Canonical Bases and Self-Evacuating Tableaux

Let g be a complex semisimple Lie algebra. In this paper, we prove that the action of a certain involution on canonical bases for irreducible g-modules deened by Lusztig agrees with the action of a special element of the associated simply connected Lie group, up to a scalar of unit absolute value. This leads to formulas for the number of xed points of the involution by means of the Weyl character formula. In the g = sl(n) case, Lusztig's involution has been proved by Berenstein and Zelevinsky to coincide with evacuation of semistandard tableaux. Thus we obtain as a corollary formulas for the number of self-evacuating semistandard tableaux of xed shape. We also prove some reenements that keep track of the xed points of the canonical basis belonging to each weight space. In particular, we obtain a formula for the number of self-evacuating standard tableaux of xed shape.