ALBANY — New York’s attorney general has the authority to investigate the fatal police shooting of a man in the upstate city of Troy last month, according to an order issued on Sunday by Gov. Andrew M. Cuomo.

The order amended a broader directive issued last year by Mr. Cuomo that empowered the attorney general, Eric T. Schneiderman, to investigate and prosecute in deadly encounters involving the police. Mr. Cuomo, a Democrat, revised the earlier order to authorize Mr. Schneiderman to explore possible “unlawful acts or omissions” related to the death in Troy of Edson Thevenin.

Last week, Mr. Schneiderman sued District Attorney Joel Abelove of Rensselaer County over case files related to the death of Mr. Thevenin, who was fatally shot on April 17 by an officer who the authorities say Mr. Thevenin had trapped between two cars after a chase.

A grand jury convened by Mr. Abelove less than a week after the killing of Mr. Thevenin declined to indict the officer. In his suit, Mr. Schneiderman said Mr. Abelove had violated the attorney general’s authority in such cases by presenting the case to a grand jury before the attorney general’s office had determined whether it would intervene in the matter.