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Abstract
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 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 (resp. subalgebra), or a natural generalization if G is of type Dn. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups.
1. Introduction
Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p. Fix a maximal torus T of G, and positive system of roots; then it is well known that every unipotent class has a representative which is a product of root group elements and that every nilpotent orbit has a representative which is a sum of root elements. However, it is not easy to extract such representatives from the existing literature and there are a few errors (some perpetrated by the second author). This note gives a definitive list when G is simple. For compactness, we provide the representatives of certain distinguished classes we call eminent classes. An eminent unipotent (resp. nilpotent) element is one which is not contained in any proper subsystem subgroup (resp. subalgebra) of G, or a natural generalization if G has a factor of type Dn—the precise definition is given in the next section. Given representatives of the eminent classes in the simple case, it is then a routine task to find representatives of all classes for G. (See Section 2.4.)
Theorem 1.1.
Let G be a simple k-group over a field of characteristic p with Lie algebra
A nilpotent element
is eminent if and only if it is G-conjugate to one of the representatives in (G of classical type) or (G of exceptional type). Moreover, no two representatives in and are G-conjugate.
Table 1. Eminent nilpotent representatives in classical types.
Table 2. Eminent nilpotent representatives in exceptional types.
A unipotent element
is eminent if and only if it is G-conjugate to one of the representatives in (G of classical type) or (G of exceptional type). Moreover, no two representatives in and are G-conjugate.
Table 3. Eminent unipotent representatives in classical types.
Table 4. Eminent unipotent representatives in exceptional types.
We note that the only time a regular element is not eminent is when G is of type Dn. Indeed, such elements are contained in generalized subsystem subgroups or subalgebras of type
We make a few remarks on the tables. An explanation of the notation used is given in Section 2.1; we observe Bourbaki doctrine when exhibiting representatives. When or
is a classical group, we also supply the action of the elements on the natural G-module V, which characterizes them up to conjugacy in G. Lastly, since we use the same representatives as [Citation17, Tables 13.3, 14.1], one can invoke [Citation17, Theorems 19.1, 19.2] to generate representatives for the classes in from those in as follows: a representative of the unipotent class with label X is
where
is the representative of the nilpotent element with label X.
It is often useful to have a means of recognizing unipotent and nilpotent elements in exceptional types from easily computable invariants. For unipotent classes, this was accomplished by Lawther [Citation15, Section 3]. Let for x a unipotent or nilpotent element. We use the following abbreviations.
JBS: List of Jordan block sizes
DS: Dimensions of the terms in the derived series of C
LS: Dimensions of the terms in the lower central series of C
ALG: Dimension of the Lie algebra of derivations of C
ALG′: Dimension of the derived subalgebra of the Lie algebra of derivations of C
NIL: Dimension of the Nilradical of C
NDS: Dimensions of the Lie normalizer of each term in the derived series of C
Theorem 1.2.
Let G be an exceptional k-group. Then gives a list of data which is sufficient to identify the class of any nilpotent or unipotent element of G.
Table 5. Recognition data for nilpotent orbits and unipotent classes.
Remark 1.3.
(i) One important application of Theorem 1.2 is its use in the proof of Theorem 1.1(i) for G of exceptional type. To find the eminent nilpotent elements in the strategy is to list all maximal subsystem subalgebras up to conjugacy; for each
in that list, construct representatives of all distinguished orbits; and then use Theorem 1.2 to identify their class in G.
(ii) We supply auxiliary tables in Section 5; whenever at least two nilpotent orbits have the same Jordan block structure on (when it exists) and
we list them in a table along with the data required to distinguish them, using the same abbreviations given before Proposition 1.2.
2. Preliminaries
2.1. Notation
Throughout G is a reductive k-group, and most often G is simple. Fix a Borel subgroup containing a maximal torus T. This defines a root system
with base
of simple roots generating the positive roots
as positive integral sums. We use the Bourbaki ordering for a Dynkin diagram a corresponding roots and follow [Citation2] in denoting roots by a string of numbers in the form of a Dynkin diagram corresponding to coefficients of a root expressed in terms of the simple roots. For example, the highest short root of an F4 system is denoted 1232.
The adjoint action of G on defines a Cartan decomposition
Corresponding to the root spaces
one has the one-parameter subgroups
where each
is a root groups. In case G is simple and simply connected, we may insist that the
form a Chevalley basis; usually, this is only recognized as canonical up to a choice of the sign of a coefficient λ for certain triples of roots
such that
These choices determine the structure constants for
[Citation3, Proposition 4.2.2]. Through exponentiation one can fix an isomorphism
in which
for all
The computations done in Section 3 were performed with GAP4 [Citation5] in the simply connected case, using the representatives for the nilpotent orbits given in [Citation21] (see the latest arXiv version for a correction in type F4). In the adjoint case we used Magma [Citation1]. Although Magma uses a different choice of signs in the structure constants compared to GAP4, the same representatives can still be used and this is justified as follows.
All representatives listed in [Citation21] are of the form where
are
-independent roots. Thus for any choice of signs
there exists a semisimple element in G that conjugates e to the nilpotent element
see for example [Citation8, Lemma 16.2 C]. In particular, the representatives in [Citation21] do not depend on the signs of structure constants in the Chevalley basis of
2.2. Eminent elements and generalized subsystem subgroups and subalgebras
Recall that a subgroup M of G is a subsystem subgroup if it is semisimple (the definition of which we assume to include connected) and normalized by a maximal torus.Footnote1 Note that the derived subgroup of any Levi subgroup is a subsystem subgroup. Define also a subsystem subalgebra to be a subalgebra with
for some subsystem subgroup M.
Define a generalized subsystem subgroup as follows. If M is a subsystem subgroup of G containing some simple factors of type
where
then there is a semisimple subgroup
of Mi of type
with
(see the proof of Lemma 2.2). Then the generalized subsystem subgroups are obtained by replacing some subset of the Mi with the subgroups
Accordingly, we define a generalized subsystem subalgebra
to be
for a generalized subsystem subgroup M.
Recall that a unipotent or nilpotent element x is distinguished if it is contained in no proper Levi of G. The following definition is more restrictive.
Definition 2.1.
A unipotent element is called eminent if it is not contained in any proper generalized subsystem subgroup of G. A nilpotent element
is called eminent if it is not contained in any proper generalized subsystem subalgebra of
For use in the proof of Theorem 1.1 we record the conjugacy classes of maximal generalized subsystem subgroups which are not Levi subgroups. The maximal generalized subsystem subalgebras follow immediately.
Lemma 2.2.
Suppose that G is a classical group. Let M be a generalized subsystem subgroup which is maximal amongst proper ones, such that M is not a Levi subgroup. Then M is conjugate to precisely one subgroup with root system as given in . Moreover, every subgroup in is a maximal generalized subsystem subgroup.
Table 6. Maximal generalized subsystem subgroups.
Proof.
For the subsystem subgroups this is a routine use of the Borel–de Siebenthal algorithm for closed subsystems [Citation18, Proposition 13.12] and [Citation18, Proposition 13.15(i),(ii)] for the 2-closed non-closed subsystems. When G has type Dn further consideration is required. The definition of generalized subsystem subgroups immediately implies that they will not be maximal unless we replace Dn itself with with
When p = 2, the subgroups of type
with
are contained in
(indeed they are subsystem subgroups of
since p = 2) and so they are not maximal. We need to check that these subgroups are unique up to conjugacy and maximal amongst proper generalized subsystem subgroups. When
a subgroup of type
is the connected stabilizer of a decomposition of V into non-degenerate orthogonal subspaces of dimensions
and
and thus is unique up to conjugacy since G acts transitively on non-degenerate orthogonal subspaces of dimension k. Moreover, this implies they are maximal amongst all connected proper subgroups of G and thus amongst all proper generalized subsystem subgroups. When p = 2, a subgroup of type
is the stabilizer of a nonsingular vector and is again unique up to conjugacy and maximal amongst all connected proper subgroups of G. □
We observe that eminence is invariant under central isogenies.
Lemma 2.3.
Let be a central isogeny of reductive k-groups.
The map
induces bijections between the unipotent varieties of G and H, and the set of generalized subsystem subgroups of G and H.
The map
induces a bijection between the nilpotent cones of G and H, and the set of generalized subsystem subalgebras of G and H.
Proof.
The bijections of unipotent varieties and nilpotent cones follow from [Citation4, Proposition 5.1.1] and [Citation11, 2.7]. Since the kernel of is central, it is contained in any maximal torus; as
is surjective with finite kernel it follows that
is a bijection on maximal tori. Thus M is a subsystem subgroup if and only if
is. If M is a generalized subsystem subgroup, the result follows from the definition. □
2.3. Unipotent classes and nilpotent orbits in classical groups
Let G be simple of classical type. In light of Lemma 2.3, for the purposes of this paper it does no harm for us to assume that or
We will choose forms preserved by G on V consistent with [Citation9]. Let V have k-basis For
and
we choose a non-degenerate alternating bilinear form on V defined by
and
for all other i, j. For
with V of odd dimension
choose a non-degenerate symmetric bilinear form on V defined by
and
for all other i, j. When p = 2, we also define a quadratic form on V by
for
such that Q has polarization
on V. Finally, for
with V of even dimension
choose a non-degenerate symmetric bilinear form on V defined by
and
for all other i, j. And in this case for p = 2, define a quadratic form on V by
for all i, with polarization
on V.
If the class of a unipotent or nilpotent element
is determined by its Jordan block sizes on V. In what follows, we will describe the unipotent classes and nilpotent orbits for
and
In this case for x a unipotent element or a nilpotent element, we have a decomposition
where the Vi are orthogonally indecomposable
-modules. Here orthogonally indecomposable means that if
as
-modules, then W = 0 or
Hesselink [Citation6, 3.5] determined the orthogonally indecomposable
-modules that can occur, and thus classified the unipotent classes and nilpotent orbits for
and
The possibilities for orthogonally indecomposable
-modules are as follows:
The
-module V(m) is non-degenerate of dimension m and x acts on V(m) with a single Jordan block of size m.
The
-module W(m) has dimension 2m, with
where Wi are totally singular of dimension m and x acts on each Wi with a single Jordan block of size m.
The
-module
only occurs when p = 2 and x is a nilpotent element. It is non-degenerate of dimension 2m and x acts with two Jordan blocks of size m. If
then
and
If
then
and
The
-module D(m) only occurs when
with
odd, p = 2 and x is a nilpotent element. It is degenerate of dimension
and x acts with two Jordan blocks of sizes m and m – 1.
The
-module R only occurs when
with
odd, p = 2 and x is a unipotent element. It is the 1-dimensional radical of V and so x acts trivially on it.
It is clear that the decomposition of into orthogonally indecomposable summands determines the class of x. There are various ways of writing down the decomposition
we will use the distinguished normal form introduced in [Citation17]. Since we are interested in the simple group
we state when a class in
meets the subgroup
and if so, whether it splits into two
-classes; see [Citation17, Lemma 3.11, Propositions 5.25, 6.22]. The following lemma is a combination of results from [Citation17, Chapters 5.2, 5.3, 5.6, 6.2, 6.8].
Lemma 2.4.
Let or
and let x be unipotent or nilpotent element of G or
, respectively. The following list states which orthogonal decompositions
can occur. Conversely, the multiset of orthogonal factors determines x uniquely up to conjugacy.
Furthermore, x is distinguished if and only if r = 0.
If
, then
where the nj are distinct, and all are even (resp. odd) when
(resp.
). For
and x unipotent, the class meets
. For
, a class splits into two
-classes if and only if s = 0 and ai = 0 for all odd mi.
If x is unipotent, p = 2 and V is of even dimension, then
where the nj are distinct and
for all j. Such a class meets
if
is even and it splits if and only if t = 0.
If x is unipotent, p = 2 and V is of odd dimension, then
where the nj are distinct and
If x is nilpotent, p = 2 and
, then
where the sequences
and
are strictly decreasing,
and for all j, k either
or qk < lj.
If x is nilpotent, p = 2 and
with V of even dimension, then
where the sequences
and
are strictly decreasing. Each class splits into two
-classes if and only if s = 0.
If x is nilpotent, p = 2 and V is of odd dimension, then
where the sequences
and
are strictly decreasing and
Remark 2.5.
Each simple k-group of classical type has a unique class of nilpotent and unipotent elements of highest dimension called regular. A representative of the regular class is well-known to be for unipotent elements [Citation20, 3.2] and
for nilpotent elements by [Citation19, 5.9] and [Citation12], where the product and sum are supported on all simple roots. From [Citation17, 3.3.6 and pp. 60–61] we see that the regular elements are usually the only ones which have a Jordan block of largest possible size. In type A, this means that x acts with a single Jordan block. For the other classical cases from the lemma, with
if
and m is even, otherwise
if
and
if
2.4. Recovering all classes from the eminent ones
We present a brief discussion of how one can find a full set of representatives of the unipotent classes and nilpotent orbits iteratively from a set of representatives of the eminent classes and orbits. (If G is of exceptional type, one can simply look at the full list in [Citation21].) Since an element x is eminent in some generalized subsystem it suffices to give a set of Chevalley generators for maximal generalized subsystem subgroups. The Borel-de Siebenthal algorithm solves the problem in the case of subsystems. Hence the following lemma completes the picture.
Lemma 2.6.
Let G be a simple k-group of type Dn with . Let
and define:
Then the subgroups H1 and H2 commute, H1 has type Bm and H2 has type
Proof.
This is proved in [Citation22, pp. 67–68], with the generators amended to reflect our choice of structure constants. □
If then Theorem 1.1 implies that the unique eminent class of x in type Bm is regular. Then in view of the lemma, an eminent unipotent element in the subsystem subgroup
can be represented by
where
Similarly, if e is a regular nilpotent element of then e is conjugate to
For even more representatives that work regardless of the choice of signs of the structure constants, see Remark 4.6.
2.5. Representatives for classical groups in small dimensions
To illustrate our main theorem, provide representatives for the unipotent classes and nilpotent orbits in classical algebraic groups of small dimension, as considered in [Citation17, Section 8].
Table 7. Representatives for nilpotent orbits and unipotent classes of
Table 8. Representatives for nilpotent orbits and unipotent classes of
Table 9. Representatives for nilpotent orbits and unipotent classes of
Table 10. Representatives for unipotent classes of
Table 11. Representatives for nilpotent orbits of
Table 12. Representatives for nilpotent orbits and unipotent classes of
Table 13. Representatives for nilpotent orbits and unipotent classes of
Type An is easily described. The class of x corresponds to a partition Viewed as an element of
and put in Jordan normal form, the elements of the superdiagonal that x is supported on correspond to the simple roots that it is supported on. More specifically, a unipotent representative is
and a nilpotent representative is
where
if
for some
and
otherwise.
In the remaining explicit examples we give, there is an injective map from the set of unipotent classes to the set of nilpotent orbits and we have chosen representatives in such a way that a representative for a unipotent class is given by where
is the corresponding representative for the corresponding nilpotent orbit. Therefore we only list representatives for nilpotent orbits, apart from
where the injection is not as natural. When
the characteristic is good for G, and so there is a Springer map, hence a fortiori a bijection from the set of unipotent classes to the set of nilpotent orbits. When p = 2 and
a unipotent class is mapped to a nilpotent orbit such that the decompositions in Lemma 2.4(ii) and (iv) coincide. For a unipotent element of
the decomposition of the corresponding nilpotent element is described in [Citation17, 6.3]. In the tables for the orthogonal groups, some representatives have two rows of decompositions. The first row corresponds to the case where
and the second row corresponds to the case where p = 2.
3. Proof of Theorem 1.2
For unipotent elements, the required result is given in [Citation15, Section 3]. We describe the calculations in the nilpotent case. First we need the following lemma, which shows that for the action of nilpotent elements on the coincidences of Jordan block sizes for nilpotent elements on different orbits are the same in the adjoint case and the simply connected case.
Lemma 3.1.
Let Gsc be a simply connected simple algebraic group over k of exceptional type, and let Gad be the corresponding simple algebraic group of adjoint type. Furthermore, let x be either a unipotent element of Gsc or a nilpotent element of . Then the Jordan block sizes of x on
and
are the same.
Proof.
For types G2, F4, and E8, we have Gsc = Gad, so it suffices to consider the case where Gsc is of type E6 or E7. In this case, we show that as Gsc-modules, from which the lemma follows. If
is simple, then
is an irreducible Weyl module and thus
and
by [Citation10, Lemma II.2.13(b)].
If is not simple (p = 3 for E6 and p = 2 for E7), then by [Citation7] there is a nonsplit short exact sequence
of Gsc-modules, where W is irreducible. Since W is self-dual, it follows that we have a nonsplit short exact sequence
Because is a Weyl module with highest weight the highest root, by [Citation10, II.2.12(4), Proposition II.2.14] we have
Furthermore, the group
is a k-vector space, and equivalence classes of extensions which are scalar multiples of each other correspond to isomorphic Gsc-modules. It follows then from
that a nonsplit extension of k by W is unique up to isomorphism of Gsc-modules, and so must be isomorphic to
By [Citation7], there is a nonsplit short exact sequence
of Gsc-modules, so we conclude that
as Gsc-modules. □
The representatives and tables of Jordan blocks on the minimal and adjoint modules is given in [Citation21]. The remaining data is supplied via in-built functions in GAP4 [Citation5] and Magma such as LieDerivedSeries. It is then straightforward to confirm there are no coincidences.
4. Proof of Theorem 1.1
We first treat the case where G is classical, making heavy use of Lemma 2.4. Throughout the section x and are unipotent elements of G or nilpotent elements of
In light of Lemma 2.3, we take G to be
or
accordingly with natural module V. The proof is completed in the following two lemmas.
Lemma 4.1.
Suppose that x is not conjugate to an element in or . Then x is not eminent.
Proof.
If x is not distinguished, then x is contained in a Levi subgroup or a Levi subalgebra and thus is not eminent. Therefore in what follows, we will assume that x is distinguished. This already completes the proof for since the only distinguished elements are the regular ones. Suppose then that
or
Comparing and with Lemma 2.4, we can assume that
has at least two summands, and r = 0 in the notation of Lemma 2.4.
If this means that
with
It follows that x is contained in the stabilizer of a non-trivial decomposition of V into non-degenerate subspaces, and thus x is contained in a proper generalized subsystem subgroup or subalgebra.
For p = 2, we must consider the unipotent and nilpotent cases separately. Suppose first that x is unipotent. If G is of type Cn, then the argument from the previous paragraph applies.
If G has type Dn, then by Lemma 2.4(ii) we have where s is even. When there are at least four summands, it follows that x is contained in a maximal subsystem subgroup of type
(one may choose
). For s = 2 there is one case remaining which is not in , namely
corresponding to the regular unipotent class. It follows from the description of unipotent classes of
in [Citation17, Section 6.8] that x is contained in the generalized subsystem subgroup of type
(conjugate to the element
in
with
).
If G has type Bn, then in the notation of Lemma 2.4(iii) we have where s > 1. Similarly to the previous cases, it follows that x is contained in a subsystem subgroup of type
(again one may choose
).
It remains to consider the case where x is nilpotent. If G has type Cn, then has at least two summands, each of them isomorphic to
or
Therefore x is contained in the stabilizer of the non-trivial decomposition of V into two non-degenerate subspaces, which is a subsystem subalgebra of type
If G has type Dn, then
either has at least two summands isomorphic to
or is isomorphic to
In the first case, x is contained in a subsystem subalgebra of type
In the second case, it follows from [Citation17, Section 5.6] that x is contained in the generalized subsystem subalgebra of type
(conjugate to the element
in
with
). Finally if G has type Bn, then
has at least one summand isomorphic to
and so x is contained in a subsystem subalgebra of type
□
Lemma 4.2.
The elements in and are eminent.
Proof.
Let x be such an element. It follows from Lemma 2.4 that x is distinguished. Therefore if x is not eminent, then it must be contained one of the maximal generalized subsystem subgroups or subalgebras described in Lemma 2.2.
Suppose first that x acts on V with a single Jordan block. Elements in the subgroups and subalgebras of Lemma 2.2 act on V with at least two Jordan blocks, so it follows that x is eminent. The remaining cases occur when p = 2 and in all of them x acts with two Jordan blocks on V. We treat them in turn.
Consider first the case where is of type Bn, and let R be the 1-dimensional radical of V. We will show that elements from the subgroups and subalgebras of Lemma 2.2 act on V with at least three Jordan blocks, which proves that x must be eminent. For M < G of type Dn, we have
where W is the natural module for
Unipotent elements of M and nilpotent elements of
act on W with an even number of Jordan blocks by Lemma 2.4 (ii) and (v), so in particular they have at least three Jordan blocks on V. The other possibility in Lemma 2.2 is M < G of type
for
In this case
where V1 is 2 m-dimensional and V2 is
-dimensional. Thus unipotent elements of M and nilpotent elements of
act on V/R with at least three Jordan blocks by Lemma 2.4(iii) and (vi), and therefore they also act on V with at least three Jordan blocks.
If of type Cn, then x is nilpotent and
for some
Such an element x acts with two Jordan blocks of size n on V. By considering each of the maximal subsystem subalgebras, we see that only those of type
(if n is even) and Dn contain elements that act with two Jordan blocks of size n. In type
the only element which does that acts as
on V, which is already in canonical form and thus not conjugate to x. In the second case, the elements
which act with two Jordan blocks in
have
for some
The definition of
for orthogonal groups in even dimension, given in [Citation17, p. 66], shows that all of these elements
are conjugate to a nilpotent element acting via W(n) in
and thus not conjugate to x.
If of type Dn, then there are both unipotent classes and nilpotent orbits to consider. In all of the cases the elements x being considered act with two Jordan blocks on V. Using Lemma 2.4 we see that every element contained in subsystem subgroups and subalgebras of type
(
) will act with at least four Jordan blocks on V. Thus they are not conjugate to x. By Lemma 2.2, what remains is to show that x is not conjugate to an element of the maximal generalized subsystem subgroup M (or subalgebra
) of type
When x is unipotent, this follows from [Citation13, Lemma 3.8], which shows that the only unipotent elements
that act with two Jordan blocks on V are those with
Such elements
are regular and not conjugate to x. Now let x be nilpotent. We consider the distinguished nilpotent elements
in
By Lemma 2.4(vi), we have
for some integers lj, nj and m. If s = 0, then
and
is a regular nilpotent element of
and therefore is not conjugate to x. When
the elements
are contained in generalized subsystem subalgebras of type
which are contained in maximal subsystem subalgebras of type
We have already seen that any nilpotent element contained in such a subalgebra will act with at least four Jordan blocks on V, and thus cannot be conjugate to x. This completes the proof of the lemma. □
The remaining task for G of classical type is to prove that the representatives of the eminent elements do indeed act with the claimed decompositions on V. When x is regular, a representative is provided by Remark 2.5. The non-regular eminent classes occur in characteristic p = 2 for types Cn and Dn, the representatives for these classes are constructed in the following sections.
4.1. Nilpotent representatives in type Cn
Let with V a k-vector space of dimension 2n. We need to prove that the nilpotent elements
in act as
on the natural module V. We do this in the next lemma, but first present our choice of Chevalley basis for
Recall the form described at the beginning of Section 2.3. For this choice of form, one checks that there is a Cartan subalgebra of diagonal matrices in
of the form
For
define maps
by
For all i, j let be the linear endomorphism on V such that
and
for
Then one checks that the endomorphisms
in are elements of
and are simultaneous eigenvectors for
Table 14. Chevalley basis for
From a dimension count, we deduce that concatenating the elements in together with a basis of gives a basis of
Moreover,
and
is a system of positive roots. We let
be our base of
corresponding to
where
for
and
We give the expressions of roots
in terms of Δ in . Lastly, checking commutator relations reveals the basis in is in fact a Chevalley basis, with positive structure constants for extraspecial pairs.
Table 15. Type Cn, expressions for positive roots in terms of base Δ.
In particular, for
and
for l < n.
Lemma 4.3.
For , the element
acts on V as
Proof.
For ease of notation we set e = el. From the description of the -module
in Section 2.3, it suffices to show that e acts on the natural 2n-dimensional module V with two Jordan blocks of size n and that
Using and , we write
and therefore the action of e on V is described as follows. We have
We immediately see that for all
Furthermore, the relations above readily imply that the kernel of e is the 2-dimensional subspace of V generated by v1 and
Thus the Jordan normal form of e has two Jordan blocks of size n.
To show that we start by observing
It remains to show that for all
Since
for all i, j, by definition of
preserving the alternating form, it suffices to show that
for all basis vectors vi.
If or
we immediately have that
since e stabilizes the totally isotropic subspaces
and
Now suppose that
Then
and so
and so
as required. □
4.2. Nilpotent and unipotent representatives in type Dn
Let with V a k-vector space of dimension 2n. We need to establish the correctness of the representatives in and . Recall we define the root
for
We mimic the process from the last section. Using the form from the beginning of Section 2.3 we check we may choose a Cartan subalgebra of the form
with
For
define maps
by
A Chevalley basis for
with positive structure constants for extraspecial pairs is given in .
Table 16. Chevalley basis of
Now is the root system of
and
is a system of positive roots. Here the base Δ of
corresponding to
is
where
for
and
We give the expressions of roots
in terms of Δ in .
Table 17. Type Dn, expressions for positive roots in terms of base Δ.
Lemma 4.4.
The element acts on V as
Proof.
From and , we have for all
where I is the identity and
is defined as before. Furthermore, we see that
Now
acts on the basis elements of V as follows:
A calculation shows that the fixed point space of u has dimension 2, and that it is spanned by v1 and Therefore the Jordan normal form of u has two Jordan blocks. To see that the Jordan block sizes are
and
it suffices to show that
and
as then the largest Jordan block size in u is
To this end, a calculation shows that
for all i and
It is now clear that and
Therefore
occurs as summand of
by [Citation14, Lemma 6.9], and we must have
as required. □
Lemma 4.5.
The element acts on V via
Proof.
The tables above yield for i < n and
so that
We need to show that e has two Jordan blocks of size n and that We start by calculating the action of e on the basis vectors of V:
One sees that the kernel of e has dimension 2, and it is spanned by v1 and Therefore the Jordan normal form of e has two Jordan blocks and a routine calculation with the basis
shows that
for all
Finally, we must show that Since
it follows that
and so
for all
It remains to prove that
for all
Since
it follows that
Therefore, it suffices to show that
for all basis vectors vi. If
or
then
for some j and so
When
we have
and so again,
□
Remark 4.6.
Interpreting the coefficients of the unipotent element ul of Lemma 4.4 as integers, reduction mod p for p > 2 gives an element acting on the natural module with Jordan block sizes Indeed, a calculation shows that the fixed point space is spanned by v1 and
and furthermore
so the largest Jordan block size of ul is
Therefore ul is regular in the subsystem of Dn and is therefore another representative alongside that provided in Lemma 2.6. Similarly one can calculate that the nilpotent element el of Lemma 4.5 has Jordan block sizes
in characteristic
Since the roots involved in ul and el are
-independent, by [Citation8, Lemma 16.2C] these representatives have the advantage that they work regardless of the choice of signs of the structure constants.
4.3. Exceptional types
In this section we prove Theorem 1.1 for G of exceptional type. If x is unipotent, this follows from work of Lawther in [Citation15]. It is routine to refine his lists to remove those distinguished unipotent classes of G which are not eminent. For each non-eminent unipotent class we list its maximal subsystem overgroups in , for completeness. We note that a representative of a unipotent class with label X is simply where
is the representative of the nilpotent element with label X given in [Citation21]. This follows from the proofs in [Citation17] (see the introduction of Sections 17 and 18 in ibid.) since the representatives in [Citation21] are deduced from [Citation17].
Table 18. Maximal subsystem overgroups of non-eminent distinguished unipotent elements in exceptional types.
In good characteristic, [Citation16, p. 24] provides enough information to find the non-eminent distinguished nilpotent orbits. As might be expected, these classes are in bijection with the non-eminent distinguished unipotent classes (sending a unipotent class to the nilpotent orbit of the same label). It remains to consider the nilpotent orbits of in bad characteristic. We follow Lawther’s approach, which entails constructing representatives of each distinguished nilpotent orbit of
for M a maximal subsystem subgroup of G and determining its G-class. To do this we make heavy use of use Theorem 1.2.
Carrying out this calculation provides a proof of Theorem 1.1. Lastly, we mention . It provides a list of distinguished, non-eminent nilpotent orbits, giving their maximal subsystem overalgebras, for all characteristics. Specializing to good characteristic, we recover the analogous result in [Citation15].
Table 19. Maximal subsystem overalgebras of non-eminent distinguished nilpotent elements in exceptional types.
Table A1. G = F4, p = 2.
Table A2. G = E6, p = 2.
Table A3. G = E6, p = 3.
Table A4. G = E7, p = 3.
Table A5. G = E7 (simply connected), p = 2.
Table A6. G = E7 (adjoint), p = 2.
Table A7. G = E8, p = 5.
Table A8. G = E8, p = 3.
Table A9. G = E8, p = 2.
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Notes
1 Unless G has a factor with root system such that the pair
is
or
this implies it has a symmetric root system which is closed under sums. In any case, the relevant root systems can be obtained from the Dynkin diagram using a version of the Borel–de Siebenthal algorithm; see [18, Proposition 13.15].
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Appendix:
Auxiliary tables for Theorem 1.2
Below follows some auxiliary data for Theorem 1.2. Let G be exceptional. In each table we use a horizontal line to separate the sets of nilpotent orbits (represented by their labels) for which the Jordan block structure on (when it exists) and
coincide. For each class we then provide the additional data required to distinguish them. See the introduction for the terminology (DS, ALG, etc.).