On a class of representations of the Virasoro algebra and a conjecture of Kac (Q755875)

The representation theory of the Virasoro algebra V plays an important role in various areas of physics. In 1982, V. Kac conjectured the following theorem: Every irreducible V-module satisfying the conditions (a) \(x_ 0\) acts semisimply, (b) the weight spaces of \(x_ 0\) are finite-dimensional, is either a highest or lowest weight V-module or has all its weight spaces of a dimension less than or equal to one. The motivation for this conjecture was a result of Kostrikin. All the irreducible V-modules with one-dimensional weight spaces were constructed by Kaplansky and Santharoubane, the conjecture was proved for unitarizable V-modules by Chari and Pressley. In this paper, the conjecture for the V-modules satisfying the conditions (a), (b) and (c) the dimensions of weight spaces are bounded is proved. As the authors note, O. Mathieu has proved the conjecture of Kac.