Structure of Certain Induced Representations of Complex Semisimple Lie Algebras

Let £ be a split semisimple Lie algebra over a field of characteristic zero and <£ = 3C+ ]C«eA«£a be the rootspace decomposition of <£ relative to a splitting Cartan subalgebra 5C, where the subset A of 3C* is the corresponding root-system. Fix a simple system of roots {c*i, «2, • • • , ai}, for which the positive (resp. negative) roots are denoted by A+ (resp. A_). For a EA let Ra be the Weyl reflection sending a into —a and fixing the elements of 3C* orthogonal to a with respect to the inverse Killing form ( , ). I t is given explicitly by \Ra=\—\(ha)<x where &a£3C is defined by requiring \(ha) = 2(a, a)(X, a) for all X£3C*. Denote the group generated by { i ? a | aEA} by W. We abbreviate Rai and hai by Ri and hi respectively. The "simple" reflections R±, R^ • • • , Ri are Coxeter generators of the Weyl group W. Let % be the universal enveloping algebra of <£, and U+ (resp. ai-) the subalgebra with identity 1 generated by <£+= ]C«eA+£a (resp. <£_= ]T)«€A-JBa). I t is an established fact that the notions of <C-module and «U-module are interchangeable. Here, and throughout, the word "module" is an abbreviation for the word "right-module." Our object in this paper is to study the structure of the JC-module 35A = lt/<3A for arbitrary A£3C*, where °U is regarded as a module under right-multiplication and $A is the right-ideal of % (i.e., submodule of %) generated by