Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8904949 | Advances in Mathematics | 2018 | 54 Pages |
Abstract
Given a connected reductive algebraic group G and a finitely generated monoid Î of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme MÎ for multiplicity-free affine G-varieties with weight monoid Î. This scheme is equipped with an action of an 'adjoint torus' Tad and has a distinguished Tad-fixed point X0. In this paper, we obtain a complete description of the Tad-module structure in the tangent space of MÎ at X0 for the case where Î is saturated. Using this description, we prove that the root monoid of any affine spherical G-variety is free. As another application, we obtain new proofs of uniqueness results for affine spherical varieties and spherical homogeneous spaces first proved by Losev in 2009. Furthermore, we obtain a new proof of Alexeev and Brion's finiteness result for multiplicity-free affine G-varieties with a prescribed weight monoid. At last, we prove that for saturated Î all the irreducible components of MÎ, equipped with their reduced subscheme structure, are affine spaces.
Keywords
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Roman Avdeev, Stéphanie Cupit-Foutou,