ar X iv : q - a lg / 9 71 10 02 v 1 3 N ov 1 99 7 Vector Coherent State Realization of Representations of the Affine Lie Algebra ŝl ( 2 )

The method of vector coherent states is generalized to study representations of the affine Lie algebra sl(2). A large class of highest weight irreps is explicitly constructed, which contains the integrable highest weight irreps as special cases. 1 INTRODUCTION The method of vector coherent states was independently developed by the research groups of Quesne and Rowe[1][2] to study representations of Lie groups appearing in physics. Applied to the irreducible representations of the compact semi-simple Lie groups, the method enables one to obtain explicit realizations of the generators of the corresponding Lie algebras in terms of holomorphic differential operators, and more importantly, the K-matrix technique [2] of the method allows the identification of the subsets of holomorphic polynomials which form the irreducible representation spaces. The method also provides a powerful machinery for constructing unitary representations of noncompact Lie groups; we refer to [2] for details on this subject. The method has also been extended to Lie superalgebras [3], and in recent years, to quantum groups and quantum supergroups [4]. A notable feature of the method of vector coherent states is that it is particularly well adapted to standard techniques in physics, thus is readily applicable to addressing concrete physical problems. Apart from its physical applications, the method is also of great mathematical interests. In particular, it is closely related [5] to the Bott-Borel-Weil theorem. The theorem is one of the hall marks in the representation theory of Lie groups. It realizes the finite dimensional irreps of compact Lie groups in terms of cohomology groups of homogeneous vector bundles. The Langlands and Kostant conjectures, proved by Schmid, generalize the theorem to noncompact semi-simple Lie groups, yielding a geometrical realization of Harish-Chandra's discrete series of representations. The Bott-Borel-Weil theory was further modified and significantly extended in the last 20 years, leading to the development of the theory of cohomological induction, which has now become a fundamental part of modern representation theory. The Bott-Borel-Weil theory also plays important roles in various other fields, most notably, geometric quantization and Penrose transforms.