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We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over
$\mathbb{Q}$
. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over
$\mathbb{Q}_{p}(x)$
.
Let
$p$
be an odd prime. We construct a
$p$
-group
$P$
of nilpotency class two, rank seven and exponent
$p$
, such that
$\text{Aut}(P)$
induces
$N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$
on the Frattini quotient
$P/\unicode[STIX]{x1D6F7}(P)$
. The constructed group
$P$
is the smallest
$p$
-group with these properties, having order
$p^{14}$
, and when
$p=3$
our construction gives two nonisomorphic
$p$
-groups. To show that
$P$
satisfies the specified properties, we study the action of
$G_{2}(q)$
on the octonion algebra over
$\mathbb{F}_{q}$
, for each power
$q$
of
$p$
, and explore the reducibility of the exterior square of each irreducible seven-dimensional
$\mathbb{F}_{q}[G_{2}(q)]$
-module.
Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If
$G= {GL }_n$
, then there is a degeneration process for obtaining from H a completely reducible subgroup
$H'$
of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup
$H'$
of G, unique up to
$G(k)$
-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for
$G= GL _n$
and with Serre’s ‘G-analogue’ of semisimplification for subgroups of
$G(k)$
from [19]). We also show that under some extra hypotheses, one can pick
$H'$
in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.
We study a relative variant of Serre’s notion of
$G$
-complete reducibility for a reductive algebraic group
$G$
. We let
$K$
be a reductive subgroup of
$G$
, and consider subgroups of
$G$
that normalize the identity component
$K^{\circ }$
. We show that such a subgroup is relatively
$G$
-completely reducible with respect to
$K$
if and only if its image in the automorphism group of
$K^{\circ }$
is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of
$G$
, as well as ‘rational’ versions over nonalgebraically closed fields.
For any prime number
$p$
and field
$k$
, we characterize the
$p$
-retract rationality of an algebraic
$k$
-torus in terms of its character lattice. We show that a
$k$
-torus is retract rational if and only if it is
$p$
-retract rational for every prime
$p$
, and that the Noether problem for retract rationality for a group of multiplicative type
$G$
has an affirmative answer for
$G$
if and only if the Noether problem for
$p$
-retract rationality for
$G$
has a positive answer for all
$p$
. For every finite set of primes
$S$
we give examples of tori that are
$p$
-retract rational if and only if
$p\notin S$
.
Order three elements in the exceptional groups of type
${{G}_{2}}$
are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.
Over an algebraically closed field, there are two conjugacy classes of order three elements in
${{G}_{2}}$
in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.
For most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras, and Waldspurger on the existence of automorphisms of
$p$
-adic classical groups that take each irreducible smooth representation to its dual.
We describe a general method for expanding a truncated
$G$
-iterative Hasse–Schmidt derivation, where
$G$
is an algebraic group. We give examples of algebraic groups for which our method works.
The maximal finite abelian subgroups, up to conjugation, of the simple algebraic group of type E8 over an algebraically closed field of characteristic 0 are computed. This is equivalent to the determination of the fine gradings on the simple Lie algebra of type E8 with trivial neutral homogeneous component. The Brauer invariant of the irreducible modules for graded semisimple Lie algebras plays a key role.
We show that the conjectural criterion of
$p$
-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties, including the norm varieties associated with symbols in Galois cohomology of arbitrary degree.
Let
$G$
be a simple simply connected algebraic group over an algebraically closed field
$k$
of characteristic
$p>0$
with
$\mathfrak{g}=\text{Lie}(G)$
. We discuss various properties of nilpotent orbits in
$\mathfrak{g}$
, which have previously only been considered over
$\mathbb{C}$
. Using computational methods, we extend to positive characteristic various calculations of de Graaf with nilpotent orbits in exceptional Lie algebras. In particular, we classify those orbits which are reachable as well as those which satisfy a certain related condition due to Panyushev, and determine the codimension of the derived subalgebra
$[\mathfrak{g}_{e},\mathfrak{g}_{e}]$
in the centraliser
$\mathfrak{g}_{e}$
of any nilpotent element
$e\in \mathfrak{g}$
. Some of these calculations are used to show that the list of rigid nilpotent orbits in
$\mathfrak{g}$
, the classification of sheets of
$\mathfrak{g}$
and the distribution of the nilpotent orbits amongst them are independent of good characteristic, remaining the same as in the characteristic zero case. We also give a comprehensive account of the theory of sheets in reductive Lie algebras over algebraically closed fields of good characteristic.
The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.
Let
$G$
be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal
$G$
-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of
$G$
. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple
$G$
, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of
$GL_{n}$
.
Let
$G$
be a simple algebraic group. A closed subgroup
$H$
of
$G$
is said to be spherical if it has a dense orbit on the flag variety
$G/B$
of
$G$
. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.
Suppose that
$\widetilde{G}$
is a connected reductive group defined over a field
$k$
, and
$\Gamma$
is a finite group acting via
$k$
-automorphisms of
$\widetilde{G}$
satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of
$\Gamma$
-fixed points in
$\widetilde{G}$
is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair
$\left( \tilde{G},\Gamma \right)$
, and consider any group
$G$
satisfying the axioms. If both
$\widetilde{G}$
and
$G$
are
$k$
-quasisplit, then we can consider their duals
$\widetilde{{{G}^{*}}}$
and
${{G}^{*}}$
. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in
${{G}^{*}}\,(k)$
to the analogous set for
$\widetilde{{{G}^{*}}}\,(k)$
. If
$k$
is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of
$G(k)$
and
$\widetilde{G}\,(k)$
, one obtains a mapping of such packets.
We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan–Kac–Vinberg decompositions of a rational vector all lie in a single rational orbit.
This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type
${{E}_{7}}$
over a field of characteristic not 2 or 3. In particular,
$\text{ed}\left( {{E}_{7}} \right)\,\le \,29$
, and
$\text{ed}\left( {{E}_{7}};\,2 \right)\,\le \,27$
.
Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.
Let
$W$
be a Weyl group,
$\sum $
a set of simple reflections in
$W$
related to a basis
$\Delta $
for the root system
$\Phi $
associated with
$W$
and
$\theta $
an involution such that
$\theta (\Delta )\,=\,\Delta $
. We show that the set of
$\theta $
- twisted involutions in
$W$
,
${{\mathcal{J}}_{\theta }}\,=\,\{w\,\in \,W\,|\,\theta (w)\,=\,{{w}^{-1}}\}$
is in one to one correspondence with the set of regular involutions
${{\mathcal{J}}_{\text{ID}}}$
. The elements of
${{\mathcal{J}}_{\theta }}$
are characterized by sequences in
$\sum $
which induce an ordering called the Richardson–Springer Poset. In particular, for
$\Phi $
irreducible, the ascending Richardson–Springer Poset of
${{\mathcal{J}}_{\theta }}$
, for nontrivial
$\theta $
is identical to the descending Richardson–Springer Poset of
${{\mathcal{J}}_{\text{ID}}}$
.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)G and k(𝔤)/k(𝔤)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(𝔤)/k(𝔤)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type An or Cn, and negative for groups of other types, except possibly G2. 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 G on itself.