QUANTIZED SYMMETRIC POWERS

Abstract

Let g = sln(C) be the Lie algebra of n×n matrices over C of trace zero, and let U be the q-analogue of its universal enveloping algebra defined by Drinfel’d [2] and Jimbo [4]. According to [7, 3.5.6, 6.2.3 & 6.3.4], for each dominant weight μ in the weight lattice of g there is an irreducible, finite-dimensional highest weight U -module V (μ) with highest weight μ. Kashiwara [5] and Lusztig [7, 14.4.12] have independently shown the existence of a certain canonical basis B(μ) for V (μ). In [5], Kashiwara proves the existence of a certain crystal basis associated with V (μ) using certain operators, and defines a graph (the crystal graph) which encodes how these operators act on the crystal basis. Using this, he defines B(μ). In §2 we define a certain subspace of the r-th tensor power of the basic module for U , for each integer r ≥ 1, and show that it is a module for U , with highest weight (r, 0, 0, . . . , 0). (We use the same numbering as [1, Planche I]). In §3 we study the irreducible finite-dimensional highest weight module for U with highest weight (r, 0, 0, . . . , 0), and find its canonical basis in terms of monomials in U acting on its highest weight vector. We then bring all of this together by showing that the module in §2 is isomorphic to the module studied in §3, and as a result we have two nice descriptions of the canonical basis for this module. Using these we determine how the generators of U act on this basis.