Open problems after Serre's embeddings of $\mathbf{PSL}_2(\mathbf{F}_p)$ into simple Lie groups: uniform prime-power lifting, conjugacy classes, $K$-forms over $\mathbf{Q}(\sqrt{p^*})$, and explicit integral constructions

Serre (Inventiones Mathematicae 124, 1996) proved that a connected semisimple adjoint algebraic group $G$ of Coxeter number $h$ over an algebraically closed field contains $\mathbf{PGL}_2(\mathbf{F}_p)$ when $h+1=p$ is prime and $\mathbf{PSL}_2(\mathbf{F}_p)$ when $2h+1=p$ is prime, by combining an integral model of Kostant's principal $\mathbf{SL}_2$-embedding with a $p$-adic lifting argument. The research program surrounding these embeddings, as recorded in Serre's own retrospective accounts (the Bourbaki exposé 864 and the 1998 Moursund lectures), leaves four linked open issues: (1) a uniform, computer-free construction for $q$ a nontrivial prime power, extending Kostant's prime-power conjecture; (2) the number of $G(\mathbf{C})$-conjugacy classes of the embedded $\mathbf{PSL}_2(\mathbf{F}_p)$-subgroups, known only in special cases; (3) the existence of a $K$-form of $G$ over $K=\mathbf{Q}(\sqrt{p^*})$, $p^*=(-1)^{(p-1)/2}p$, containing the lifted subgroup, and more generally the determination of the subfields over which the embeddings are defined; and (4) more explicit constructions of the lifts. The state of the literature is: existence for prime powers was completed case-by-case by 1993 (before the paper), so the surviving core there is a uniform method; conjugacy class counts are known only in special cases ($G_2$ with $L_2(13)$ and $E_8$ with $L_2(61)$ unique, three classes of $\mathbf{PGL}_2(31)$ in $E_8$, class data for $L_2(41)$ and $L_2(49)$); the $K$-form and rationality questions are resolved in the $G_2$ case only; and the explicit-construction program has progressed only in type $E_8$ (Kostant's construction from Springer's regular elements), while the integral lifts remain untouched. A precise nonempty open core remains.

Background

Let $k$ be an algebraically closed field and let $G$ be a connected semisimple algebraic group over $k$ which is adjoint, meaning that its center is trivial (for example $\mathbf{PGL}_n$ rather than $\mathbf{SL}_n$). Two families of numerical invariants of $G$ drive everything below. Choosing a maximal torus with root system $\Phi$ and Weyl group $W$, the Coxeter number $h$ is the order of a Coxeter element of $W$, that is, the product of all simple reflections with respect to a fixed base of $\Phi$; it is the largest of the degrees $d_1,\dots,d_r$ of $W$, i.e. the degrees of homogeneity of a set of algebraically independent homogeneous polynomial generators of the ring of $W$-invariant polynomial functions on the reflection representation, where $r$ is the rank. Writing $m_i=d_i-1$, the $m_i$ are the exponents of $W$.

Kostant's classical theory of the principal three-dimensional subgroup attaches to such a $G$ a homomorphism $\mathbf{SL}_2\to G$, unique up to conjugacy, characterized by the fact that, as a module for the image of $\mathbf{sl}_2$, the Lie algebra of $G$ decomposes as $\operatorname{Lie}G\simeq\bigoplus_{i=1}^r V(2m_i)$, where $V(n)$ denotes the irreducible $\mathbf{sl}_2$-module of dimension $n+1$. The finite groups entering the problem are $\mathbf{PSL}_2(\mathbf{F}_q)=\mathbf{SL}_2(\mathbf{F}_q)/\{\pm I\}$ and $\mathbf{PGL}_2(\mathbf{F}_q)=\mathbf{GL}_2(\mathbf{F}_q)/(\text{scalar matrices})$, the projective special and general linear groups over the finite field of $q$ elements.

Kostant conjectured in his 1983 AMS colloquium lecture that if $2h+1$ is a prime power then $G$ has a subgroup isomorphic to $\mathbf{PSL}_2(\mathbf{F}_{2h+1})$. For $G$ of classical type this follows from the character table, as Kostant noted. For the exceptional types it was established case-by-case by 1993: Meurman treated $G_2$ ($h=6$, subgroup $L_2(13)$, 1982); Cohen and Wales treated $F_4$ and $E_6$ ($h=12$, subgroup $L_2(25)$, which sits inside the Tits group ${}^2F_4(2)'$, the derived subgroup of the Ree group of type ${}^2F_4$ over $\mathbf{F}_2$, itself contained in $E_6$ and in a conjugate of $F_4$); Kleidman and Ryba treated $E_7$ ($h=18$, subgroup $L_2(37)$, 1993); and Cohen, Griess and Lisser treated $E_8$ ($h=30$, subgroup $L_2(61)$, 1993), completing the conjecture. These exceptional-case proofs relied on substantial computer calculations, and Cohen, Griess and Lisser moreover proved that $E_8(\mathbf{C})$ has a unique conjugacy class of $L_2(61)$-subgroups; before that, uniqueness was known only in the $G_2$ case.

Serre's 1996 Inventiones paper gave the prime cases a uniform, computer-free proof: if $h+1=p$ is prime then $G(k)$ contains $\mathbf{PGL}_2(\mathbf{F}_p)$ (apart from a small exceptional case in characteristic 2 when $h=2$, that is type $A_1$), and if $2h+1=p$ is prime then $G(k)$ contains $\mathbf{PSL}_2(\mathbf{F}_p)$. The method runs as follows. The principal homomorphism is defined over $\mathbf{Z}$, hence over $\mathbf{Z}_{(p)}$, the integers localized at $p$ (all primes different from $p$ inverted); reducing modulo $p$ gives a subgroup of the corresponding group over $\overline{\mathbf{F}}_p$, which is then lifted $p$-adically back to characteristic zero. For $p=h+1$ the lifting rests on the vanishing of a second cohomology group $H^2(A,\operatorname{Lie}G)$ for the relevant finite subgroup $A$; for $p=2h+1$ one must work over the ring $\mathbf{Z}_p[\sqrt{p^*}]$, where $p^*=(-1)^{(p-1)/2}p$, so that $\mathbf{Q}(\sqrt{p^*})$ is the unique quadratic subfield of the cyclotomic field $\mathbf{Q}(\zeta_p)$, where $\zeta_p$ denotes a primitive $p$-th root of unity: the conjugacy classes of the elements of the lifted subgroup are rational precisely over this field. The argument for $p=2h+1$ passes through a deformation scheme parametrizing lifts of a Sylow $p$-subgroup, lifts of Borel subgroups (maximal connected solvable subgroups), an averaging (fixed-point) argument, and the cohomology of a strongly $p$-embedded subgroup of the finite group (a proper subgroup containing a Sylow $p$-subgroup whose intersection with each conjugate outside it has order prime to $p$); the transfer to arbitrary positive characteristic uses the Bruhat-Tits building of the group over a local field (the contractible polyhedral complex, equipped with an action of the group, whose simplices correspond to the parahoric subgroups over that field).

Serre's later Bourbaki exposé (1999) summarizes the state of the art by a criterion (proposition 2.1.1 there, due to several authors): a homomorphism that is non-degenerate, meaning that its kernel is contained in the centre of $\mathbf{SL}_2(\mathbf{F}_q)$, and of toral type, meaning that the image of its unipotent upper-triangular subgroup (a Sylow $p$-subgroup, writing $q=p^e$) is contained in a maximal torus of $G$, exists if and only if $q-1$ divides one of the integers $2d_i$. The sufficiency direction for the exceptional types rests on the computer-assisted work just cited, and Serre remarks there that computer calculations have the drawback of not being step-by-step verifiable the way a proof must be, while conceding that in some cases they established uniqueness of the embedding up to conjugacy. The same exposé lists what remains open: a general (uniform, computer-free) proof of the criterion, describing all embeddings up to conjugacy (done in some cases, not all), the rationality of the embeddings over subfields of $k$ (settled for $G_2$ and $L_2(13)$, where the relevant form of $G_2$ is the non-split $\mathbf{Q}$-form attached to the usual octonions, and $L_2(13)$ embeds in the split form over $k'$ if and only if $k'$ contains $\sqrt{13}$ and $-1$ is a sum of four squares in $k'$), and the desirability of more explicit and more uniform constructions. These are the four linked open issues recorded here.

A Chevalley group is the split group attached to the root system with a fixed pinning, constructed over $\mathbf{Z}$; in it the Steinberg representatives are canonical lifts of the elements of the Weyl group to the normalizer of a maximal torus.

Problem Statement

Let $G$ be a connected semisimple adjoint algebraic group over an algebraically closed field, with Coxeter number $h$, exponents $m_1,\dots,m_r$ and degrees $d_i=m_i+1$. Serre (1996) proved that $\mathbf{PGL}_2(\mathbf{F}_p)\subset G(k)$ when $p=h+1$ is prime and $\mathbf{PSL}_2(\mathbf{F}_p)\subset G(k)$ when $p=2h+1$ is prime. The research program surrounding these embeddings, as recorded in Serre's own retrospective accounts (Bourbaki exposé 864, §2.3 and §2.5; 1998 Moursund lectures), leaves four linked open questions, whose common objective is to complete the arithmetic and structural theory of these principal $\mathbf{L}_2$-embeddings:

(1) Uniform prime-power lifting. Existence of subgroups $\mathbf{PSL}_2(\mathbf{F}_q)\subset G(k)$ for $q=2h+1$ a prime power is known (Kostant's conjecture, completed case-by-case by 1993, computer-assisted in the exceptional types, by character tables in the classical types). Give a uniform, computer-free construction of these subgroups, and a uniform proof of the sufficiency of Serre's divisibility criterion: a homomorphism that is non-degenerate (kernel contained in the centre of $\mathbf{SL}_2(\mathbf{F}_q)$) and of toral type (the image of a Sylow $p$-subgroup, $q=p^e$, is contained in a maximal torus of $G$) $\mathbf{SL}_2(\mathbf{F}_q)\to G$ exists if and only if $q-1$ divides some $2d_i$ (stated for $G$ almost simple; the semisimple case follows componentwise).

(2) Conjugacy classes. For every $G$ with $p=2h+1$ prime, determine the number of $G(\mathbf{C})$-conjugacy classes of subgroups isomorphic to $\mathbf{PSL}_2(\mathbf{F}_p)$; this all-$G$ determination is the required target, and the cases $E_7$ ($p=37$) and the classical groups are its still-unsettled instances rather than the full scope (a single class is proven for $G_2$, $p=13$, and $E_8$, $p=61$). In addition, for every $G$ with $q=2h+1$ a non-prime prime power, determine the analogous counts; the cases $L_2(25)\subset F_4(\mathbf{C})$ and $E_6(\mathbf{C})$ are instances of this class.

(3) $K$-forms and rationality. For $p=2h+1$ prime, the conjugacy classes of the elements of the lifted subgroup $A\simeq\mathbf{PSL}_2(\mathbf{F}_p)$ are rational over the quadratic field $K=\mathbf{Q}(\sqrt{p^*})$, $p^*=(-1)^{(p-1)/2}p$. Does there exist a $K$-form $G'$ of $G$ (an algebraic group over $K$ that becomes isomorphic to $G$ over the algebraic closure) such that $A\subset G'(K)$? More generally, in the formulation of Serre's Bourbaki exposé (§2.5.3), determine the subfields over which the embeddings are defined: for which subfields $k'$ of the ground field is $A$ embeddable in the split form over $k'$? This question is closely tied to the conjugacy-class question, since the torsor of embeddings descends to the ground field only when uniqueness up to conjugacy is available. Both are resolved for $G_2$ ($p=13$), where the relevant form is the non-split $\mathbf{Q}$-form of $G_2$ and $L_2(13)$ embeds in the split form over $k'$ if and only if $k'$ contains $\sqrt{13}$ and $-1$ is a sum of four squares in $k'$, and open for $E_7$ ($p=37$), $E_8$ ($p=61$) and in general.

(4) Explicit constructions. Replace the deformation-theoretic existence argument for the prime cases $p=h+1$ and $p=2h+1$ by explicit constructions of the integral lifts: for each simple type for which $h+1$ or $2h+1$ is prime, exhibit the lift explicitly, integrally, and uniformly (one description covering all such types, not a case-by-case or computer-assisted argument). This part is additional to, and not implied by, (1): (1) asks for a uniform construction of the subgroups for the values $q=2h+1$ that are prime powers, together with a uniform proof of the sufficiency of the divisibility criterion for arbitrary prime powers $q=p^e$, whereas (4) asks for the explicit integral lifts themselves.

Parts (1)-(4) are independent open problems of different kinds.

References

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