Finiteness of z-classes in reductive groups

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.

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.

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

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive group of type C_2 or G_2.

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 first characterization of the isomorphism classes of k -involutions for any reductive algebraic group defined over a perfect field was given in On the Classification of k -involutions I using 3 invariants.In Involutions of SL(n, k) , (n > 2) a full classification of all k -involutions on SL(n, k) for k algebraically closed, the real numbers, the p -adic numbers or a finite field was provided.In Isomorphy Classes of Involutions of SP(2n, k) , (n > 2) , a similar classification was given for all k -involutions of SP(2n, k) .In this paper, we find analogous results to develop a detailed characterization of the k -involutions of SO(n, k, ) , where is any non-degenerate symmetric bilinear form and k is any field not of characteristic 2 .We use these results to characterize the isomorphy classes of k -involutions of SO(n, k, ) for all bilinear forms, when char(k) = 2, 3 , and for some bilinear forms when char(k) = 3 .When n = 3, 4, 6, 8 , then the characterization considers all involutions.If n = 3, 4, 6, 8 , then the characterization only considers inner involutions.

A first characterization of the isomorphism classes of \(k\)-involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of \(\operatorname{SL}(n, k)\) with \(k\) any field of characteristic not equal to \(2\). We classify the isomorphism classes of involutions for \(k\) algebraically closed, the real numbers, the \(\mathfrak{p}\)-adic numbers and finite fields. We also determine in which cases the corresponding fixed point group \(H\) is \(k\)-anisotropic. In those cases the corresponding symmetric \(k\)-variety consists of semisimple elements.

On the theory of reductive algebraic groups over a perfect field By Ichiro SATAKE

Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with an appendix by Jessica Fintzen)

Let X X be a complete smooth variety defined over a number field K K and let i i be an integer. The absolute Galois group G a l K \mathrm {Gal}_K of K K acts on the i i th étale cohomology group H e ´ t i ( X K ¯ , Q ℓ ) H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell ) for all primes ℓ \ell , producing a system of ℓ \ell -adic representations { Φ ℓ } ℓ \{\Phi _\ell \}_\ell . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ \Phi _\ell admits a reductive Q \mathbb {Q} -form that is independent of ℓ \ell if X X is projective. Denote by Γ ℓ \Gamma _\ell and G ℓ \mathbf {G}_\ell respectively the monodromy group and the algebraic monodromy group of Φ ℓ s s \Phi _\ell ^{\mathrm {ss}} , the semisimplification of Φ ℓ \Phi _\ell . Assuming that G ℓ 0 \mathbf {G}_{\ell _0} satisfies some group theoretic conditions for some prime ℓ 0 \ell _0 , we construct a connected quasi-split Q \mathbb {Q} -reductive group G Q \mathbf {G}_{\mathbb {Q}} which is a common Q \mathbb {Q} -form of G ℓ ∘ \mathbf {G}_\ell ^\circ for all sufficiently large ℓ \ell . Let G Q s c \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}} be the universal cover of the derived group of G Q \mathbf {G}_{\mathbb {Q}} . As an application, we prove that the monodromy group Γ ℓ \Gamma _\ell is big in the sense that Γ ℓ s c ≅ G Q s c ( Z ℓ ) \Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell ) for all sufficiently large ℓ \ell .

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Letkbe a field of characteristic zero, letGbe a connected reductive algebraic group overkand let 𝔤 be its Lie algebra. Letk(G), respectively,k(𝔤), be the field ofk-rational functions onG, respectively, 𝔤. The conjugation action of Gon itself induces the adjoint action of Gon 𝔤. We investigate the question whether or not the field extensionsk(G)/k(G)Gandk(𝔤)/k(𝔤)Gare purely transcendental. We show that the answer is the same fork(G)/k(G)Gandk(𝔤)/k(𝔤)G, and reduce the problem to the case whereGis simple. For simple groups we show that the answer is positive ifGis split of typeAnorCn, and negative for groups of other types, except possiblyG2. 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 Gon itself.

Restriction theorems for principal bundles in arbitrary characteristic

Let K1 and K2 be extension fields over a field K with char K = p > 0. Assume L = K1(x1) = K2(x2) ⊃ K where xi is transcendental over Ki, for i = 1, 2. In this paper we prove that if K1 is a perfect field, then K1 = K2.

In this paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear algebraic groups over some fields of zero characteristic possess the R ∞ ${R_{\infty}}$ and S ∞ ${S_{\infty}}$ properties.