SUBSINGULAR VECTORS AND CONDITIONALLY INVARIANT (Q-DEFORMED) EQUATIONS

We give a systematic discussion of the relation between subsingular ve ctors of Verma modules over semisimple Lie algebras G and differential equations which are conditionally G-invariant. This is extended to th e Drinfeld-Jimbo q-deformation Ug(G) of G. We treat in detail the conf ormal algebra su(2, 2), its complexification sl(4) and their q-deforma tions. The conditionally invariant equations are the d'Alembert equati on and a new equation arising from a subsingular vector proposed by Be mstein-Gel'fand-GeI'fand. We also give the q-difference analogues of t hese equations.