06 v 2 2 2 N ov 1 99 5 Crystallized Peter - Weyl type decomposition for level 0 part of modified quantum algebra Ũ q ( ŝl 2 ) 0

Modified quantum algebra Ũq(g) is an algebra which is ’modified’ the Cartan part of the underlying quantum algebra Uq(g) as in (3.1) and (3.2). Modified quantum algebra Ũq(g) holds the remarkable property that modified quantum algebra Ũq(g) affords the commutative two crystal structures. The first one is the usual crystal structure, which was found by Lusztig ([8]). Another crystal structure was discovered by Kashiwara ([4]), which is called right crystal structure. In [4], it is shown that Ũq(g) is stable by the action of the antiautomorphism ∗ (see 2.1) and moreover, crystal base B(Ũq(g)) is also stable by the action of ∗. By using ∗, right crystal structure is constructed. The commutativity of those crystal structures motivated us to consider Peter-Weyl type decomposition on the crystal base of Ũq(g). In [4], Kashiwara gave the Peter-Weyl type decomposition for the crystal base of modified quantum algebra of finite type and affine type of non-zero level part. But the Peter-Weyl type decomposition for affine type with level 0 part is still unclear. In this paper, we shall give some criteria for the existence of the Peter-Weyl type decomposition: