Moduli of equivariant algebraic vector bundles over affine cones with one dimensional quotient (Q1917012)

The main result of this paper is to extend a result of \textit{G. Schwarz} [see C. R. Acad. Sci., Paris, Sér. I 309, No. 2, 89-94 (1989; Zbl 0688.14040)] to a more general setting. Given a reductive group \(G\) over the complex numbers \(\mathbb{C}\) let \(X\) be an algebraic \(G\)-variety with a distinguished \(G\)-fixed point \(x_0\), and let \(Q\) be a \(G\)-module. Denote by \(\text{VEC}_G(X,Q)\) the set of isomorphism classes of \(G\)-vector bundles with base \(X\) and whose fiber over \(x_0\) is \(Q\). In general, it is not known how to classify this set. However, Schwarz showed that if \(X=P\) is a \(G\)-module with a one-dimensional quotient, this can be done. More specifically, in this case, \(\text{VEC}_G(P,Q)\) is isomorphic to a vector space \(\mathbb{C}^p\), and the addition in \(\mathbb{C}^p\) can be interpreted using the Whitney sum of vector bundles. In the present article, the author shows that this result also holds in the case where \(X\) is a weighted \(G\)-cone with a smooth one-dimensional quotient.