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Abstract This paper studies the "reduction mod p " method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group U ζ at a p r -th root of unity. The G -modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between Ext G n (V , W) and Ext U ζ n (V ′ , W ′) , when V , W are obtained from V ′ , W ′ by reduction mod p. Since the dimensions of Ext n -spaces for U ζ -modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G -modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p -th root of unity case and a general p r -th root of unity case are essentially different. [ABSTRACT FROM AUTHOR]

Abstract We show that for every conjugacy class O in a connected semisimple algebraic group G over an algebraically closed field of characteristic good for G one can find a special transversal slice Σ to the set of conjugacy classes in G such that O intersects Σ and dim O = codim Σ. [ABSTRACT FROM AUTHOR]

We introduce a class of strongly étale difference algebras , whose role in the study of difference equations is analogous to the role of étale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's decomposition theorem and we present applications to difference algebraic groups and the compatibility problem. [ABSTRACT FROM AUTHOR]

Let Q be a connected algebraic group with Lie algebra q . Symmetric invariants of q , i.e., the Q -invariants in the symmetric algebra S ( q ) of q , is a first approximation to the understanding of the coadjoint action ( Q : q ⁎ ) and coadjoint Q -orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If G is a semisimple group with Lie algebra g and V is G -module, then we define q to be the semi-direct product of g and V . Then we are interested in the case, where the generic isotropy group for the G -action on V is reductive and commutative. It turns out that in this case symmetric invariants of q can be constructed via certain G -equivariant maps from g to V (“covariants”). [ABSTRACT FROM AUTHOR]

Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R . Let H be a simply connected semisimple R -subgroup of G . We consider the homogeneous space X = G / H . We ask: how many connected components has X ( R ) ? We give a method of answering this question. Our method is based on our solutions of generalized Reeder puzzles. [ABSTRACT FROM AUTHOR]

Abstract: We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra in characteristic 3 and in the dual of exceptional Lie algebra in characteristic 2 are finite. We determine the closure relation among nilpotent coadjoint orbits in the dual of Lie algebras of type in characteristic 2 and in the dual of Lie algebra of type in characteristic 3. In each case we give an explicit description of the nilpotent pieces in the dual defined in [2], which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case-by-case definition in [11,26] in the case of classical groups and have nice properties independent of the characteristic of the base field. This completes the classification of nilpotent coadjoint orbits in the dual of Lie algebras of reductive algebraic groups and the determination of closure relation among such orbits in all characteristic. [Copyright &y& Elsevier]

Abstract: The purpose of this short note is to establish the existence of ∂-parameterized Picard–Vessiot extensions for systems of linear difference-differential equations over difference-differential fields with an algebraically closed field of constants. [Copyright &y& Elsevier]

Abstract: For a simply-connected complex algebraic group G of type , , or , we prove (see C. Geiss et al. (2008) , Conjecture 19.2) a new family of total positivity criteria for partial flag varieties , where P is a parabolic subgroup of G. [Copyright &y& Elsevier]

Abstract: Let G be a simple algebraic group. Associated with the finite-dimensional rational representation of G there is the monoid and the projective -embedding . One can identify the cases where is rationally smooth; and in such cases it is desirable to calculate the H-polynomial, H, of . In this paper we consider the situation where ρ is irreducible. We then determine H explicitly in terms of combinatorial invariants of ρ. Indeed, there is a canonical cellular decomposition for . These cells are defined in terms of idempotents, -orbits and other natural quantities obtained from . Furthermore, H is obtained by recording the dimension of each of these cells in terms of the descent system of . As a special case we reacquire the well-known formula for the Poincaré polynomial of a “wonderful embedding” of a simple algebraic group of adjoint type. [Copyright &y& Elsevier]

Abstract: We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. In this paper we give a classification of all triples where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic , G is a closed disconnected positive dimensional subgroup of Y and V is a nontrivial rational irreducible KY-module on which G acts irreducibly. [Copyright &y& Elsevier]

Abstract: We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our treatment of the problem is via differential Hopf algebras and Deligne''s fibre functor construction [P. Deligne, Catégories tannakiennes, in: The Grothendieck Festschrift, vol. II, in: Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195]. [Copyright &y& Elsevier]

Abstract: There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon–Hochschild–Serre spectral sequence of a split extension of an integral lattice L by a group G. These characteristic classes exist in the rth page of the spectral sequence provided that the differentials for all . When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands. [Copyright &y& Elsevier]

Abstract: A recollement of triangulated categories describes one such category as being “glued together” from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a differential graded algebra in terms of two other such categories. [Copyright &y& Elsevier]

Let G be a semi-simple algebraic group and let H be a spherical subgroup. The ground field k is algebraically closed and of characteristic zero. This article is concerned with projective embeddings Y of spherical homogeneous spaces G/H. Our approach in the study of such a variety Y is to realize them as quotients under the action of H of projective embeddings of G. First, we give a more precise sense to this project by defining the quotient of a G-variety by a spherical subgroup H. Then, we give a condition, in terms of G-invariant valuations, under which Y can be obtained by quotient of an embedding of G. Finally, if the index of H in its normalizer is finite, we show that an important class of embeddings of G/H (toroidal and liftable) geometric quotients of embeddings of G. [Copyright &y& Elsevier]