Affine pavings of Hessenberg varieties for semisimple groups

In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of $\mathfrak{gl }_n(\mathbb C )$ are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology.     

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Under suitable hypotheses, we prove that a form of a projective homogeneous variety G/P defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre’s Conjecture II in Galois cohomology for function fields over an algebraically closed field.     

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety. Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000     

Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all -rank one lattices in semisimple groups of -rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.     

Low dimensional actions of semisimple groups

For a simple non-compact Lie groupG with finite center we determine the smallest integern(G) such thatG has an almost effective action on a compact manifold of dimensionn(G) and characterize the compact manifolds of dimensionn(G) on whichG acts. We study actions of a semisimple groupG on compact manifolds of dimensionn(G)+1 and determine the orbit structure of the action ofG and its maximal compact subgroup. We give several examples to illustrate the results. This work was supported by an NSF postdoctoral Research Fellowship. An erratum to this article is available at .     

On subsemigroups of semisimple Lie groups

In this paper we classify the subsemigroups of any connected semisimple Lie groupG which areK-bi-invariant, whereG=KAN is an Iwasawa decomposition ofG.     

Prime spectra of quantum semisimple groups

We study the prime ideal spaces of the quantized function algebras , for a semisimple Lie group and an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar's strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature -- it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of . In the final section the results are specialized to the case , where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine -space satisfy our axiom scheme when the group generated by the parameters is torsionfree.

     

A characterization of linearly semisimple groups

Let G = SpecA be an affine K-group scheme and à = {w ∈ A*: dim _K A*·w < ∞, dim _K w· A* < ∞}. Let 〈?,?〉: A* × Ã → K, 〈w, \(\tilde w\)〉:=tr(w~w), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A*.     

Problems in R-equivalence of semisimple groups

R-equivalence on the unimodular group of a simple algebra is studied in detail. A rather complete characterization is obtained for the R-equivalence group of semisimple groups over the field of -adic numbers. In this case all the Manin groups are commutative, and for simply connected groups, they are trivial. It is shown how R-equivalence can be applied to the weak approximation problem.Translated from Zapiskii Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 86, pp. 49–65, 1979.     

Totally nonnegative and oscillatory elements in semisimple groups

We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F. R. Gantmacher, M. G. Krein, A. Whitney, C. Loewner, M. Gasca, and J. M. Peña to the case of an arbitrary complex semisimple Lie group.     

Generating sets for compact semisimple Lie groups

Let be a compact connected semisimple Lie group. We prove that the subset of consisting of pairs which topologically generate is Zariski open.