Musical Fathers and Sons Note: Following in their fathers' footsteps, a number of classical musicians are beginning to carve careers for themselves. Vladimir Ashkenazy, son of, well, Vladimir Ashkenazy, will make his professional debut in September in London, playing Bartok's Sonata for Two Pianos and percussion with his father . . . Andrew Marriner, son of Neville Marriner, founder-conductor of the Academy of St. Martin-in-the-Fields, has just recorded a performance of Mozart's clarinet quintet. . . . Claudio Abbado's son Roberto will make his professional debut conducting a Rossini opera at the Edinburgh Festival . . .

* Casting Note: Casting has been completed for the highly anticipated New York production of "Cats," the musical based on T.S. Eliot's book "Old Possum's Book of Practical Cats." The play has been a smash in London and will be directed by Trevor Nunn, who directed there. Harry Groener, Ken Page and Betty Buckley will have the leading roles among the 23-person company . . .

* Gimme Another Chance Note: John Fernald is a theatrical director. Thirty years ago he was working on a play when he got into a dispute over casting and quit, which is not an uncommon occurrence in the theater. The gesture probably cost him a bit in royalties. The play he quit was "The Mousetrap," which has been running ever since. Now at the age of 76, he is being given a second chance. Sir Peter Saunders has hired him to direct the play again beginning in November, the 30th anniversary of its opening . . .