The United States really is a federal republic and the individual states have more autonomy than people sometimes realize. Every state government has its own legislative system, but most or all of them mimic the Federal legislative system, with two "houses," a senate and a body of "representatives" (which, confusingly, is often called "the legislature.")
The novel is referring to an historical event, and--as I guessed--real names were used: http://politicalgraveyard.com/geo/WV/ofc/stsen1950s.html
, "Members of the West Virginia State Senate: 1st District: William A. Hannig 1943-54."
As one of perhaps 34 state senators, Hannig would have represented perhaps 100,000 constituents, and would be a "local dignitary" of perhaps the same importance as the mayor of Wheeling
Hannig stood up and invited the group to sing together with him. As Gary says, it could have been a hymn, but I am thinking it was probably something patriotic.
Personally, strictly a guess, I am imagining "My Country, 'Tis of Thee."
I imagine about half of the attendees actually sung, mostly in unison, many quite badly, and there are always one or two show-off harmonizing. He was probably not a trained chorus director. He probably "led the group" by standing up and saying something like "And now, I ask you to join me in singing [whatever]," and then sang loudly, slowly, and waving his hands vaguely.
The purpose of group singing is to involve an audience through participation, "warm them up" for the speech, and create a feeling of community.