The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext-groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.
"Exts and vertex operators." Duke Math. J. 161 (9) 1797 - 1815, 15 June 2012. https://doi.org/10.1215/00127094-1593380