The piano music furnished to and played by Pvt. Sheridan is "Piano Concerto for the Left Hand in D major", composed by Maurice Ravel between 1929-30. The story of the piece's creation as told by Maj. Winchester (David Ogden Stiers) is all true.
In telling Pvt. Sheridan about the piano music, Charles mentions a pianist named Paul Wittgenstein, who lost his right arm in World War I. He is the brother of renowned philosopher Ludwig Wittgenstein.
During the scene with Winchester and Pvt. Sheridan at the Officers' club, actor James Stephens appears to be tackling the difficult Ravel Concerto for the Left Hand when in fact, the actual playing is that of a real pianist, with the camera moving deftly between the actor's face and the actual pianist's left hand. This is called an "insert", for obvious reasons. (You never see James' hand and his face together during the playing; only at rest.)