Highest weight modules over W1+∞ algebra and the bispectral problem

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 each β ∈ C N we defined a tau-function τβ(t) which we called Bessel tau-function. We proved that it is a highest weight vector for a representation Mβ of the algebra W1+∞ with central charge N . In [6] we introduced a version of Darboux transformation, which we called monomial , on the corresponding wave functions Ψβ(x, z) (see also Subsect. 1.2) and showed that the resulting wave functions are bispectral. For example all bispectral operators from [9, 22] can be obtained in this way. The present paper establishes closer connections between W1+∞ and the bispectral problem. Our first result (Theorem 2.1) shows that a tau-function is a monomial Darboux transformation of a Bessel tau-function if and only if it belongs to one of the modules Mβ . This type of connection between the representation theory (of W1+∞) and the bispectral problem is, to the best of our knowledge, new even for the bispectral tau-functions of Duistermaat and Grünbaum [9]. The second of the questions we try to answer in the present paper originates from Duistermaat and Grünbaum [9]. They noticed that their rank 1 bispectral operators are invariant under the KdV-flows and asked if there is a hierarchy of symmetries for the rank 2 bispectral operators. The latter question was answered affirmatively by Magri and Zubelli [17] who showed that the algebra V ir+ (the subalgebra of the Virasoro algebra spanned by the operators of non-negative weight) is tangent to