Demazure’s character formula for arbitrary Kac-Moody Lie algebra was given by S.Kumar and O.Mathieu independently ([6],[8]) by using geometric methods. In 1995, P.Littelmann gave some conjecture (partially solved by himself) about the relation between Demazure’s character formula and crystal bases [7], which was solved affirmatively by M.Kashiwara [3]. Then it gave purely algebraic proof for De...

Assume that G is a nilpotent Lie group. Denote by [Formula: see text] the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class [Formula: see text] for some [Formula: see text] and D is the dimension at infinity of G. Let [Formula: see text] be the Riesz transform associated with L. In this paper we obtain some estimates for the...

Abstract We show that if a nilpotent group