The Murder Room by P. D. James

When Commander Dalgliesh is persuaded by an old friend to visit the Dupayne, a small private museum on the edge of Hampstead Heath, he can have no idea that he will return to it one week later under very different circumstances. One of the family trustees has been horribly murdered and Dalgliesh and his team are called in to investigate a death which, from the first, is fraught with complications. Even before the murder, the museum was in tumult. A new lease is due to be signed and two of the trustees are determined to keep the museum open, the third passionately determined on its closure.