Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras

摘要: Let G be a reductive algebraic group defined over an algebraically closed field k. We fix Borel subgroup B, and for dominant weight λ let Lλ the associated line bundle on generalized flag variety G/B. In series of articles, Lakshmibai, Musili Seshadri initiated program to construct basis space H0(G/B,Lλ) with some particularly nice geometric properties. The purpose is extend Hodge-Young standard monomial theory SL(n) case any semisimple and, more generally, Kac-Moody algebras. refer [3], [7], [10], [14] survey subject applications. provide new approach which completes avoids by considerations earlier articles. fact, method works all symmetrizable most important tools we need in our are combinatorial language path model representation [11], [12], quantum groups at root unity. Uv(g) `-th unity v. use Frobenius map [15] “contract” certain Uv(g)-modules so that they become G-modules. corresponding between dual spaces can seen as kind splitting power H(G/B,Lλ) → H(G/B,L`λ), s 7→ s. For simplicity us assume simply laced case. Vλ Weyl module highest λ, Mλ Uv(g). There canonical way attach tensor product bπ := bν1 ⊗ . .⊗ bν` extremal vectors bνj ∈ M∗ each L-S π shape [11] appropriate ` (recall characterized collection weights rational numbers). To = V ∗ , contraction embed into (Mλ) ⊗`. Denote pπ image under (M λ) ⊗` show pπ, form Further, H0(G/B,L`λ) pν1 · pν` pνi H0(G/B,Lλ), plus linear combination elements “bigger” partial order. given compatible restriction H0(X,Lλ) Schubert X it has “standard property”.