Jamieson, Fausset and Brown on Isaiah 48:11

11. how should my name--Maurer, instead of "My name" from Is 48:9, supplies "My glory" from the next clause; and translates, "How (shamefully) My glory has been profaned!" In English Version the sense is, "I will refrain (Is 48:9, that is, not utterly destroy thee), for why should I permit My name to be polluted, which it would be, if the Lord utterly destroyed His elect people" (Eze 20:9)?

not give my glory unto another--If God forsook His people for ever, the heathen would attribute their triumph over Israel to their idols; so God's glory would be given to another.