Let $T$ be a maximal torus of a semisimple complex algebraic group, $\mathrm{BS}(s)$ be the Bott-Samelson variety for a sequence of simple reflections $s$ and $\mathrm{BS}(s)^T$ be the set of $T$-fixed points of $\mathrm{BS}(s)$. We prove the tensor product decompositions for the image of the restriction $H^\bullet_T(\mathrm{BS}(s),k)\to H_T^\bullet(X,k)$, where $X\subset\mathrm{BS}(s)^T$ is defined by some special not overlapping equations $γ_iγ_{i+1}\cdotsγ_j=w_{i,j}$ with right-hand sides belonging to the Weyl group.
We give a topological explanation of the main results of V.Shchigolev, Categories of Bott-Samelson Varieties, Algebras and Representation Theory, 23 (2), 349-391, 2020. To this end, we consider some subspaces of Bott-Samelson varieties invariant under the action of the maximal compact torus $K$ and study their topological and homological properties. Moreover, we describe multiplicative generators of the equivariant cohomologies of Bott-Samelson varieties.
We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measu…
We prove that the Gromov width of any Bott-Samelson variety birational to a Schubert variety and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.
Let $C$ be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to $C$. We prove that the decorated double Bott-Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson-Thomas transformations on double Bott-Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock-Goncharov duality conjecture in these cases.…
In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott-Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.
Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the $\infty$-category of D-modules on the loop group of $G$ is equivalent to the monoidal colimit of the $\infty$-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. W…
Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced express…
Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety $G/B$. For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott-Samelson resolution of Schubert varieties we prove a BGG-type recursion for the elliptic classes, and study the Hecke algebra of our ellipti…
Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${\mathcal{A}}_{\rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $π_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${\mathcal{A}}_{\rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, …
Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ i…
We define the Barbasch-Evens-Magyar varieties. We show they are isomorphic to the smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely to symmetric orbit closures, thereby giving resolutions of singularities in certain cases. Our definition parallels [P.~Magyar '98]'s construction of the Bott-Samelson varieties [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces a graphical description in type $A$, stratification into closed subvarieties of the same kind, and determination of the torus-fixed points. Moreover, we explain how these ma…
Expanding an idea of Raoul Bott, we propose a construction of canonical bases for unitary representations that comes from big torus actions on families of Bott-Samelson manifolds. The construction depends only on the choices of a maximal torus, a Borel subgroup ,and a reduced expression for the longest element of the Weyl group. It relies on a conjectural vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles on the total spaces of the families, hence the question mark in the title.
An analogue of Bott's theorem for Schubert varieties-related to torus semistable points
设$G$是复数域上的单连通代数群.本文给出了Schubert簇$X(τ)$的所有高上同调$H^{i}(X(τ), E)$为零的充分条件，即当$G/B$的切丛的限制$E$为X（τ）时，$. We further show that the global sections $ H^{0}（X（τ），E）$ is the adjoint representation of $ G $ when $ G$是单系的.
