Can you play Carnatic music on the saxophone?

Yes: Kadri Gopalnath (1949 to 2019) played Carnatic music on the alto saxophone, which he modified for the purpose. He first learned the nadaswaram from his father, turned to the saxophone after hearing the Mysore Palace band, and trained in Carnatic music with N. Gopalakrishna Iyer and T. V. Gopalakrishnan. In 1994 he became the first Carnatic musician invited to the BBC Proms.