Federally-recognized Indian tribes exercise sovereign authority over the reservations. Consequently, there is no single answer to the question as to whether state police have jurisdiction to investigate crimes committed on reservations, as there are "open" and "closed" reservations, the latter being more isolated from local or state jurisidiction. The 1953 passage of Public Law 83-280 (commonly referred to as Public Law 280) conferred authority on five specific states -- California, Minnesota, Nebraska, Oregon, and Wisconsin -- to exercise jurisdiction over the Indian reservations within their borders. In the cases of Minnesota, Oregon, and Wisconsin, however, exceptions are made in the law for closed reservations, specifically, the Red Lake, Warm Springs, and Meonimee Reservations in each of those three states, respectively. While Public Law 280 did not provide those states with regulatory authority over the open reservations within their borders, they do allow for enforcement of state laws on the reservations. In fact, the basis of Public Law 280 was the removal of the federal role in law enforcement in most instances and its replacement by state authorities.
In addition to those five states, other states subsequently signed on to the provisions of Public Law 280, including Arizona, Florida, Idaho, Montana, Nevada, North Dakota, South Dakota, Utah, and Washington. The most recent such adddition of a state to the provisions of Public Law 280 was Utah's 1971 accession to the law's provisions. The Bureau of Indian Affairs, however, has reported that, over time, some of those states have withdrawn from the pact and ceded authority for investigating crimes back to the federal government. Another element in the back-and-forth history of law enforcement jurisidiction on Indian reservations is the growth over the years of tribal police departments that enforce laws within the confines of the reservations.
In conclusion, then, there are instances in which state police investigate crimes on Indian or Native American reservations. It is all, however, contingent upon the individual state and the individual reservation.