### Book chapter

## Modality of representations and geometry of θ-groups

We first establish several general properties of modality of algebraic group

actions. In particular,we introduce the notion of a modality-regular action and prove

that every visible action is modality-regular. Then, using these results, we classify irreducible

linear representations of connected simple algebraic groups of every fixed

modality < 3. Next, exploring a finer geometric structure of linear actions, we generalize

to the case of any cyclically graded semisimple Lie algebra the notion of a

packet (or a Jordan/decomposition class) and establish the properties of packets.

### In book

After an introductory chapter that provides an overview to theoretical issues in tense, aspect, modality and evidentiality, this volume presents a variety of original contributions that are firmly empirically-grounded based on elicited or corpus data, while adopting different theoretical frameworks. Thus, some chapters rely on large diachronic corpora and provide new qualitative insight on the evolution of TAM systems through quantitative methods, while others carry out a collostructional analysis of past-tensed verbs using inferential statistics to explore the lexical grammar of verbs. A common goal is to uncover semantic regularities and variation in the TAM systems of the languages under study by taking a close look at context. Such a fine-grained approach contributes to our understanding of the TAM systems from a typological perspective. The focus on well-known Indo-European languages (e.g. French, German, English, Spanish) and also on less commonly studied languages (e.g. Hungarian, Estonian, Avar, Andi, Tagalog) provides a valuable cross-linguistic perspective.

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final chapters we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices; an application to the correspondence between Tate Harish-Chandra modules with complementary central charges is worked out; and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined.

We study the PBW-filtration on the highest weight representations V(λ) of the Lie algebras of type A n and C n . This filtration is induced by the standard degree filtration on . In previous papers, the authors studied the filtration and the associated graded algebras and modules over the complex numbers. The aim of this paper is to present a proof of the results which holds over the integers and hence makes the whole construction available over any field.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.