Standard Monomial Theory for Bott–Samelson Varieties

Bott–Samelson varieties are an important tool in geometric representation theory 1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the properties of Schubert varieties. They have an action of a Borel subgroup, and the projective coordinate ring of a Bott–Samelson variety splits into certain generalized Demazure modules (which also appear in other contexts 22, 23]). Standard Monomial Theory, developed by Seshadri and the first author 15, 16], and recently completed by the second author 20], gives explicit bases for the Demazure modules associated to Schubert varieties. In this paper, we extend the techniques of 20] to give explicit bases for the generalized Demazure modules associated to Bott–Samelson varieties, thus proving a strengthened form of the results announced by the first and third authors in 12] (see also 13]). We also obtain more elementary proofs of the cohomology vanishing theorems of Kumar 10] and Mathieu 25]; of the projective normality of Bott–Samelson varieties; and of the Demazure character formula.