Merging divisorial with colored fans

The Cox ring of a spherical embedding

According to Sakellaridis, many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings. We pursue this perspective by developing a local counterpart and try to explicate the functional equations. These constructions are also related to the L 2 -spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of p-adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore, we explain the doubling method and show that it fits into the paradigm of L-monoids developed by L. Lafforgue, B. C. Ngo et al., by reviewing the constructions of Braverman and Kazhdan (2002). In the global setting, we give certain speculations about global zeta integrals, Poisson formulas and their relation to period integrals.

First we give a formula of spherical functions on certain spherical homogeneous spaces. Then, applying it, we complete the theory of the spherical functions on the space X of nondegenerate unramified hermitian forms on a p-adic number field. More precisely, we give an explicit expression for the spherical functions, prove theorems on the spherical Fourier transforms on the space of Schwartz-Bruhat functions on X, and parametrize of all spherical functions on X. Finally, as an application, we give explicit expressions of local densities of representations of hermitian forms.

The aim of this paper is to study further the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus $T^k$. As the torus action in this case has only isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In that context for homogeneous $SU$-spaces we prove the well known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any $T^k$-equivariant Hirzebruch genus given by an odd power series vanishes. Related to the problem of multiplicativity we provide construction of stable complex $T^k$-fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces.

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 measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed $2$-forms.

A spherical homogeneous spaceG/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism.

This chapter focuses on Einstein metrics which are Hermitian with respect to some integrable complex structure on a compact complex surface. These will, for the sake of brevity, sometimes be referred to as Einstein Hermitian metrics, so it is worth warning the reader that these are not a priori Hermite-Einstein in the sense of the theory of holomorphic vector bundles. The chapter explores a local result concerning the conformal curvature of Einstein Hermitian metrics. The identity component of the isometry group of an extremal Kahler metric is a maximal compact subgroup of the identity component of the complex automorphism group; and since the maximal compact is unique up to conjugation, a suitable change of homogeneous coordinates will make the extremal Kahler metric g invariant under the torus action. An extremal Kahler metric in the anti-canonical class would have to be a Kahler-Einstein metric because the relevant Futaki invariant vanishes, and so coincide with the metric of Siu.

This article studies components of Springer fibers for 𝔤𝔩(n) that are associated to closed orbits of GL(p)×GL(q) on the flag variety of GL(n),n=p+q. These components occur in any Springer fiber. In contrast to the case of arbitrary components, these components are smooth varieties. Using results of Barchini and Zierau we show these components are iterated bundles and are stable under the action of a maximal torus of GL(n). We prove that if ℒ is a line bundle on the flag variety associated to a dominant weight, then the higher cohomology groups of the restriction of ℒ to these components vanish. We derive some consequences of localization theorems in equivariant cohomology and K-theory, applied to these components. In the appendix we identify the tableaux corresponding to these components, under the bijective correspondence between components of Springer fibers for GL(n) and standard tableaux.

We prove the LeBrun–Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold $X$ of dimension $2n+1$ that has reductive automorphism group of rank at least $n-2$ is necessarily homogeneous. This implies that any positive quaternion-Kähler manifold of real dimension at most $16$ is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most $9$ with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Białynicki-Birula decomposition and equivariant Riemann–Roch theorems. From the point of view of $T$ varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most $\frac{1}{2} (\operatorname{dim}X+5)$ with $\operatorname{dim}$ arbitrarily high, but we require this special (contact) structure of $X$. Previous methods for studying $T$-varieties in general usually only apply for complexity at most $2$ or $3$.

Under study are equivariant projective compactifications of reductive groups that can be obtained as the closure of the image of the group in the space of projective linear operators of a representation. The structure and the mutual position of the orbits of the action of the direct square of the group acting by left/right multiplication and the local structure of the compactification in the neighbourhood of a closed orbit are described. Several conditions for the normality and smoothness of a compactification are obtained. The methods used are based on the theory of equivariant embeddings of spherical homogeneous spaces and reductive algebraic semigroups.

Chapter 1. A classification of Curtis-Tits amalgams.- Chapter 2. The use of valuations for classifying point-line geometries.- Chapter 3. An outline of polar spaces: basics and advances.- Chapter 4. Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties.- Chapter 5. Generation of Lie Incidence Geometries: A Survey.- Chapter 6. Witt-Type Theorems for Subspaces of Lie Geometries: A Survey.- Chapter 7. Embeddings of cotriangular spaces.- Chapter 8. Unipotent Overgroups in Simple Algebraic Groups.- Chapter 9. The axes of a Majorana representation of A12.- Chapter 10. GIT related problems of the flag variety for the action of a maximal torus.- Chapter 11. Characterizations of trialities of type Iid in buildings of type D4.- Chapter 12. On the isomorphism problem for Coxeter groups and related topics.- Chapter 13. Lectures on Artin groups and the K(pi 1) conjecture. Chapter 14. Algebraic codes and geometry of some classical generalized polygons.- Chapter 15. Some Weyl modules of the algebraic groups of type E_6.- Problem Set.

We compute the automorphism group of the quotient of a generalized Grassmannian G/P by the action of a maximal torus of the semisimple group G. We classify the twisted forms of such quotients, that is, varieties isomorphic to these quotients over an algebraic closure of the base field. It is proved that all such forms are unirational.Bibliography: 20 titles.

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad (G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

Let G be a connected, complex reductive group and B, a Borel subgroup. The homogeneous space G/B is called the (full) flag variety of G; for GL n this is the ordinary variety of flags of subspaces in ℂ n . The properties of this variety have many implications in the study of reductive groups and their representations. For example, all the irreducible finite dimensional G-modules can be obtained as global sections of line bundles on the flag variety. More generally, any of its (G-linearized) line bundles has at most one non-vanishing sheaf-cohomology space, having a natural structure of a simple G-module if G is simply connected. This is part of the content of the Borel-Weil-Bott theorem and can be seen as a geometric interpretation of E. Cartan’s theory of highest weights.

In [GT], Goldin and Tolman extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99,Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class alpha(p) in the equivariant cohomology of the manifold M for each fixed point p is an element of M. When they exist, canonical classes form a natural basis of the equivariant cohomology of M. In particular, when M is a flag variety, these classes are the equivariant Schubert classes. It is a long-standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class alpha(p) to a fixed point q. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.