Flag Manifolds Research Articles

Laplace Eigenfunctions on Riemannian Symmetric Spaces and Borel–Weil Theorem

AbstractWe identify a geometric relation between the Laplace–Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel–Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating with each compact Riemannian symmetric space, via Marsden–Weinstein reduction, a generalized flag manifold that covers the space parametrizing all of its maximal totally geodesic tori. In the process, we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in detail the examples of the classical simply connected spaces of rank one and the space $SU(3)/SO(3)$. In the 2nd part of the paper, with the aid of harmonic polynomials, we induce Laplace–Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. In the examples we consider, we show that our construction provides all of the eigenfunctions.

On the stability of homogeneous Einstein manifolds II

For any G $G$ -invariant metric on a compact homogeneous space M = G / K $M=G/K$ , we give a formula for the Lichnerowicz Laplacian restricted to the space of all G $G$ -invariant symmetric 2-tensors in terms of the structural constants of G / K $G/K$ . As an application, we compute the G $G$ -invariant spectrum of the Lichnerowicz Laplacian for all the Einstein metrics on most generalized Wallach spaces and any flag manifold with b 2 ( M ) = 1 $b_2(M)=1$ . This allows to deduce the G $G$ -stability and critical point types of each of such Einstein metrics as a critical point of the scalar curvature functional.

An Atlas Adapted to the Toda Flow

Abstract We describe an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra and on its Hessenberg-type submanifolds. We show that in the local coordinates of the atlas the Toda flow becomes linear. The local coordinates are also used to show that the Toda flow on the manifold of full flags is Morse–Smale, which generalizes the result for traceless matrices in [27] to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory.

A web basis of invariant polynomials from noncrossing partitions

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.

Wonderful asymptotics of matrix coefficient D-modules

Beilinson–Bernstein localization realizes representations of complex reductive Lie algebras as monodromic D-modules on the “basic affine space” G/N, a torus bundle over the flag variety. A doubled version of the same space appears as the horocycle space describing the geometry of the reductive group G at infinity, near the closed stratum of the wonderful compactification G‾, or equivalently in the special fiber of the Vinberg semigroup of G. We show that Beilinson–Bernstein localization for Ug-bimodules arises naturally as the specialization at infinity in G‾ of the D-modules on G describing matrix coefficients of Lie algebra representations. More generally, the asymptotics of matrix coefficient D-modules along any stratum of G‾ are given by the matrix coefficient D-modules for parabolic restrictions. This provides a simple algebraic derivation of the relation between growth of matrix coefficients of admissible representations and n-homology. The result is an elementary consequence of the compatibility of localization with the degeneration of affine G-varieties to their asymptotic cones; analogous results hold for the asymptotics of the equations describing spherical functions on symmetric spaces.

A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds

Abstract We establish a noncommutative generalisation of the Borel–Weil theorem for the Heckenberger–Kolb calculi of the irreducible quantum flag manifolds ${\mathcal {O}}_q(G/L_S)$, generalising previous work for the quantum Grassmannians ${\mathcal {O}}_q(\textrm {Gr}_{n,m})$. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings $S_q[G/L_S]$ of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces ${\mathcal {O}}_q(G/L^{\,\textrm {s}}_S)$.

Reduced arc schemes for Veronese embeddings and global Demazure modules

We consider arc spaces for the compositions of Plücker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A1 (which corresponds to the Veronese curve), we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.

Euler characteristic of stable envelopes

In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.

Contravariant pairings between standard Whittaker modules and Verma modules

We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category N have the structure of highest weight categories and we establish a BGG reciprocity theorem for N.

Remarks on the signatures of invariant pseudo-Kähler metrics on generalized flag manifolds

A pseudo-Kähler manifold is a natural generalization of a Kähler manifold. It is well-known that any generalized flag manifold has pseudo-Kähler metrics. Moreover, there exists a T-root system corresponding to a generalized flag manifold. In this paper, we investigate the signatures of invariant pseudo-Kähler metrics on a generalized flag manifold of which the T-root system becomes one of the irreducible reduced root systems (in general, a T-root system is not an irreducible reduced root system).

Cominuscule points and Schubert varieties

We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions.

K$K$‐Motives and Koszul duality

We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K $K$ -theory. For this, we introduce and study categories DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ of S $\mathcal {S}$ -constructible K $K$ -motivic sheaves on varieties X $X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S $\mathcal {S}$ -constructible mixed sheaves D S m i x ( X ) $\hbox{D}^{mix}_\mathcal {S}(X)$ to DK S ( X ) $\hbox{DK}_\mathcal {S}(X)$ . We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K $K$ -motivic sheaves on Langlands dual flag varieties.

Generalized Permutahedra and Schubert Calculus

We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time.

Fargues–Rapoport conjecture for $p$-adic period domains in the non-basic case

We prove the Fargues–Rapoport conjecture for p -adic period domains in the non-basic case with minuscule cocharacter. More precisely, we give a group-theoretical criterion for the cases when the admissible locus and weakly admissible locus coincide. This generalizes the result of Hartl (2013) for the group \operatorname{GL}_n . In the last section, we also give a conjecture about the intersection of weakly admissible locus and the Newton strata in the flag variety.

Area-Minimizing Cones Over Grassmannian Manifolds

It is a well-known fact that there exists a standard minimal embedding map for the Grassmannians of n-planes \(G(n,m;{\mathbb {F}})({\mathbb {F}}={\mathbb {R}},{\mathbb {C}},{\mathbb {H}})\) and Cayley plane \({\mathbb {O}}P^2\) into Euclidean spheres, then a natural question is that if the cones over these embedded Grassmannians are area-minimizing? In this paper, detailed descriptions for this embedding map are given from the point of view of Hermitian orthogonal projectors which can be seen as a direct generalization of Gary R. Lawlor’s( [23]) original considerations for the case of real projective spaces, then we re-prove the area-minimization of those cones which was gradually obtained in [18, 19], and [30] from the perspectives of isolated orbits of adjoint actions or canonical embedding of symmetric R-spaces, all based on the method of Gary R. Lawlor’s Curvature Criterion. Additionally, area-minimizing cones over almost all common Grassmannians have been given by Takahiro Kanno, except those cones over oriented real Grassmannians \({\widetilde{G}}(n,m;{\mathbb {R}})\) which are not Grassmannians of oriented 2-planes. The second part of this paper is devoted to complement this result, a natural and key observation is that the oriented real Grassmannians can be considered as unit simple vectors in the exterior vector spaces, we prove that all their cones are area-minimizing except \({\widetilde{G}}(2,4;{\mathbb {R}})\).

Quantum cohomology of symplectic flag manifolds

We compute the quantum cohomology of symplectic flag manifolds. Symplectic flag manifolds can be described by non-abelian GLSMs with superpotential. Although the ring relations cannot be directly read off from the equations of motion on the Coulomb branch due to complication introduced by the non-abelian gauge symmetry, it can be shown that they can be extracted from the localization formula in a gauge-invariant form. Our result is general for all symplectic flag manifolds, which reduces to previously established results on symplectic Grassmannians and complete symplectic flag manifolds derived by other means. We also explain why a (0, 2) deformation of the GLSM does not give rise to a deformation of the quantum cohomology.

A Gallery Model for Affine Flag Varieties via Chimney Retractions

This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat–Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat–Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety.

SOLUTIONS OF THE tt*-TODA EQUATIONS AND QUANTUM COHOMOLOGY OF MINUSCULE FLAG MANIFOLDS

AbstractWe relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological–antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest–Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput–Manivel–Perrin and Golyshev–Manivel.

On secant dimensions and identifiability of flag varieties

We investigate the secant dimensions and the identifiability of flag varieties parametrizing flags of subspaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.

Polynomial representations of classical Lie algebras and flag varieties

Recently we have started a program to describe the action of Lie algebras associated with Dynkin-type diagrams on generic Verma modules in terms of polynomial vector fields. In this paper we explain that the results for the classical ABCD series of Lie algebras coincide with the more conventional approach, based on the knowledge of the entire algebra, not only the simple roots. We apply the coset description, starting with a large representation and then reducing it with the help of the algebra, commuting with the original one. The irreducible representations are then obtained by gauge fixing this residual symmetry.