This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equations. Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions.
For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.
We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small …
We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Psi from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.
Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.
Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$ encoding, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. We show that the domains of convergence and meromorphic continuation of these zeta functions of groups $\mathbf{G}(\mathcal{O})$ are independent of the number fi…
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.
This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class 2 generalising the Heisenberg group of three by three unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral …
Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$, the fixed point subgroup of $F$. Let $\bfP$ be an $F$-stable parabolic subgroup of $\bfG$ and let $\bfU$ be the unipotent radical of $\bfP$; set $P = \bfP^F$ and $U = \bfU^F$. Let $G_\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\uni$ is given by a polynomial in $q$ with integer coefficien…
Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients.
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…
Let $G$ be a connected reductive algebraic group defined over the finite field $\FF_q$, where $q$ is a power of a good prime for $G$. We write $F$ for the Frobenius morphism of $G$ corresponding to the $\FF_q$-structure, so that $G^F$ is a finite group of Lie type. Let $P$ be an $F$-stable parabolic subgroup of $G$ and $U$ the unipotent radical of $P$. In this paper, we prove that the number of $U^F$-conjugacy classes in $G^F$ is given by a polynomial in $q$, under the assumption that the centre of $G$ is connected. This answers a question of J. Alperin in \cite{alperin}. In order to prove the…
We extend the notion of induced conjugacy classes in reductive groups, introduced by Lusztig and Spaltenstein for unipotent classes, to arbitrary classes. We study properties of equivariant fibrations of prehomogeneous affine spaces, especially the existence of relative invariants. We also detect prehomogeneous affine spaces as subquotients of canonical parabolic subgroups attached to elements of reductive groups in the sense of Jacobson-Morozov. These results are prerequisites for making the geometric expansion of the Arthur-Selberg trace formula more explicit.
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module $W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 + \varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$ has highest weight $\varpi_2$, we determine for every unipotent element $u \in G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As…
Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and $H_1/R_u(H_1)$ and $H_2/R_u(H_2)$ are semisimple, then $H_1$ and $H_2$ are $G$-conjugate. Moreover, we show that if $H$ is a semisimple linear algebraic group with maximal unipotent subgroup $U$ then for any algebraic group homomorphism $σ\colon U\rightarrow G$, there are only finitely many $G$-conjugacy classes of algebraic group homomorphisms $ρ\colon H\rightarr…
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.
Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checki…
When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.
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…
Markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic. As a corollary of these methods one obtains a probabilistic proof of Steinberg's count of unipotent matrices and generalizations of formulas of Rudvalis and Shinoda.
Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p \geqslant 0$ which is not algebraic over a finite field. Let $\mathcal{C}_1, \ldots, \mathcal{C}_t$ be non-central conjugacy classes in $G$. In earlier work with Gerhardt and Guralnick, we proved that if $t \geqslant 5$ (or $t \geqslant 4$ if $G = G_2$), then there exist elements $x_i \in \mathcal{C}_i$ such that $\langle x_1, \ldots, x_t \rangle$ is Zariski dense in $G$. Moreover, this bound on $t$ is best possible. Here we establish a more refined version of this result in the spec…
Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…
Describing the conjugacy classes of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ uniformly (for all or many values of $n$ and $q$) is a nearly impossible task. This paper takes on the related problem of describing the normal subgroups of $\mathrm{UT}_{n}(\mathbb{F}_{q})$. For $q$ a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from $\mathbb{F}_{q}^{\times}$. Each pair comprises a loopless binary matroid and a tight splice, an apparently new kind of combinatorial object which interpolates between nonnesti…
Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which h…
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…
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…
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 …
We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.
We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of $\mathbf{F}_q$-points of the groups under consideration depend polynomially on $q$. Our approach combines group theory, graph theory, toric geometry, and $p$-adic integration. Our uniformity results are in line …
In 1965, Steinberg's study of conjugacy classes in connected reductive groups led him to construct an affine subspace parametrising regular conjugacy classes, which he noticed is also a cross section for the conjugation action by the unipotent radical of a Borel subgroup on another affine subspace. Recently, generalisations of this slice and its cross section property have been obtained by Sevostyanov in the context of quantum group analogues of W-algebras and by He-Lusztig in the context of Deligne-Lusztig varieties. Such slices are often thought of as group analogues of Slodowy slices. In th…
Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very go…
Define a module representation to be a linear parameterisation of a collection of module homomorphisms over a ring. Generalising work of Knuth, we define duality functors indexed by the elements of the symmetric group of degree three between categories of module representations. We show that these functors have tame effects on average sizes of kernels. This provides a general framework for and a generalisation of duality phenomena previously observed in work of O'Brien and Voll and in the predecessor of the present article. We discuss applications to class numbers and conjugacy class zeta func…
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. We give in this paper some sufficient conditions for the existence and the conjugacy of Levi factors of G. Let 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 …
