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, making $(G/Q, π_{G/Q}, {\mathcal{A}}_{\rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T \subset G$ is a maximal torus, $B \subset G$ a Borel subgroup, and $N$ the uniradical of $B$.
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.
We give an exact algorithm to calculate (under some GKM-restriction) the matrix describing the embedding $\B(\s)_x\subset\B(\s)^x$, where the first module is the costalk and the second one is the stalk at $x$ of a Bott-Samelson module (sheaf) $\B(\s)$. This allows us to calculate the first few terms of the decomposition of $\B(\s)$ into a sum of indecomposable modules (sheaves) and to calculate the characters of Braden--MacPherson sheaves in some previously unknown cases.
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…
Let G be a connected complex semisimple Lie group. Webster and Yakimov have constructed partitions of the double flag variety G/B x G/B_, where (B, B_) is a pair of opposite Borel subgroups of G, generalizing the Deodhar decompositions of G/B. We show that these partitions can be better understood by constructing cell decompositions of a product of two Bott-Samelson varieties Z_{u, v}, where u and v are sequences of simple reflections. We construct coordinates on each cell of the decompositions and in the case of a positive subexpression, we relate these coordinates to regular functions on a p…
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 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.
We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from that used previously in the literature. The polytope that arises is a special case of the Grossberg-Karshon twisted cubes studied by Grossberg and Karshon in connection to character formulae for irreducible $G$-representations and also studied previously by the authors in relation to certain toric varieties associated to 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.…
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 id…
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 use the theory of Mori dream spaces to prove that the global Okounkov body of a Bott-Samelson variety with respect to a natural flag of subvarieties is rational polyhedral. In fact, we prove more generally that this holds for any Mori dream space which admits a flag of Mori dream spaces satisfying a certain regularity condition. As a corollary, Okounkov bodies of effective line bundles over Schubert varieties are shown to be rational polyhedral. In particular, it follows that the global Okounkov body of a flag variety $G/B$ is rational polyhedral. As an application we show that the asymptot…
Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we …
We construct combinatorial bases of the $T$-equivariant ($T$ is the maximal torus) cohomology $H^\bullet_T(Σ,k)$ of the Bott-Samelson variety $Σ$ under some mild restrictions on the field of coefficients $k$. This bases allow us to prove the surjectivity of the restrictions $H^\bullet_T(Σ,k)\to H^\bullet_T(π^{-1}(x),k)$ and $H^\bullet_T(Σ,k)\to H^\bullet_T(Σ\setminusπ^{-1}(x),k)$, where $π:Σ\to G/B$ is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of $H^\bullet_T(Σ,k)$ and restricting them to $π^{-…
We consider all Bott-Samelson varieties ${\rm BS}(s)$ for a fixed connected semisimple complex algebraic group with maximal torus $T$ as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms ${\rm BS}(s)\hookrightarrow{\rm BS}(s')$, where $s$ is a subsequence of $s'$. Every morphism of the new category induces a map between the $T$-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded $H^\bulle…
Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there are two models for the flag space of $G$: the complex quotient $X=G/B$ and the real quotient $K/T$. These models are smoothly equivalent via the map $\tilde{\mathbf k}\colon G/B\to K/T$ induced by factorization in $G$ relative to the Iwasawa decomposition $G=KAN$, where $N$ is the nilradical of $B$ and $H=TA$. Likewise, there are two models for resolutions …
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…
Let $G$ be a connected complex semi-simple Lie group, and let $Z_{\bf u}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure $π_n$ on $Z_{\bf u}$ defined by a standard multiplicative Poisson structure $π_{\rm st}$ on $G$. We explicitly express $π_n$ on each of the $2^n$ affine coordinate charts, one for every subexpression of ${\bf u}$, in terms of the root strings and the structure constants of the Lie algebra of $G$. We show that the restriction of $π_n$ to each affine coordinate c…
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 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 give a construction of MV-polytopes of a complex semisimple algebraic group G in terms of the geometry of the Bott-Samelson variety and the affine building. This is done by using the construction of dense subsets of MV-cycles by Gaussent and Littelmann. They used LS-gallery to define subsets in the Bott-Samelson variety that map to subsets of the affine Grassmannian, whose closure are MV-cycles. Since points in the Bott-Samelson variety correspond to galleries in the affine building one can look at the image of a point in such a special subset under all retractions at infinity. We prove tha…
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…
