Tambarization of a Mackey functor and its application to the Witt–Burnside construction

For an arbitrary group G, a (semi-)Mackey functor is a pair of covariant and contravariant functors from the category of G-sets, and is regarded as a G-bivariant analog of a commutative (semi-)group. In this view, a G-bivariant analog of a (semi-)ring should be a (semi-)Tambara functor. A Tambara functor is firstly defined by Tambara, which he called a TNR-functor, when G is finite. As shown by Brun, a Tambara functor plays a natural role in the Witt–Burnside construction.It will be a natural question if there exist sufficiently many examples of Tambara functors, compared to the wide range of Mackey functors. In the first part of this article, we give a general construction of a Tambara functor from any Mackey functor, on an arbitrary group G. In fact, we construct a functor from the category of semi-Mackey functors to the category of Tambara functors. This functor gives a left adjoint to the forgetful functor, and can be regarded as a G-bivariant analog of the monoid-ring functor.In the latter part, when G is finite, we investigate relations with other Mackey-functorial constructions — crossed Burnside ring, Elliott?s ring of G-strings, Jacobson?s F-Burnside ring — all these lead to the study of the Witt–Burnside construction.     

Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties

Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| _H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| _H of discrete series π for a symmetric space G/H is also given. Oblatum 2-X-1996 & 10-III-1997     

Mackey functors and bisets

For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.     

On the enright functor in the highest weight category

LetG be a complex semisimple Lie group,B its Borel subgroup andX a flag variety ofG. We define a functor on the category ofB-equivariantD _X-modules that corresponds, under the global section functor, to the Enright functor on the highest weight category. We use this to lift Enright functor to the mixed version of the highest weight category. As an application we obtain that the socle and the cosocle filtration of a primitive quotient of the enveloping algebra coincide.     

The slice classification of categories of coalgebras for comonads

We prove that the category _G Set of all G-coalgebras in s-equivalent to the category Alg(1) of all unary algebras iff the functional part G of G is not a product of the identity functor I and a constant functor. Received June 11, 1998; accepted in final form January 16, 1999.     

Real Closed Graded Fields

We investigate homogeneous orderings on G-graded rings where G is an arbitrary ordered abelian group. For this we introduce the notion of real closed graded fields. We generalize the Artin–Schreier characterization of real closed fields to the graded context. We also characterize real closed graded fields in terms of the group G and in terms of its homogeneous elements of degree 0. Supported by DFG-project KN202/5-1.     

On injective rational coefficient systems

LetG be a finite group. By a rational coefficient system forG we mean a contravariant functor from the category of canonical orbits ofG andG-maps into the category of -vector spaces. In this paper we study injective objects in the category of rational coefficient systems forG.     

On certain domains in cycle spaces of flag manifolds

The affine homogeneous space associated to a real semi-simple Lie group G with maximal compact subgroup K contains a number of naturally defined{\it G}-invariant neighborhoods of its real points which are of interest from various points of view. Here the universal Iwasawa domain is introduced from the point of view of incidence geometry and certain of its properties are derived, e.g., it is Stein, Kobayashi hyperbolic and contains the domain introduced by Akhiezer and Gindikin which is now known to be equivalent to the maximal domain of definition of the adapted complex structure associated to the Killing metric in the tangent bundle . One of the main goals of the paper is to develop methods which lead to a better understanding of the Wolf domain of cycles in an open G-orbit D in a flag manifold . The key is the Schubert domain which is defined by Schubert cycles of complementary dimension to the cycles. These are defined by a Borel subgroup containing an Iwasawa factor AN and consequently and are closely related. Received: 28 May 2001 / Revised version: 19 November 2001 / Published online: 23 May 2002     

Principe de Hasse pour les espaces principaux homogènes sous les groupes classiques sur un corps de dimension cohomologique virtuelle au plus 1

Letk be a perfect field of virtual cohomological dimension at most 1. LetG be a semi-simple, simply connected classical group. In this paper it is proved that a principal homogeneous space underG which is trivial over every real closure ofk is trivial.      

Operator K-Theory and Functor N 0. Some Applications

In this paper, we present a general introduction to the K-theory of C ^*-algebras and survey of our previous papers, where the functor N ₀ from the category of von Neumann algebras to the category of Abelian groups was defined. We investigate the properties of this functor (in particular, its interrelation with the functor K ₀) and point out some applications of the functor N ₀ in noncommutative geometry. In addition, we recall facts of theory of C ^*-algebras, von Neumann algebras, and Hilbert C ^*-modules.     

The spectrum of the Burnside Tambara functor of a cyclic group

We derive a family of prime ideals of the Burnside Tambara functor for a finite group G. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups.     

An Embedding Theorem for Automorphism Groups of Cartan Geometries

We study the automorphism group of a Cartan geometry, and prove an embedding theorem analogous to a result of Zimmer for automorphism groups of G-structures. Our embedding theorem leads to general upper bounds on the real rank or nilpotence degree of a Lie subgroup of the automorphism group. We prove that if the maximal real rank is attained in the automorphism group of a geometry of parabolic type, then the geometry is flat and complete.     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Equivariant first order differential operators for parabolic geometries

Let G be a real semisimple Lie group, P a parabolic subgroup, V and W irreducible representations of P, G × p V and G × p W the associated homogeneous vector bundles. The G-equivariant first order differential operators from the first to the second bundle are determined and described using methods of Lie theory.     

Measures with real spectra

Summary We solve a problem of Y. Katznelson: ifG is a locally compact abelian group and if a measure inM(G) has real spectrum, does it follow that the range of its Fourier-Stieljes transform is dense in its spectrum? We give a general construction of continuous measures with real spectrum and singular convolution powers. It is shown that the real spectrum property is strongly related to conditions of quasi-invariance under convolutors and that, in the simple case of quasi-invariance under translations, the range of the Fourier-Stieltjes transform is dense in the spectrum. However, a construction inM ₀(T) provides a negative answer to Katznelson's question.     

Tannakian Approach to Linear Differential Algebraic Groups

Tannaka’s theorem states that a linear algebraic group G is determined by the category of finite-dimensional G-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category corresponding to their representations and show how this category determines such a group. This work was partially supported by NSF Grant CCR-0096842 and by the Russian Foundation for Basic Research, project no. 05-01-00671.     

On the modularity of a lattice of τ-closed <Emphasis Type="Italic">n</Emphasis>-multiply ω-composite formations

Let n ≥ 0, let ω be a nonempty set of prime numbers and let τ be a subgroup functor (in Skiba’s sense) such that all subgroups of any finite group G contained in τ (G) are subnormal in G. It is shown that the lattice of all τ-closed n-multiply ω-composite formations is algebraic and modular.     

Deformation of proper actions on reductive homogeneous spaces

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of standard compact quotients of G/H, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Γ keeps acting properly discontinuously and cocompactly on G/H. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n, 2)/U(n, 1) by Zariski-dense discrete subgroups of SO(2n, 2) acting properly discontinuously.     

Bundles on non-proper schemes: representability

Let X be a proper scheme over a field k which satisfies Serre’s condition S ₂ and G a reductive group over k. We prove that the functor of principal G-bundles, defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin.