There's also an invisible being present--a soul that suffuses this production: Balanchine's, especially Georgi Balanchivadze's. As a young student in the renowned dance academy of St. Petersburg's Mariinsky Ballet (Pavlova's school, Nijinsky's, Nureyev's, Baryshnikov's), Balanchine played the Nutcracker Prince. Decades later, creating his own production for the New York City Ballet, he co-opted from the Russian version the mime monologue with which the young Prince explains to the Sugarplum Fairy the dramatic events that led to his arrival with Marie in the Land of Sweets. During his lifetime, as one Prince succeeded another, Balanchine personally coached any boy new to the role. (Two casts alternate in the children's roles, and the youngsters must abandon parts for which they've outgrown the costumes.) Now that Balanchine is gone, the children's ballet mistress, Garielle Whittle, often tells her charges in rehearsal, "Mr. Balanchine used to say . . ."