A New Approach to Kostant’s Problem

For every involution w of the symmetric group Sn we establish, in terms ofa special canonical quotient of the dominant Verma module associated with w, an effective criterion, which allows us to verify whether the universal enveloping algebra U(sln) surjects onto the space of all ad-finite linear transformations of the simple highest weight module L(w). An easy sufficient condition derived from this criterion admits a straightforward computational check for example using a computer. All this is applied to get some old and many new results, which answer the classical question of Kostant in special cases, in particular we give a complete answer for simple highest weight modules in the regular block of sln, n ≤ 5.