New York's congressional primaries on Tuesday will be a key test of strength for the left-wing movement challenging the Democratic Party's establishment.Why it matters: Several of these races pit the interests of House Minority Leader Hakeem Jeffries (D-N.Y.) against those of democratic socialist New York City Mayor Zohran Mamdani.Mamdani, one of the left's most prominent figures both in New York and nationwide, is backing three left-wing insurgents vying for NYC-based House seats.Two of those insurgents are running against incumbent Democrats — and Jeffries almost always supports his incumbents.State of play: In the state's 10th district, Jeffries is supporting Rep. Dan Goldman (D-N.Y.) while Mamdani is backing NYC comptroller Brad Lander.In the 13th district, Jeffries-backed Rep. Adriano Espaillat (D-N.Y.) faces a spirited challenge from Mamdani-endorsed democratic socialist Darializa Avila-Chevalier.Mamdani is also supporting democratic socialist State Assembly member Claire Valdez in the 7th district over Brooklyn Borough President Antonio Reynoso, though Jeffries hasn't endorsed in that race.Zoom in: Lander, who famously cross-endorsed with Mamdani in last year's NYC mayoral race, is seen as an extremely formidable foe to Goldman and polls have generally shown him with a large lead.In NY-13, outside groups are spending heavily in support of both Espaillat and Avila Chevalier, though Espaillat's allies have a clear financial edge.Mamdani has cut multiple ads supporting Lander, Avila Chevalier and Valdez and campaigned heavily for all three.Yes, but: Mamdani has avoided more direct confrontations with Jeffries, most notably opposing City Councilman Chi Ossé's aborted attempt to primary the Democratic House leader.These also aren't perfectly clear-cut battles of left vs. center: Goldman and Espaillat are both members of the Congressional Progressive Caucus, but have come under fire from the left in part for their support of Israel.Things get even more complicated in NY-7, where Reynoso is backed by retiring progressive Rep. Nydia Velázquez and the Working Families Party, which also supports Lander.Zoom out: The other big contest in New York on Tuesday is the 12th district, where several major Democratic candidates are duking it out to succeed Rep. Jerry Nadler (D-N.Y.).The race includes high-profile figures such as Jack Schlossberg, the grandson of former President John F. Kennedy, and prominent anti-Trump figure George Conway.But the two frontrunners are Assembly member Micah Lasher, who is backed by Nadler and former NYC Mayor Mike Bloomberg, and Assembly member Alex Bores.The race has attracted tons of outside spending for and against Bores, an architect of New York's AI regulations, with OpenAI-connected super PAC Think Big spending nearly $8 million to oppose him and Anthropic-affiliated Jobs and Democracy PAC spending $9 million to defend him.