New generalized Verma modules and multilinear intertwining differential operators (Q1299144)

Let \({\mathcal G}\) denote a complex semi-simple finite-dimensional Lie algebra or a split real semisimple Lie algebra with a fixed triangular decomposition \({\mathcal G}={\mathcal G}^-\oplus{\mathcal H}\oplus{\mathcal G}^+\). The author proposes a new generalization of Verma modules, which he calls \(k\)-Verma modules (\(k\) is a positive integer). As a vector space such module is isomorphic to the symmetric tensor product of \(k\) copies of \(U({\mathcal G}^-)\). For \(k=1\) one gets the classical Verma modules, whereas for \(k>2\) such modules are always reducible and contain more than one singular (= primitive) vector. For \(2\)-Verma modules the author constructs a class of singular (or primitive) vectors, which exhausts all such vectors in the case \({\mathcal G}\simeq sl(2)\). With each singular vector of a \(k\)-Verma module the author associates a multilinear intertwining differential operator. He gives an explicit formulae for all bilinear (\(k=2\)) intertwining differential operators for \({\mathcal G}=sl(2,{\mathbb R})\) and \({\mathcal G}= SL(2, {\mathbb R})\) (noting the difference between the algebra and group invariants). Many examples are given in detail.