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Marc Besson

Publications and source records attributed to Marc Besson.

10 recordsLinked to original sources
arXiv2026

Derived Enhancements of $T$-fixed subschemes

For $X$ a conical affine symplectic singularity with $\mathbb{T}=T \times \mathbb{G}_m$-action, the fixed scheme $X^T$ and the map $X^T \rightarrow X$ carry much information about the geometry of $X$. In general, $X^T \rightarrow X$ fails to be a complete intersection. Thus, we study a derived intersection whose classical locus is the $T$-fixed subscheme $X^T$. We show that the structure of the symplectic singularity on $X$ produces a duality theorem for the structure sheaf of the derived intersection. The duality theorem allows us to study the structure of such derived intersections; in particular we describe their cohomological amplitude. An important source of symplectic singularities with $\mathbb{T}$-action are affine Grassmannian slices $\overline{W}^{\lambda}_{\mu}$. We pay particular attention to these slices when $G=\mathrm{SL}_{n+1}$, and we use the previously developed theory to characterize when $(\overline{W}^{\lambda}_{\mu})^T \rightarrow \overline{W}^{\lambda}_{\mu}$ is a complete intersection.

math.AG↗
arXiv2026

Reduction rules for Demazure modules

For $G$ a complex reductive group and $B \subseteq G$ a Borel subgroup, we provide a reduction rule for certain weight multiplicities in Demazure modules $V_\lambda^w$: given a weight $\mu$ on a face of the associated weight polytope $P_\lambda^w$, we reduce the computation of the dimension of the weight space $V_\lambda^w(\mu)$ to a similar problem of computing the weight space dimension for a Demazure module of a Levi subgroup of $G$.

math.RT↗
arXiv2024

Geometric invariant theory and stretched Kostka quasi-polynomials

For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $\lambda, \mu$, we consider the dimensions of weight spaces $V_\lambda(\mu)$ of weight $\mu$ in the irreducible, finite-dimensional highest weight $\lambda$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{N\lambda}(N\mu)$ is a quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we realize the degree of this quasi-polynomial as the dimension of a certain GIT quotient. As a result, we resolve a conjecture of Gao and Gao on an explicit formula for this degree. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period.

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arXiv2024

Affine Demazure Weight Polytopes and Twisted Bruhat Orders

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible highest-weight representations of $\mathfrak{g}$. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the $\mathfrak{h}$-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders.

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arXiv2022

Weight polytopes and saturation of Demazure characters

For $G$ a reductive group and $T\subset B$ a maximal torus and Borel subgroup, Demazure modules are certain $B$-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of $G$. In order to describe the $T$-weight spaces that appear in a Demazure module, we study the convex hull of these weights - the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that $G$ is simple of classical Lie type. Specializing to $G=GL_n$, we recover results of Fink, M\'esz\'aros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.

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arXiv2020

Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules

Let $\mathscr{G}$ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb{C}$, excluding the absolutely special case of $A_{2\ell}^{(2)}$. Using the methods and results of Zhu, we prove a duality theorem for general $\mathscr{G}$ : there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for $\mathscr{G}$. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of $\mathscr{G}$. In particular, this confirms a conjecture of Haines and Richarz.

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arXiv2020

Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones

Let $\mathscr{K}(G)$ be the rational cone generated by pairs $(\lambda, \mu)$ where $\lambda$ and $\mu$ are dominant integral weights and $\mu$ is a nontrivial weight space in the representation $V_{\lambda}$ of $G$. We produce all extremal rays of $\mathscr{K}(G)$ by considering the vertices of corresponding intersection polytopes $IP_{\lambda}$, the set of points in $\mathscr{K}(G)$ with first coordinate $\lambda$. We show that vertices of $IP_{\varpi_i}$ arise as lifts of vertices coming from cones $\mathscr{K}(L)$ associated to simple Levi subgroups possessing the simple root $\alpha_i$. As corollaries we obtain a complete description of all extremal rays, as well as polynomial formulas describing the numbers of extremal rays depending on type and rank.

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arXiv2019

Multiplicity in root components via Geometric Satake

In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands $V(\lambda+\mu-N\beta) \subset V(\lambda) \otimes V(\mu)$ for $G^\vee=SL_{n+1}$, where $N\ge 1$ and $\beta$ is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when $\beta$ is not a simple root. Finally, we demonstrate that not all such top-dimensional components can be realized as closures of orbits.

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arXiv2019

A combinatorial study of affine Schubert varieties in affine Grassmannian

Let $\overline{\mathtt{X}}_\lambda$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_\lambda$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $\lambda$ is a coweight of $G$. We find a simple algorithm which describes the set $\Psi(\lambda)$ of all $\mathtt{I}$-orbits in $\overline{\mathtt{X}}_\lambda$ in terms of coweights. We introduce $R$-operators (associated to positive roots) on the coweight lattice of $G$, which exactly describe the closure relation of $\mathtt{I}$-orbits. These operators satisfy Braid relations generically on the coweight lattice. We also establish a duality between the set $\Psi(\lambda)$ and the weight system of the level one affine Demazure module $\hat{\mathscr{D}}_\lambda$ of $^L\tilde{\mathfrak{g}}$ indexed by $\lambda$, where $^L\tilde{\mathfrak{g}}$ is the affine Kac-Moody algebra dual to the affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}$ associated to the Lie algebra $\mathfrak{g}$ of $G$.

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