Is Galatians 4:6 speaking of the Holy Spirit? 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.