians:
John 15:26 KJV But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall testify of me:
Paul is actually telling them that when we accept Jesus the Holy Spirit then enters our body and we become as pure in God's eyes as Jesus himself is. We are then the adopted children of God and can share in the Glory of Heaven just as they do.
So yes your impression the Spirit of Galatians 4:6 is the Holy Spirit is absolutely correct.
