Irreducible Modules of Quantized Enveloping Algebras at Roots of 1, II

A central topic in representation theory is to understand the irreducible modules. In this paper we are concerned with nite dimensional irreducible modules of quantized hy-peralgebras and of hyperalgebras. Recently, big progress has been made on the characters of the irreducible modules, namely the Lusztig's conjectural formula for the characters is true for quantized hyperalgebras and is true for hyperalgebras when the characteristic of the groud eld is suuciently large, see AJS, KL2, KT, L6]. In X2] we see that certain monomials in a quantized hyperalgebra U (resp. a hy-peralgebra U k) at a root of 1 are interesting in realizing the nite dimensional irreducible U-modules (resp. U k-modules). The realization seems useful in understanding the irre-ducible modules, for example, based on the realization and assuming chark=2, Xu and Ye constructed explicitly certain bases for nite dimensional irreducible U k-modules for type A 4 ; B 3 ; C 3 ; A 5 , see XY, Y1, Y2] and then get the irreducible characters. Until now it seems diicult to get the information by other methods. One purpose of this paper is to give a uniied proof and to romove a restriction for the results in X2, x5]. Due to technical reasons, the proof in X2] is lengthy and involves case by case analysis, what is worse, it doesnot work for type G 2. In this paper we will give a simple and uniied proof for the main results in X2, x5], the proof works for all types. To do this we work in the category O of Bernstein-Gelfand-Gelfand, see x2. Another purpose is to give some applications of the main results in X2], such as (a) the description of the socle of a quantized Frobenius kernel (see 3.3), (b) a realization of 1 the tensor product of two irreducible modules of a quantized Frobenius kernel (see 3.5), (c) calculation of the dimensions of some irreducible modules for type A (see 4.5-6). In X4-5] we show that the main results in X2] can be used to nd out some (quasi) primitive elements in a Verma modules. For type A 2 ; B 2 we can nd all (quasi) primitive elements in a Verma module of a quantized Frobenius kernel and in a Weyl module, see X4-5]. The contents of the paper are as follows. In section 1 we introduce the basic notation and recall some basic facts. In section 2 …