WASHINGTON — The Supreme Court on Tuesday avoided taking up a series of new cases on the scope of the right to bear arms and left in place an Illinois law that bans assault-style weapons such as the AR-15 semiautomatic rifle, which has been used in various high-profile mass shootings.

The decision not to hear the multiple cases challenging the Illinois law means it remains in effect. Litigation over the Illinois ban and similar laws enacted by other states is ongoing, and the issue is…