Abstract. A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the NewtonOkounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara's involution ( * -operation) corresponds to a change of valuations on the rational function field.

A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (resp., the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (resp., the Nakashima-Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott-Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (resp., by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation.Comment: 21 pages, to appear in J. London Math. Soc. (2

Let g be a complex semisimple Lie algebra. For a regular element x in g and a Hessenberg space H ⊆ g, we consider a regular Hessenberg variety X(x, H) in the flag variety associated with g. We take a Hessenberg space so that X(x, H) is irreducible, and show that the higher cohomology groups of the structure sheaf of X(x, H) vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.

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A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this generalizes a Newton polytope for a toric variety. This convex body has various kinds of information about the original projective variety; for instance, Kaveh showed that the string polytopes from representation theory are examples of Newton-Okounkov bodies for Schubert varieties. In this paper, we extend the notion of string polytopes for Demazure modules to generalized Demazure modules, and prove that the resulting generalized string polytopes are identical to the Newton-Okounkov bodies for Bott-Samelson varieties with respect to a specific valuation. As an application of this result, we show that these are indeed polytopes. Contents 1. Introduction 1 2. Newton-Okounkov bodies 3 3. Upper crystal bases and upper global bases 8 4. String polytopes for Demazure modules 11 5. String polytopes for generalized Demazure modules 13 6. Upper global bases of tensor product modules 18 7. Main result 20 8. Similar results for another valuation 22 Appendix A. Proof of Proposition 5.15 25 Appendix B. Some examples of string polytopes for generalized Demazure modules 27 References 30