Illinois voters on Tuesday will weigh in on a host of House races that are unlikely to affect the balance of power in Congress but will showcase some of the deeper divides within the Democratic Party. In many of these crowded Democratic primaries, voters are choosing among veteran lawmakers, political outsiders and upstart progressives. And because the setting is the deep-blue Chicago area, where Republicans are expected to fare poorly in November, Tuesday’s contests will essentially determine the next class of Illinois House members. Many of the candidates are vying for open seats in districts where longtime House members have...