We prove that Ad-semisimple conjugacy classes in a connected Lie group $G$ are closed embedded submanifolds of $G$. We also prove that if $α:H\to G$ is a homomorphism of connected Lie groups such that the kernel of $α$ is discrete in $H$, then for an Ad-semisimple conjugacy class $C$ in $G$, every connected component of $α^{-1}(C)$ is a conjugacy class in $H$. Corresponding results for adjoint orbits in real Lie algebras are also proved.
We determine the conjugacy classes of semisimple elements in the symplectic groups ${\rm Sp}(2m,F)$, where $F$ is an arbitrary field of characteristic not $2$. This note was originally a letter dated 23 March, 2006, from G.E. Wall to Cheryl E. Praeger, and has been reproduced with his kind permission.
Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $Γ$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely, that there are infinitely $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. Also, we show that any lattice in $G$ has the $R_\infty$-property, extending our earlier result for irreducible lattices.
Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special orthogonal groups the results are new.
Given a group automorphism $ϕ:Γ\to Γ$, one has an action of $Γ$ on itself by $ϕ$-twisted conjugacy, namely, $g.x=gxϕ(g^{-1})$. The orbits of this action are called $ϕ$-conjugacy classes. One says that $Γ$ has the $R_\infty$-property if there are infinitely many $ϕ$-conjugacy classes for every automorphism $ϕ$ of $Γ$. In this paper we show that any irreducible lattice in a connected semi simple Lie group having finite centre and rank at least 2 has the $R_\infty$-property.
The existence of closed orbits of real algebraic groups on real algebraic varieties is established. As an application, it is shown that if G is a real reductive linear group with Iwasawa decomposition G= KAN, then every unipotent subgroup of G is conjugate to a subgroup of N.
Let $G$ be an algebraic group over a field $k$. We call $g\in G(k)$ {\bf real} if $g$ is conjugate to $g^{-1}$ in $G(k)$. In this paper we study reality for groups of type $G_2$ over fields of characteristic different from 2. Let $G$ be such a group over $k$. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in $G(k)$ is real if and only if it is a product of two involutions in $G(k)$. Every unipotent element in $G(k)$ is a product of two involutions in $G(k)$. We discuss reality for $G_2$ over special fields and construct examples to show that re…
We study quasi-semisimple elements of disconnected reductive algebraic groups over an algebraically closed field. We describe their centralizers, define isolated and quasi-isolated quasi-semisimple elements and classify their conjugacy classes.
Random walk on the chambers of hyperplanes arrangements is used to define a family of card shuffling measures $H_{W,x}$ for a finite Coxeter group W and real $x \neq 0$. By algebraic group theory, there is a map from the semisimple orbits of the adjoint action of a finite group of Lie type on its Lie algebra to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the Weyl group. For types A, B, and the identity conjugacy class of W for all types, it is proved that for q very good, this me…
Let $p$ be an odd prime, $m\in {\mathbb N}$ and set $q=p^m$, $G=\operatorname{PSL}_n(q)$. Let $θ$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and $\operatorname{Fr}_q$ be the Frobenius endomorphism of $\operatorname{PSL}_n(\overline{{\mathbb F}_q})$. We show that every $(d\circ θ)$-conjugacy class of a $(d\circ θ,p)$-regular element in $G$ is represented in some $\operatorname{Fr}_q$-stable maximal torus and that most of them are of type D. We write out the possible exceptions and show that, in particular, if $n\geq5$ is either odd or a multiple of $4$ and $q>7$, th…
Let $G$ be an algebraic group of type $G_2$ over a field $k$ of characteristic $\neq 2,3$. In this paper we calculate centralizers of semisimple elements in anisotropic $G_2$. Using these, we show explicitly that there are six conjugacy classes of centralizers in the compact real form of $G_2$.
Solomon's descent algebra is used to define a family of signed measures M(W,x) for a finite Coxeter group W and non-zero x. The measures corresponding to W of types A and B are known to arise from the theory of card shuffling and to be related to the Poincare-Birkhoff-Witt theorem and splittings of Hochschild homology. Formulas for these measures are obtained in special cases. The eigenvalues of the associated Markov chains are computed. By elementary algebraic group theory, choosing a random semisimple orbit on a Lie algebra corresponding to a finite group of Lie type G^F induces a measure on…
For a simple algebraic group $G$ over an algebraically closed field, we study products of normal subsets. For this we mark the nodes of the Dynkin diagram of $G$. We use two types of labels, a binary marking and a labeling with non-negative integers. The first is used to recognize large conjugacy classes which appear in a product of two conjugacy classes while the second is used to keep track of multiplicities of regular diagrams. In particular, we formulate sufficient conditions in terms of marked diagrams, for a product of normal subsets in $G$ to contain regular semisimple elements.
Let a split element of a connected semisimple Lie group act on one of its flag manifolds. We prove that each connected set of fixed points of this action is itself a flag manifold. With this we can obtain the generalized Bruhat decomposition of a semisimple Lie group by entirely dynamical arguments.
Let $G$ be an algebraic group defined over a field $k$. We call $g\in G$ {\bf real} if $g$ is conjugate to $g^{-1}$ and $g\in G(k)$ as {\bf $k$-real} if $g$ is real in $G(k)$. An element $g\in G$ is {\bf strongly real} if $\exists h\in G$, $h^{2}=1$ (i.e. $h$ is an {\bf involution}) such that $hgh^{-1}=g^{-1}$. Clearly, strongly real elements are real and are product of two involutions. Let $G$ be a connected adjoint semisimple group over a perfect field $k$, with -1 in the Weyl group. We prove that any strongly regular $k$-real element in $G(k)$ is strongly $k$-real (i.e. is a product of two …
Let G be a simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.
First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every $x$ in $L = \operatorname{Lie}(G)$ can be written as $x =[a, b]$ for some $a$, $b$ in $L$. By using the same method, we give a new proof of the following theorem (thus avoiding the classification tables of fundamental weights): in compact semisimple Lie algebras, orthogonal Cartan subalgebras always exist (where one of them can be chosen arbitrarily). Some of the consequences of this theorem are the following. $(i)$ If $L=\operatorname{Lie}(G)$ is such a Li…
For any noncompact semisimple real Lie group $G$, we construct a group of affine transformations of its Lie algebra $\mathfrak{g}$ whose linear part is Zariski-dense in $\operatorname{Ad} G$ and which is free, nonabelian and acts properly discontinuously on $\mathfrak{g}$.
We show that the sheets for a connected reductive algebraic group G over an algebraically closed field in good characteristic acting on itself by conjugation are in bijection with G-conjugacy classes of triples (M, Z(M)^\circ t, O) where M is the connected centralizer of a semisimple element in G, Z(M)^\circ t is a suitable coset in Z(M)/Z(M)^\circ and O is a rigid unipotent conjugacy class in M. Any semisimple element is contained in a unique sheet S and S corresponds to a triple with O={1}. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples fo…
It is well-known that an element of the linear group ${\rm GL}_n(\C)$ is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms.
Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $p \geq 0$. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements $a, a' \in G$ are conjugate in $G$ if and only if $f(a)$ and $f(a')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$. Steinberg showed that the conjecture holds if $a$ and $a'$ are semisimple, and also proved the conjecture when $p = 0$. In this paper, we give a counterexample to Steinberg's conjecture. S…
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
The purpose of this paper is to consider when two maximal subalgebras of a finite-dimensional solvable Lie algebra $L$ are conjugate, and to investigate their intersection.
We give a short elementary proof of the fact that connected semisimple real Lie groups satisfy property RD. The proof is based on a process of linearization.
Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group $G$, each connected component of the set $E_n(G)$ of all elements of order $n$ in $G$ is a conjugacy class in $G$. In particular, all conjugacy classes of finite order in $G$ are closed. Some properties of connected components of $E_n(G)$ are also given.
In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conje…
The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of card shuffling.
We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n,q), any finite-dimensional pointed Hopf algebra H with G(H) = PSL(n,q) or SL(n,q) is isomorphic to a group algebra.
Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.
Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra $\mathfrak{n}$ that preserves its restricted root space decomposition is of the form $\text{ad}( W)$, where $W \in \mathfrak{m}\oplus .
By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type $A$ are found. This leads to new models of card shuffling and has interesting combinatorial and numb…
We study locally compact group topologies on semisimple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $σ$-compact group $Γ$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.
We compute L2-invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel-Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for Novikov-Shubin numbers and vanishing L2-torsion for lattices in groups with even deficiency. We discuss applications to Gromov's Zero-in-the-Spectrum Conjecture as well as to a proportionality conjecture for the L2-torsion of measure equivalent groups.
We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov's question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a conseque…
We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a rack collapses.
We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $Σ$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify the $σ_{mod}$-regularity for convex projective structures and positive representations. Then we show that the holonomies of convex projective structures and positive representations on $Σ$ are all primitive stable if $Σ$ has one boundary component.
We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.
We prove that if $Γ$ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of $Γ$ equals its proper geometric dimension.
We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.
The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras --extended affine Lie algebras-- that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schem…
The set $\mathcal S(G)$ of compact subgroups of a Hausdorff topological group $G$ can be equipped with the Vietoris topology. A compact subgroup $K\in\mathcal S(G)$ is isolated up to conjugacy if there is a neighborhood $\mathcal U\subseteq\mathcal S(G)$ of $K$ such that every $L\in\mathcal U$ is conjugate to $K$. In this paper, we characterize compact subgroups of a Lie group that are isolated up to conjugacy. Our characterization depends only on the intrinsic structure of $K$, the ambient Lie group $G$ and the embedding of $K$ into $G$ are irrelevant. In addition, we prove that any continuou…
We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.
