Asymptotics for singular vectors in Verma modules over the Virasoro algebra
Let \(M(h,c)\) be the Verma module over the Virasoro algebra with the highest weight \((h,c)\) and highest weight vector \(v\). For every \(k,l \in {\mathbb Z}\) and \(t \in {\mathbb C} \setminus 0\), let \[ h_{k,l}(t) = { {1-k^2} \over {4} } t+{ { 1-k l}\over {2} }+{ {1-l^2} \over {4} } t^{-1},\qquad c(t) = 6 t+13+6 t^{-1}. \] Then for \(k, l > 0\) and \(t \in {\mathbb C} \setminus 0\) the Verma module \(M( h_{k,l}(t), c(t))\) contains a singular vector \(S_{k,l} (t) v\) of degree \(k l\) [cf. \textit{D. Fuchs}, Adv. Sov. Math. 17, 65-74 (1993; Zbl 0827.17028)]. In the paper under review, the authors investigate the terms of extremal degrees of \(S_{k,l} (t)\). In particular, they prove that the highest and the lowest degrees of \(S_{k,l} (t)\) in \(t\) are equal, respectively, to \((k-1) l\) and \(-(l-1)k\).
scientific article; zbMATH DE number 1054060