Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

New classes of parameterized differential Galois groups

This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method of patching over fields with a suitable version of Galois descent to prove that certain groups do occur as parameterized differential Galois groups over k((t))(x). This class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups.

     

Schubert cycles, differential forms and D-modules on varieties of unseparated flags

Proper homogeneous G-spaces (where G is semisimple algebraic group) over positive characteristic fields k can be divided into two classes, the first one being the flag varieties G/P and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the D -affinity of Beilinson and Bernstein fails for varieties of unseparated flags.     

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.     

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k ₁, ..., k _r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.     

A Note on the Double Affine Hecke Algebra of Type GL2

We express the double affine Hecke algebra ? associated to the general linear group GL₂(k) (here, k is a field with char(k) ≠ 2) as an amalgamated free product of quadratic extensions over the three-dimensional quantum torus 𝒪_q((k^×)³). With an eye towards proving ring-theoretic results pertaining to ?, a general treatment of amalgamated products of Ore and quadratic extensions is given. We prove an analogue of the Hilbert Basis Theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. Furthermore, we determine sufficient conditions which imply Q is a principal ideal ring. Finally, we construct an explicit isomorphism from ? to the amalgamated free product ring of quadratic extensions over 𝒪_q((k^×)³), a ring known to be noetherian. Therefore, it follows that ? is noetherian.     

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group withLie algebra . We extend scalars to the power series field in one variable C(()), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of C(()),i.e. fixed point varieties on the full affine flag manifold. We definerepresentations of the affine Weyl group in the homology of these varieties,generalizing Kazhdan and Lusztig's topological construction of Springer'srepresentations to the affine context.     

The wedderburn-malcev theorem for comodule algebras

Let H be a Hopf algebra over a field k. Under some assumptions on H we state and prove a generalization of the Wedderburn-Malcev theorem for i7-comodule algebras. We show that our version of this theorem holds for a large enough class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups, finite dimensional Hopf algebras over fields of characteristic 0 and group algebras. Some dual results are also included.     

Appendix: On parahoric subgroups

We give the proofs of some simple facts on parahoric subgroups and on Iwahori Weyl groups used in [T. Haines, The base change fundamental lemma for central elements in parahoric Hecke algebras, preprint, 2008; G. Pappas, M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008) 118–198; M. Rapoport, A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005) 271–318].     

Homotopy invariance of coherent Witt groups

 We prove the following general homotopy invariance theorem for coherent Witt groups: Let be a flat morphism of separated Gorenstein schemes of finite Krull dimension with affine fibers,i.e. π⁻¹(y) is an affine space over the residue field k(y) for all yY, then the induced homomorphism of coherent Witt groups is an isomorphism. As an application we calculate the (classical) Witt group of the affine hyperbolic sphere over a regular local ring. Received: 22 August 2001; in final form: 22 June 2002 / Published online: 1 April 2003     

From moment graphs to intersection cohomology

We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks. Received: 9 November 2000 / Published online: 24 September 2001     

K-theory of twisted Grassmannians

We give a computation of the K-groups of Grassmannians and flag varieties over an arbitrary Noetherian base scheme. We also compute the K-groups of forms of Grassmannians and flag varieties associated to a sheaf of Azumaya algebras. One ingredient in the computation is the extension of the Bott theorem on the cohomology of line bundles on the flag variety (over Q) to a K ₀-Bott theorem valid over arbitrary Noetherian base schemes.Partially supported by the NSF.     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

The groups of points on abelian varieties over finite fields

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f _A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f _A (1 − t).     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

We prove that for every field k and every positive integer n there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n we have S(F_q, n)→1 as q→∞ over the prime powers.     

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL_n-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n=1 our definitions and results reduce to those of classical affine algebraic geometry.     

Foliations in deformation spaces of local G-shtukas

We study local G-shtukas with level structure over a base scheme whose Newton polygons are constant on the base. We show that after a finite base change and after passing to an étale covering, such a local G-shtuka is isogenous to a completely slope divisible one, generalizing corresponding results for p-divisible groups by Oort and Zink. As an application we establish a product structure up to finite surjective morphism on the closed Newton stratum of the universal deformation of a local G-shtuka, similarly to Oort?s foliations for p-divisible groups and abelian varieties. This also yields bounds on the dimensions of affine Deligne–Lusztig varieties and proves equidimensionality of affine Deligne–Lusztig varieties in the affine Grassmannian.     

Affine Kac-Moody Lie algebras as torsors over the punctured line

We interpret and develop a theory of loop algebras as torsors (principal homogeneous spaces) over Spec (k[t, t⁻¹]). As an application, we recover Kac's realization of affine Kac-Moody Lie algebras.