Here is the unanimous opinion from Justice Breyer.

An excerpt:

The question presented is whether an Indian tribe’s police officer has authority to detain temporarily and to search a non-Indian on a public right-of-way that runs through an Indian reservation. The search and detention, we assume, took place based on a potential violation of state or federal law prior to the suspect’s transport to the proper nontribal authorities for prosecution.
We have previously noted that a tribe retains inherent sovereign authority to address “conduct [that] threatens or has some direct effect on . . . the health or welfare of the tribe.” Montana v. United States, 450 U. S. 544, 566 (1981); see also Strate v. A–1 Contractors, 520 U. S. 438, 456, n. 11 (1997). We believe this statement of law governs here. And we hold the tribal officer possesses the authority at issue.

Another excerpt:

More broadly, cross-deputization agreements are difficult to reach, and they often require negotiation between other authorities and the tribes over such matters as training, reciprocal authority to arrest, the “geographical reach of the agreements, the jurisdiction of the parties, liability of officers performing under the agreements, and sovereign immunity.” Fletcher, Fort, & Singel, Indian Country Law Enforcement and Cooperative Public Safety Agreements, 89 Mich. Bar J. 42, 44 (2010).

Here are the briefs and other background materials.