The R ∞ and S ∞ properties for linear algebraic groups

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

Reductive subgroups of reductive groups in nonzero characteristic

We prove a uniform upper estimate on the number of cuspidal eigenvalues of the Γ-automorphic Laplacian below a given bound when Γ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each Γ in the family is assumed to contain a principal congruence subgroup whose index in Γ does not exceed a fixed number. The bound we prove depends linearly on the covolume of Γ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice Γ.

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

The six operations in equivariant motivic homotopy theory

Reduction theorems for groups of matrices

Let G → GL ( V ) be a representation of a reductive linear algebraic group G on a finite-dimensional vector space V , defined over an algebraically closed field of characteristic zero. The categorical quotient X = V // G carries a natural stratification, due to D. Luna. This paper addresses the following questions: (i) Is the Luna stratification of X intrinsic? That is, does every automorphism of V // G map each stratum to another stratum? (ii) Are the individual Luna strata in X intrinsic? That is, does every automorphism of V // G map each stratum to itself? In general, the Luna stratification is not intrinsic. Nevertheless, we give positive answers to questions (i) and (ii) for interesting families of representations.

Let X be a smooth irreducible projective curve. In this note, we generalize the main result of Pauly and Peon-Nieto (Geometriae Dedicata 1–6, 2018) to principal G-bundles for any reductive linear algebraic group G. After defining very stability of principal G-bundles, we show that this definition is equivalent to the fact that the Hitchin fibration restricted to the space of Higgs fields on that principal bundle is finite. We also study the relation between very stability and other stability conditions in the case of $$\text {SL}_2$$-bundles.

Let be a rational finite dimensional complex representation of a reductive linear algebraic group G, and let be a system of generators of the algebra of invariant poly- nomials ℂ[V]G. We study the problem of lifting mappings over the mapping of invariants . Note that ¾(V) can be identified with the cate- gorical quotientV//G and its points correspond bijectively to the closed orbits inV. We prove that if f belongs to a quasianalytic subclass satisfying some mild closedness properties that guarantee resolution of singularities in 𝒞 e.g., the real analytic class, then f admits a lift of the same class 𝒞 after desingularization by local blow-ups and local power substitutions. As a consequence we show that f itself allows for a lift that belongs to SBVloc, i.e., special functions of bounded variation. If ρ is a real representation of a compact Lie group, we obtain stronger versions.

The Franke filtration is a finite filtration of the space of automorphic forms on a connected reductive linear algebraic group defined over an algebraic number field. The main feature of the filtration is that its quotients can be described in terms of parabolically induced representations using the main values of derivatives of Eisenstein series and the residues of these. The goal of this paper is to provide a complete explicit description of the Franke filtration of the space of automorphic forms on the symplectic group of rank two. The approach is different from the original approach of Franke, and takes into account the full cuspidal support of automorphic forms, that is, the cuspidal automorphic representation from which the Eisenstein series is built and the evaluation point at which it is evaluated. This considerably simplifies the exposition and makes it possible to obtain very explicit results and reveal the phenomena present in the filtration. The considered low rank case exhibits many of the properties and phenomena present in the cases of arbitrary rank. The explicit description of the Franke filtration in this case has important implications and applications in cohomology of congruence subgroups related to the Hilbert–Siegel modular forms of degree two, the Hilbert–Siegel modular variety of degree two and the corresponding Shimura variety.

In this paper we determine the stringy motivic volume of log terminal horospherical G-varieties of complexity one, where G is a connected reductive linear algebraic group. The stringy motivic volume of a log terminal variety is an invariant of singularities which was introduced by Batyrev and plays an important role in mirror symmetry for Calabi-Yau varieties. A horospherical G-variety of complexity one is a normal G-variety which is equivariantly birational to a product C×G/H, where C is a smooth projective curve and the closed subgroup H contains a maximal unipotent subgroup of G. The simplest example of such a variety is a normal surface with a non-trivial ?*-action. Our formula extends the results of Batyrev-Moreau [BM13] on stringy invariants of horospherical embeddings. The proof involves the study of the arc space of a horospherical variety of complexity one and a combinatorial description of its orbits. In contrast to [BM13], the number of orbits is no longer countable, which adds significant difficulties to the problem. As a corollary of our main theorem, we obtain a smoothness criterion using a comparison of the stringy and usual Euler characteristics.

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

Restriction theorems for principal bundles in arbitrary characteristic