JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 3, July 1998, Pages 551–567 S 0894-0347(98)00268-9 CONTRACTING MODULES AND STANDARD MONOMIAL THEORY FOR SYMMETRIZABLE KAC-MOODY ALGEBRAS PETER LITTELMANN Introduction Let G be a reductive algebraic group defined over an algebraically closed field k. We fix a Borel subgroup B, and for a dominant weight λ let L λ be the as- sociated line bundle on the generalized flag variety G/B. In a series of articles, Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space H 0 (G/B, L λ ) with some particularly nice geometric properties. The purpose of the program is to extend the Hodge-Young standard monomial theory for the group SL(n) to the case of any semisimple algebraic group and, more generally, to Kac-Moody algebras. We refer to [3], [7], [10], [14] for a survey of the subject and applications. We provide a new approach which completes the program and which avoids the case by case considerations of the earlier articles. In fact, the method works for all symmetrizable Kac-Moody algebras. The most important tools we need in our approach are the combinatorial language of the path model of a representation [11], [12], and quantum groups at a root of unity. Let U v (g) be the quantum group associated to G at an ℓ-th root of unity v. We use the quantum Frobenius map [15] to “contract” certain U v (g)-modules so that they become G-modules. The corresponding map between the dual spaces can be seen as a kind of splitting of the power map H 0 (G/B, L λ ) → H 0 (G/B, L ℓλ ), s → s ℓ . For simplicity let us assume we are in the simply laced case. Let V λ be the Weyl module of G of highest weight λ, and let M λ be the corresponding Weyl module of U v (g). There is a canonical way to attach a tensor product b π := b ν1 ⊗ ... ⊗ b ν ℓ of extremal weight vectors b νj ∈ M ∗ λ to each L-S path π of shape λ [11] for an appropriate ℓ (recall that an L-S path can be characterized by a collection of extremal weights and rational numbers). To construct a basis of H 0 (G/B, L λ )= V ∗ λ , we use the contraction map to embed V λ into (M λ ) ⊗ℓ . Denote by p π the image of b π in V ∗ λ under the dual map (M ∗ λ ) ⊗ℓ → V ∗ λ . We show that the vectors p π , π an L-S path of shape λ, form a basis of V ∗ λ . Further, the ℓ-th power p ℓ π ∈ H 0 (G/B, L ℓλ ) is a product of extremal weight vectors p ν1 ··· p ν ℓ , p νi ∈ H 0 (G/B, L λ ), plus a linear combination of elements which are “bigger” in some partial order. The basis given by the p π is compatible with the restriction map H 0 (G/B, L λ ) → H 0 (X, L λ ) to a Schubert variety X , and it has the “standard monomial property”. Received by the editors July 17, 1997. 1991 Mathematics Subject Classification. Primary 17B10, 17B67, 20G05, 14M15. Key words and phrases. Path model, quantum Frobenius map, standard monomial theory. c 1998 American Mathematical Society 551 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use