WASHINGTON — Prime Minister Narendra Modi’s official state visit turned the nation’s capital into a microcosm of Indian politics on Thursday. Thousands of South Asians of every creed and community flooded the city’s landmarks — some to support the controversial leader, others to protest his visit, while many attended to simply take in the historic moment. Chants of “Go Modi” and “Jai Hind” (“Long live India”), juxtaposed against “Killer Modi” and “no justice, no peace,” echoed through the streets and buildings. The South Asian American diaspora cares about Indian politics like never before, experts say, and the common denominator is...