This paper is the last of a series of papers devoted to the bispectral problem [3]–[6]. Here we examine the connection between the bispectral operators constructed in [6] and the Lie algebra W1+∞ (and its subalgebras). To give a more detailed idea of the contents of the present paper we briefly recall the results of [4]–[6] which we need. In [4] we built large families of representations of W1+...

For a linear semisimple Lie group we obtain a necessary and sufficient condition for a highest weight module with non-singular infinitesimal character to be unitarizable.