# Some reality properties of finite simple orthogonal groups

Jiwon Kim  107, Haedoji-ro, 6-4101
Yeonsu-gu, Incheon, 21977
Republic of Korea<EMAIL_ADDRESS>,  Stephen Trefethen  Department of
Mathematics, William & Mary
Williamsburg, VA, 23187
USA<EMAIL_ADDRESS><EMAIL_ADDRESS>and  C. Ryan Vinroot

###### Abstract.

We prove several reality properties for finite simple orthogonal groups. For
any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are
strongly real in the simple groups $\mathrm{P}\Omega^{\pm}(4m+2,q),m\geq 1$,
except in the case $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q\equiv
3(\mathrm{mod}\;4)$, and we construct weakly real classes in this exceptional
case for any $m$. We also show that no irreducible complex character of
$\mathrm{P}\Omega^{\pm}(n,q)$ can have Frobenius–Schur indicator $-1$, except
possibly in the case $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q\equiv
3(\mathrm{mod}\;4)$.

2020 AMS Mathematics Subject Classification: 15B10, 20G40, 20C33

##  1\. Introduction

Let $\mathrm{O}(V)$ be the full orthogonal group corresponding to a non-
degenerate symmetric form on a finite-dimensional vector space $V$ over a
field of characteristic which is not 2. It is a result of M. Wonenburger [22]
that every element of $\mathrm{O}(V)$ is the product of two involutions from
$\mathrm{O}(V)$. This is equivalent to the fact that every element of
$\mathrm{O}(V)$ may be conjugated to its inverse by an involution in
$\mathrm{O}(V)$. That is, every conjugacy class of the orthogonal group
$\mathrm{O}(V)$ is _strongly real_. For certain subgroups of orthogonal
groups, this property is further explored by F. Knüppel and G. Thomsen [13],
who classify when the commutator subgroup $\Omega(V)$ of $\mathrm{O}(V)$, and
the kernel of the spinor norm on $\mathrm{O}(V)$, are strongly real.

If $\mathrm{O}(V)$ is now the orthogonal group corresponding to a non-
defective quadratic form on $V$ over a field which is characteristic 2, it was
shown by E. Ellers and W. Nolte [3] and R. Gow [6] that $\mathrm{O}(V)$ is
also strongly real. In the case that $V$ has dimension divisible by $4$ and
$V$ is defined over a finite field of characteristic $2$, it is proven by J.
Rämö [14] that the commutator subgroup $\Omega(V)$ of $\mathrm{O}(V)$ is
strongly real. This is not the case if the dimension of $V$ is 2 modulo $4$,
by the work of P. H. Tiep and A. E. Zalesski [17].

For an element (or conjugacy class) of a group to be strongly real, it is
necessary that it is _real_ , or conjugate to its inverse within the group. In
the representation theory of finite groups, it is of interest to understand
which conjugacy classes of a group are real, because the number of real
conjugacy classes of a finite group is equal to the number of real-valued
irreducible complex characters of that group. The classification of finite
simple groups with the property that all elements are real is completed by P.
H. Tiep and A. E. Zalesski [17]. The classification of finite simple groups
such that all elements are strongly real is completed by J. Rämö [14], and E.
P. Vdovin and A. A. Gal′t [18], with the somewhat surprising result that all
elements of a finite simple group are real if and only if they are all
strongly real.

In this paper, we focus on the finite simple orthogonal groups, and the
question of when all real elements are strongly real. By the results mentioned
above, the answer to this question is understood in the case when all elements
are real, and so we study the remaining cases. We give a complete answer to
the question in the case that the underlying vector space has dimension which
is 2 modulo 4.

The connection between the strongly real classes of a finite group and the
irreducible complex characters of that group is still mysterious in the
general case, and one motivation for our study is to further understand this
connection in the case of finite simple orthogonal groups. In particular, in
some classes of finite groups there is a connection between strongly real
classes and the irreducible complex characters which are afforded by a real
representation. For example, it is proven in [20] that for a finite simple
group, all conjugacy classes are strongly real if and only if all irreducible
complex characters can be afforded by a real representation. In this paper we
gather evidence that for finite simple orthogonal groups, all real classes are
strongly real if and only if all real-valued irreducible complex characters
can be afforded by real representations.

This paper is organized as follows. In Section 2, we give some standard
information on orthogonal groups over finite fields and their important
subgroups. In Section 3, after giving some previous results, we prove our main
results in the case of odd characteristic, which are reality results for
$\Omega(V)$ and $\mathrm{P}\Omega(V)$ in the remaining cases when
$\dim(V)\equiv 2($mod $4)$. In particular, in the cases when these groups are
known to have classes which are not real, we show that all real elements are
strongly real, in Proposition 3.1 and Theorem 3.5, except in the case of
$\Omega^{-}(4m+2,q)$ and $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q\equiv 3($mod
$4)$. In that exceptional case, we construct elements which are real but not
strongly real in Theorem 3.6 (with technical base case constructions given in
the Appendix). In Section 4, we prove that all real elements are strongly real
in $\Omega^{\pm}(4m+2,q)$ when $q$ is a power of $2$, and we classify all real
elements, in Theorem 4.1. In Section 5, we prove in Theorem 5.1 that all real-
valued irreducible characters of $\Omega^{\pm}(n,q)$ and
$\mathrm{P}\Omega^{\pm}(n,q)$ can be afforded by real representations, except
possibly in the case $\Omega^{-}(4m+2,q)$ and $\mathrm{P}\Omega^{-}(4m+2,q)$
with $q\equiv 3($mod $4)$, and we give some concluding remarks.

Acknowledgments.  Much of this work was done when the first-named author was a
visiting assistant professor at William & Mary. The third-named author was
supported in part by a grant from the Simons Foundation, Award #713090.

##  2\. Finite orthogonal groups

In this section we state some well-known facts about finite orthogonal groups
and some of their subgroups. All of these results may be found in [8] or [1,
Chapter 16].

Let $q$ be the power of a prime, and $\mathbb{F}_{q}$ a finite field with $q$
elements. Fix a finite-dimensional $\mathbb{F}_{q}$-vector space $V$, say
$\dim_{\mathbb{F}_{q}}(V)=n$, and we assume $V$ carries a quadratic form $Q$.
In the case that $q$ is odd, we assume that the bilinear form $B$ associated
with $Q$ is non-degenerate, and in the case that $q$ is even we assume that
$n$ is even and $Q$ is non-defective. That is, when $q$ is odd we have, for
all $u,v\in V$,

$Q(u+v)=Q(u)+2B(u,v)+Q(v),$

for a non-degenerate symmetric bilinear form $B$, while if $q$ is even we have

$Q(u+v)=Q(u)+B(u,v)+Q(v),$

for a non-degenerate symplectic form $B$. We may then consider the group
$\mathrm{O}(V)$ of isometries of the quadratic form $Q$, which is the
orthogonal group. When $q$ is odd, this is the same as the group of isometries
of the associated symmetric bilinear form $B$.

For all $n$ and $q$ considered above, there are precisely two equivalence
classes of quadratic forms $Q$. For $v\in V$, we take $v$ to have coordinate
representation $v=(x_{1},\ldots,x_{n})$ with respect to some basis. When
$n=2m$ is even, one form may be taken to be

$Q(v)=x_{1}x_{2}+x_{3}x_{4}+\cdots+x_{2m-1}x_{2m},$

which we say is a split form, or $+$-type. When $q$ is odd and $n=2m+1$ is
odd, the split, or $+$-type, form is given by

$Q(v)=x_{1}x_{2}+x_{3}x_{4}+\cdots+x_{2m-1}x_{2m}+x_{2m+1}^{2}.$

The second equivalence class of quadratic forms (which we do not formulate
explicitly here) in the cases under consideration will be called non-split
forms, or $-$-type. We will denote by $\mathrm{O}^{+}(n,q)$ or
$\mathrm{O}^{-}(n,q)$ the orthogonal groups corresponding to a quadratic form
of split or non-split type, respectively, and we will write
$\mathrm{O}^{\pm}(n,q)$ for the generic case. In the case that $n$ and $q$ are
odd, we in fact have $\mathrm{O}^{+}(n,q)\cong\mathrm{O}^{-}(n,q)$, but these
groups are not isomorphic in the general case.

One group of primary interest in this paper is the derived (or commutator)
subgroup of the orthogonal group. Letting $X^{\prime}$ denote the derived
subgroup of a group $X$, we may define

$\Omega^{\pm}(n,q)=\mathrm{O}^{\pm}(n,q)^{\prime}.$

When $q$ is even, $\Omega^{\pm}(n,q)$ is an index two subgroup of
$\mathrm{O}^{\pm}(n,q)$ and is a simple group. When $q$ is odd,
$\Omega^{\pm}(n,q)$ is an index $4$ subgroup of $\mathrm{O}^{\pm}(n,q)$, and
may or may not contain the central involution of $\mathrm{O}^{\pm}(n,q)$.
Whether the central involution is contained in $\Omega^{\pm}(n,q)$ may be
determined by $n$, $q$, and the type $\pm$ of the underlying form, as
described in [1, pg. 230]. If $Z$ is the center of $\Omega^{\pm}(n,q)$, then
$\mathrm{P}\Omega^{\pm}(n,q)=\Omega^{\pm}(n,q)/Z$ is a finite simple group
(with a finite number of exceptions when $n$ is small).

We now assume that $q$ is the power of an odd prime until Section 4. In this
case, we may define the special orthogonal groups $\mathrm{SO}^{\pm}(n,q)$ to
be the elements of $\mathrm{O}^{\pm}(n,q)$ with determinant $1$. Then
$\Omega^{\pm}(n,q)$ is an index $2$ subgroup of $\mathrm{SO}^{\pm}(n,q)$, and
$\mathrm{SO}^{\pm}(n,q)$ is an index $2$ subgroup of $\mathrm{O}^{\pm}(n,q)$.
One may also define the spinor norm on $\mathrm{O}^{\pm}(n,q)$, which is a
multiplicative homomorphism

$\theta:\mathrm{O}^{\pm}(n,q)\longrightarrow\mathbb{F}_{q}^{\times}/(\mathbb{F}_{q}^{\times})^{2},$

and the explicit definition can be found in [8, Chapter 9]. Then
$\ker(\theta)$ is an index $2$ subgroup of $\mathrm{O}^{\pm}(n,q)$, and we
write $\ker(\theta)=\mathrm{K}^{\pm}(n,q)$. Also $\Omega^{\pm}(n,q)$ is an
index 2 subgroup of $\mathrm{K}^{\pm}(n,q)$, and in fact we have

$\Omega^{\pm}(n,q)=\mathrm{SO}^{\pm}(n,q)\cap\mathrm{K}^{\pm}(n,q).$

There is a third index two subgroup of $\mathrm{O}^{\pm}(n,q)$ containing
$\Omega^{\pm}(n,q)$, consisting of $\Omega^{\pm}(n,q)$ together with the coset
in $\mathrm{O}^{\pm}(n,q)$ disjoint from both $\mathrm{K}^{\pm}(n,q)$ and
$\mathrm{SO}^{\pm}(n,q)$. We define this subgroup to be
$\mathrm{T}^{\pm}(n,q)$, that is,

$\displaystyle\mathrm{T}^{\pm}(n,q)$
$\displaystyle=\Omega^{\pm}(n,q)\cup\\{g\in\mathrm{O}^{\pm}(n,q)\,\mid\,g\not\in\mathrm{K}^{\pm}(n,q)\cup\mathrm{SO}^{\pm}(n,q)\\}$
$\displaystyle=\mathrm{ker}(\theta\times\det).$

We denote the _discriminant_ of the symmetric form $B$ carried by $V$, defined
in [8, Chapter 2], by
$dV\in\mathbb{F}_{q}^{\times}/(\mathbb{F}_{q}^{\times})^{2}$.

The discriminant, spinor norm, and determinant each behave as a direct product
map over an orthogonal direct sum of the underlying space
$V=\mathbb{F}_{q}^{n}$. That is, suppose $V=\oplus_{i}V_{i}$ is an orthogonal
direct sum with respect to $B$, so that we may consider the non-degenerate
symmetric form $B$ restricted to the subspace $V_{i}$, denoted $B_{V_{i}}$.
Then we have $dV=\sum_{i}dV_{i}$. If $g\in\mathrm{O}(V)$ and each $V_{i}$ is
$g$-invariant, then we may consider $g$ with restricted action on each
subspace $V_{i}$, and denote this by $g_{i}=g_{V_{i}}\in\mathrm{O}(V_{i})$.
Then we have that the spinor norm and determinant maps satisfy

$\theta(g)=\sum_{i}\theta(g_{i})\quad\text{ and
}\quad\det(g)=\prod_{i}\det(g_{i}).$

When restricting the symmetric form $B$ to a subspace $V_{i}$, the $\pm$ type
of the corresponding quadratic form may be determined by the values of $q$,
$\dim(V_{i})$, and the discriminant $dV_{i}$, as described in [1, pgs.
221-222].

##  3\. Odd characteristic

We take $q$ to be the power of an odd prime, $V=\mathbb{F}_{q}^{n}$ endowed
with a quadratic form $Q$, and $B$ the associated non-degenerate symmetric
bilinear form on $V$. Let $\mathrm{O}^{\pm}(n,q)$ be the corresponding
orthogonal group.

###  3.1. Preliminary Results

In this section we give previous results which will be repeatedly used in what
follows. If $G$ is a finite group, then we say $g\in G$ is _real in $G$_ if
there is an element $h\in G$ such that $hgh^{-1}=g^{-1}$. If all elements of
$G$ satisfy this property, then we say $G$ is a _real group_. We say $g\in G$
is _strongly real in $G$_ if there is an element $s\in G$ such that $s^{2}=1$
and $sgs^{-1}=g^{-1}$, or equivalently, if there are elements $s_{1},s_{2}\in
G$ such that $s_{1}^{2}=s_{2}^{2}=1$ and $g=s_{1}s_{2}$. If all elements of
$G$ are strongly real in $G$, then we say $G$ is a _strongly real group_.

We have the following result of Wonenburger [22] mentioned in the introduction
(see also [13, Proposition 4.5]).

######  Theorem 3.1.

If $q$ is the power of an odd prime and $n\geq 2$, then
$\mathrm{O}^{\pm}(n,q)$ is strongly real.

The following essentially comes from [22], but also follows from [15].

######  Theorem 3.2.

If $q$ is the power of an odd prime and $n\geq 3$, then
$\mathrm{SO}^{\pm}(n,q)$ is strongly real if and only if it is real, and this
group is real if and only if $n\not\equiv 2($mod $4)$.

The following is the result [13, Theorem 8.6] of Knüppel and Thomsen,
specialized to the case of finite fields.

######  Theorem 3.3.

If $q$ is the power of an odd prime, and $n\geq 3$ (and $q>3$ when $n<6$),
then $\mathrm{K}^{\pm}(n,q)$ is strongly real if and only if:

  1. (i)

$q\equiv 1(\mathrm{mod}\;4)$, or

  2. (ii)

$q\equiv 3($mod $4)$ and is one of $\mathrm{K}^{+}(4m+2,q)$,
$\mathrm{K}^{+}(4m+3,q)$, $\mathrm{K}^{-}(4m,q)$, or $\mathrm{K}^{-}(4m+1,q)$,
or

  3. (iii)

$n=8$ or $n=9$.

Knüppel and Thomsen also prove the following in [13, Theorem 8.5] (see also
[5]).

######  Theorem 3.4.

If $q$ is the power of an odd prime, and $n\geq 3$ (and $q>3$ when $n<6$),
then $\Omega^{\pm}(n,q)$ is strongly real if and only if:

  1. (i)

$q\equiv 1($mod $4)$ and $n\not\equiv 2($mod $4)$, or

  2. (ii)

$q\equiv 3($mod $4)$ and it is $\Omega^{-}(4m,q)$, or

  3. (iii)

$n=8$ or $n=9$.

Note that for $n<6$ we have $\Omega^{\pm}(n,q)$ is isomorphic to other well-
understood finite groups, by [12, Proposition 2.9.1].

Recall that if $F$ is a field with fixed algebraic closure $\overline{F}$,
given a monic non-constant polynomial with nonzero constant $f(t)\in F[t]$,
the _reciprocal polynomial_ of $f(t)$, denoted $f^{*}(t)\in F[t]$, is the
polynomial with the property that for any zero
$\alpha\in\overline{F}^{\times}$ of $f(t)$, $\alpha^{-1}$ is a zero of
$f^{*}(t)$ with the same multiplicity. Then $f(t)$ is _self-reciprocal_ if
$f(t)=f^{*}(t)$.

We next describe an important orthogonal decomposition of the underlying space
$V$. Given any $g\in\mathrm{O}^{\pm}(n,q)$, then as in [13, Section 3], $V$
may be decomposed as an orthogonal (with respect to $B$) direct sum, $V=\oplus
V_{i}$, such that each $V_{i}$ is an orthogonally indecomposable $g$-invariant
subspace of $V$, where each $V_{i}$ is of one of the following forms:

  1. (1)

$V_{i}$ is a (non-orthogonal) direct sum, $V_{i}=U_{i}\oplus W_{i}$ of
degenerate totally isotropic $g$-invariant cyclic subspaces (with
$\dim_{\mathbb{F}_{q}}(U_{i})=\dim_{\mathbb{F}_{q}}(W_{i})$), and $g$ has a
single elementary divisor on both $U_{i}$ and $W_{i}$ which is either
$(t-1)^{2e}$ (_type $1^{-}$_) or $(t+1)^{2e}$ (_type $1^{+}$_);

  2. (2)

$V_{i}$ is $g$-cyclic, such that $g$ has a single elementary divisor
$f(t)^{e}$ on $V_{i}$, where $f(t)$ is irreducible and self-reciprocal, and
either $f(t)\neq t\pm 1$ (_type $2^{*}$_), or $e$ is odd and $f(t)=t-1$ (_type
$2^{-}$_) or $f(t)=t+1$ (_type $2^{+}$_);

  3. (3)

$V_{i}$ is $g$-cyclic, and $V_{i}$ is a (non-orthogonal) direct sum
$V_{i}=U_{i}\oplus W_{i}$ of totally isotropic $g$-invariant subspaces (with
$\dim_{\mathbb{F}_{q}}(U_{i})=\dim_{\mathbb{F}_{q}}(W_{i})$), such that $g$
has a single elementary divisor $f(t)^{e}$ on $U_{i}$, and a single elementary
divisor $f^{*}(t)^{e}$ on $W_{i}$, such that $f(t)$ is irreducible and
$f(t)\neq f^{*}(t)$ (_type $3$_).

In case (1), we say that $V_{i}$ is _bicyclic_ with respect to $g$.

###  3.2. Reality results

We begin with the following observation.

######  Proposition 3.1.

Let $G$ be one of the finite simple groups $G=\Omega^{-}(4m+2,q)$ with
$q\equiv 1(\mathrm{mod}\;4)$ or $G=\Omega^{+}(4m+2,q)$ with $q\equiv
3(\mathrm{mod}\;4)$, where $m\geq 1$. Then all real classes of $G$ are
strongly real.

###### Proof.

We have $G$ is a subgroup of $\mathrm{SO}^{\pm}(4m+2,q)$, and all real classes
of this group are strongly real by Theorem 3.2. Let $g\in G$ be real in $G$,
which is then real in $\mathrm{SO}^{\pm}(4m+2,q)$. So we may write
$g=s_{1}s_{2}$, where $s_{1}^{2}=s_{2}^{2}=1$, and we may assume
$s_{1},s_{2}\in\mathrm{SO}^{\pm}(4m+2,q)\setminus G$. We have $-I\not\in G$,
and so the involutions $-s_{1},-s_{2}\in G$ with $g=(-s_{1})(-s_{2})$. Thus
$g$ is strongly real in $G$. ∎

The goal for this section is to prove that the same statement holds true in
the case $G=\Omega^{+}(4m+2,q)$ with $q\equiv 1($mod $4)$ and for the simple
group $G/Z=\mathrm{P}\Omega^{+}(4m+2,q)$ with $q\equiv 1($mod $4)$, but in
contrast we also show that the groups $\Omega^{-}(4m+2,q)$ and
$\mathrm{P}\Omega^{-}(4m+2,q)$ with $q\equiv 3($mod $4)$ always have weakly
real classes.

######  Lemma 3.1.

Let $g\in\mathrm{T}^{\pm}(n,q)$, with $V=\mathbb{F}_{q}^{n}$. Let $V=\oplus
V_{i}$ be the orthogonal decomposition corresponding to $g$. Then $g$ is
strongly real in $\mathrm{T}^{\pm}(n,q)$ if:

  1. (1)

there exists a subspace $V_{i}$ such that $\dim(V_{i})\equiv
2(\mathrm{mod}\;4)$ which is type $2^{*}$ or $3$, or

  2. (2)

there exists a subspace $V_{i}$ which is type $2^{\pm}$ and
$dV_{i}=(\mathbb{F}_{q}^{\times})^{2}$.

###### Proof.

First, since $g\in\mathrm{O}^{\pm}(n,q)$, then $g$ is the product of two
involutions in $\mathrm{O}^{\pm}(n,q)$ by Theorem 3.1. Let us take $g=sh$ with
$s^{2}=h^{2}=1$, and we assume
$s,h\in\mathrm{O}^{\pm}(n,q)\setminus\mathrm{T}^{\pm}(n,q)$, otherwise $g$ is
already strongly real in $\mathrm{T}^{\pm}(n,q)$. Write $s_{i}=s_{V_{i}}$ for
each subspace $V_{i}$, and by the proof of [13, Proposition 4.5], we may
assume $s_{i}\in\mathrm{O}(V_{i})$. First suppose that there is a subspace
$V_{i}$ with $\dim(V_{i})\equiv 2($mod $4)$ which is of type $2^{*}$ or type
$3$. We consider the spinor norm on $\mathrm{O}(V_{i})$, and by [13, Lemma
5.8] and its proof, for any non-square
$\alpha\in\mathbb{F}_{q}^{\times}\setminus(\mathbb{F}_{q}^{\times})^{2}$, we
may replace $s_{i}$ with some involution $s_{i}^{\prime}$ such that
$\theta(s_{i}^{\prime})=\alpha\theta(s_{i})$, where $s_{i}^{\prime}$ is a
$\mathrm{GL}(V_{i})$-conjugate of $s_{i}$ so that
$\det(s_{i})=\det(s_{i}^{\prime})$. Then we define

$s^{\prime}=s_{i}^{\prime}\oplus\bigoplus_{j\neq i}s_{j},$

and by construction $s^{\prime}\in\mathrm{T}^{\pm}(n,q)$, and $s^{\prime}$ is
an inverting involution for $g$.

Next assume that there is a subspace $V_{i}$ which is type $2^{\pm}$ and has
discriminant $dV_{i}=(\mathbb{F}_{q}^{\times})^{2}$. Then we define

$s^{\prime}=(-s_{i})\oplus\bigoplus_{j\neq i}s_{j},$

so that $s^{\prime}$ is an involution in $\mathrm{O}^{\pm}(n,q)$. Since
$dV_{i}=(\mathbb{F}_{q}^{\times})^{2}$, then
$-I_{V_{i}}\in\mathrm{ker}(\theta_{i})$ (by [1, pgs. 221-222, 230]), and so
$\theta(s^{\prime})=\theta(s)$. Since $V_{i}$ is type $2^{\pm}$, then
$\dim(V_{i})$ is odd, and so $\det(s^{\prime})=-\det(s)$. So by construction
we now have $s^{\prime}\in\mathrm{T}^{\pm}(n,q)$ and $s^{\prime}$ is an
inverting involution for $g$. ∎

######  Proposition 3.2.

Let $n\geq 3$ and $q\equiv 1(\mathrm{mod}\;4)$. Then $\mathrm{T}^{\pm}(n,q)$
is strongly real.

###### Proof.

Let $g\in\mathrm{T}^{\pm}(n,q)$ with $V=\mathbb{F}_{q}^{n}=\oplus V_{i}$ the
orthogonal decomposition corresponding to $g$. By Lemma 3.1, we may assume
that any $V_{i}$ of type $2^{*}$ or $3$ has dimension divisible by 4, and any
$V_{i}$ of type $2^{\pm}$ has discriminant
$dV_{i}\neq(\mathbb{F}_{q}^{\times})^{2}$.

For any subspace $V_{i}$ of type $1^{\pm}$ or $2^{-}$, it follows from [13,
Lemma 4.7(c)] that $g_{i}=g_{V_{i}}\in\Omega(V_{i})$. Since $q\equiv 1($mod
$4)$ and $\mathrm{dim}(V_{i})$ must be either odd or divisible by 4 in these
cases, it follows from Theorem 3.4 that $g_{i}$ is strongly real in
$\Omega(V_{i})$.

Consider a subspace $V_{i}$ of type $2^{+}$. Then $\mathrm{det}(g_{i})=-1$ by
definition, and by assumption $dV_{i}\neq(\mathbb{F}_{q}^{\times})^{2}$. From
[13, Lemma 4.7(d)], we also have
$\theta(g_{i})=dV_{i}\neq(\mathbb{F}_{q}^{\times})^{2}$, and it follows that
we have $g_{i}\in\mathrm{T}(V_{i})\setminus\Omega(V_{i})$. By [13, Lemma 5.1],
$g_{i}$ can be inverted by an orthogonal involution $s_{i}$ such that either
$\mathrm{det}(s_{i})=1$ and
$\theta(s_{i})=\pm(\mathbb{F}_{q}^{\times})^{2}=(\mathbb{F}_{q}^{\times})^{2}$
(since $q\equiv 1($mod $4)$), or such that $\det(s_{i})=-1$ and $\theta(s)=\pm
dV_{i}\neq(\mathbb{F}_{q}^{\times})^{2}$ (by our assumption on $dV_{i}$ and
since $q\equiv 1($mod $4)$). In either case, we have
$s_{i}\in\mathrm{T}(V_{i})$.

Now let $Y$ be the orthogonal direct sum of all subspaces of type $2^{*}$ or
type $3$, and so $Y$ has dimension divisible by $4$, and let $X$ be the
orthogonal direct sum of all other subspaces $V_{i}$. If
$g\in\Omega^{\pm}(n,q)$, then the number of subspaces of type $2^{+}$ must be
even, and it follows from the previous two paragraphs that
$g_{X}\in\Omega(X)$. Then we must also have $g_{Y}\in\Omega(Y)$. If
$g\in\mathrm{T}^{\pm}(n,q)\setminus\Omega^{\pm}(n,q)$, then
$g\not\in\mathrm{SO}^{\pm}(n,q)$ and so the number of subspaces of type
$2^{-}$ must be odd. Again, from the above it then follows that
$g_{X}\in\mathrm{T}(X)\setminus\Omega(X)$, and so we must have
$g_{Y}\in\Omega(Y)$. In any case, we have $g_{Y}\in\Omega(Y)$ with
$\mathrm{dim}(Y)$ divisible by $4$ and $q\equiv 1($mod $4)$, and so $g_{Y}$ is
strongly real in $\Omega(Y)$ by Theorem 3.4. We now have $g=g_{X}\oplus g_{Y}$
with $g_{X}$ strongly real in $\mathrm{T}(X)$ and $g_{Y}$ strongly real in
$\Omega(Y)$, so that $g$ is strongly real in $\mathrm{T}^{\pm}(n,q)$. ∎

######  Theorem 3.5.

Let $q\equiv 1(\mathrm{mod}\;4)$ and $m\geq 1$. In either of the groups
$G=\Omega^{+}(4m+2,q)$ or $G/Z=\mathrm{P}\Omega^{+}(4m+2,q)$, every real class
is strongly real.

###### Proof.

From Theorem 3.3 and Proposition 3.2, we have both $\mathrm{K}^{+}(4m+2,q)$
and $\mathrm{T}^{+}(4m+2,q)$ are strongly real. Letting
$g\in\Omega^{+}(4m+2,q)$, we have $g$ can be inverted by an involution
$\sigma\in\mathrm{K}^{+}(4m+2,q)$ and by an involution
$s\in\mathrm{T}^{+}(4m+2,q)$. We may assume
$\sigma,s\not\in\Omega^{+}(4m+2,q)$.

Let $V=\mathbb{F}_{q}^{4m+2}$ and $V=\oplus V_{i}$ be the orthogonal
decomposition of $V$ corresponding to $g$. Suppose that $V$ has a subspace
$V_{j}$ of type $2^{\pm}$. Then $V_{j}$ has odd dimension and
$\det(-I_{V_{j}})=-1$. If $dV_{j}=(\mathbb{F}_{q}^{\times})^{2}$, then by [1,
pgs. 230, 222] we have $\theta(-I_{V_{j}})=(\mathbb{F}_{q}^{\times})^{2}$. If
we write $\sigma=\oplus\sigma_{i}$ with $\sigma_{i}=\sigma_{V_{i}}$, then
define $\sigma^{\prime}=(-\sigma_{j})\oplus\bigoplus_{i\neq j}\sigma_{i}$.
Then we have $\det(\sigma^{\prime})=-\det(\sigma)$ and
$\theta(\sigma^{\prime})=\theta(\sigma)$, so that
$\sigma^{\prime}\in\Omega^{+}(4m+2,q)$ and $\sigma^{\prime}$ is an involution
which inverts $g$, so $g$ is strongly real in $\Omega^{+}(4m+2,q)$. If
$dV_{j}\neq(\mathbb{F}_{q}^{\times})^{2}$, then
$\theta(-I_{V_{j}})\neq(\mathbb{F}_{q}^{\times})^{2}$, again from [1, pgs.
230, 222]. Writing $s=\oplus s_{i}$, we define
$s^{\prime}=(-s_{j})\oplus\bigoplus_{i\neq j}s_{i}$. Now we have
$\det(s^{\prime})=-\det(s)$ and $\theta(s^{\prime})\neq\theta(s)$, so that
$s^{\prime}\in\Omega^{+}(4m+2,q)$ is an inverting involution for $g$. Thus if
$V$ has a subspace of type $2^{\pm}$, then $g$ is strongly real in
$\Omega^{+}(4m+2,q)$.

Now suppose $g\in\Omega^{+}(4m+2,q)$ is a real element in this group. In
particular, $g$ is then real as an element of $\mathrm{SO}^{+}(4m+2,q)$. By
[15, Theorem 7.2], this implies that $g$ has an elementary divisor of the form
$(t\pm 1)^{e}$ with $e$ odd, which means that $V$ has a subspace of type
$2^{\pm}$. Thus $g$ is strongly real in $\Omega^{+}(4m+2,q)$, concluding the
proof of the first claim.

To prove the statement in the group $G/Z=P\Omega^{+}(4m+2,q)$, where $Z=\\{\pm
I\\}$, we must also consider elements $g\in\Omega^{+}(4m+2,q)=G$ such that $g$
is conjugate to $-g^{-1}$ in $G$, but $g$ is not real in $G$ (otherwise we
already know $g$ is strongly real in $G$). Since $g$ and $-g^{-1}$ are
conjugate, then for any elementary divisor $f(t)^{e}$ of $g$, there is also an
elementary divisor $\tilde{f}(t)^{e}$ of $g$ with the same multiplicity, where
if the roots of $f(t)$ in an algebraic closure $\overline{\mathbb{F}}_{q}$ are
given by the Frobenius orbit of $\alpha\in\overline{\mathbb{F}}_{q}^{\times}$,
then the roots of $\tilde{f}(t)$ are given by the Frobenius orbit of
$-\alpha^{-1}$ (noting that the action $\alpha\mapsto\alpha^{q}$ commutes with
$\alpha\mapsto-\alpha^{-1}$).

Continue to write $V=\oplus_{i}V_{i}$ for the orthogonal decomposition of $V$
corresponding to $g$. Since we are assuming $g$ is not real in $G$, then from
the above we may assume $V$ has no subspaces of type $2^{\pm}$. Note also that
subspaces of $V$ which are type $1^{\pm}$ have dimension divisible by 4. We
now consider subspaces $V_{i}$ of type $2^{*}$ or type $3$.

Consider a subspace $V_{i}$ of type $2^{*}$, which then corresponds to an
elementary divisor $f(t)^{e}$ on $V_{i}$, where $f(t)$ is irreducible in
$\mathbb{F}_{q}[t]$ and is self-reciprocal, with $f(t)\neq t\pm 1$. That is,
the set of roots of $f(t)$ in $\overline{\mathbb{F}}_{q}$ is invariant under
the map $\alpha\mapsto\alpha^{-1}$. In particular, $f(t)$ has even degree.
Then also $\tilde{f}(t)^{e}$ is an elementary divisor, where $\tilde{f}(t)$ is
also self-reciprocal, corresponding to a subspace $\tilde{V}_{i}$ of type
$2^{*}$ with the same dimension as $V_{i}$. If $f(t)\neq\tilde{f}(t)$, then
$V_{i}\oplus\tilde{V}_{i}$ has dimension divisible by $4$. If
$f(t)=\tilde{f}(t)$, then the set of roots of $f(t)$ is invariant under both
the maps $\alpha\mapsto\alpha^{-1}$ and $\alpha\mapsto-\alpha^{-1}$, and so is
also invariant under $\alpha\mapsto-\alpha$. This implies the only nonzero
coefficients of $f(t)$ are those of even powers of $t$, so that
$f(t)=f_{1}(t^{2})$ for some irreducible self-reciprocal polynomial
$f_{1}(t)\in\mathbb{F}_{q}[t]$. Also $f_{1}(t)\neq t\pm 1$ since neither of
$t^{2}\pm 1$ are irreducible (since $q\equiv 1($mod $4)$), and so $f_{1}(t)$
must have even degree. Then $f(t)=f_{1}(t^{2})$ has degree divisible by $4$,
and it follows that the direct sum of all subspaces of $V$ of type $2^{*}$ has
dimension divisible by $4$.

Finally, consider a subspace $V_{i}$ of type $3$, so $V_{i}$ is a non-
orthogonal direct sum, $V_{i}=U_{i}\oplus W_{i}$, and $g$ has elementary
divisors $f(t)^{e}$ on $U_{i}$ and $f^{*}(t)^{e}$ on $W_{i}$. If
$\tilde{f}(t)\neq f(t),f^{*}(t)$, then $V$ also has a subspace
$\tilde{V}_{i}=\tilde{U}_{i}\oplus\tilde{W}_{i}$ with elementary divisors
$\tilde{f}(t)^{e}$ and $\tilde{f}^{*}(t)^{e}$. In this case, $V_{i}$ and
$\tilde{V}_{i}$ each have the same even dimension, and
$V_{i}\oplus\tilde{V}_{i}$ has dimension divisible by $4$. If either
$f(t)=\tilde{f}(t)$ or $f^{*}(t)=\tilde{f}(t)$, then the set of roots of
$f(t)$ is invariant under one of the order 2 actions $\alpha\mapsto-\alpha$ or
$\alpha\mapsto-\alpha^{-1}$, and so $f(t)$ must have even degree. Then $U_{i}$
and $W_{i}$ each have even degree, and so $V_{i}$ has dimension divisible by
$4$.

Through exhausting all cases, it follows that if $g$ is not real but $g$ is
conjugate to $-g^{-1}$ in $G$, then $\dim(V)$ is divisible by $4$. This
contradicts the fact that $\dim(V)=4m+2$, and so $G$ has no such elements.
Thus all real classes of $G/Z=\mathrm{P}\Omega^{+}(4m+2,q),\,q\equiv
1(\mathrm{mod}\;4)$, are strongly real. ∎

We now assume $q\equiv 3($mod $4)$ and $m\geq 1$, with $G=\Omega^{-}(4m+2,q)$
and $G/Z=\mathrm{P}\Omega^{-}(4m+2,q)$, and we show both $G$ and $G/Z$ always
have _weakly real_ elements, that is, elements which are real but not strongly
real.

######  Theorem 3.6.

Let $q\equiv 3(\mathrm{mod}\;4)$ and $m\geq 1$. Then
$\mathrm{P}\Omega^{-}(4m+2,q)$ contains weakly real elements.

###### Proof.

In Lemma A.1 of the Appendix, we construct an element $h\in\Omega^{-}(6,q)$
such that $hZ$ is weakly real in $\mathrm{P}\Omega^{-}(6,q)$, where the
elementary divisors of $h$ are $(t-1)^{2},(t-1)^{2},t+1,t+1$. Now consider an
element $\eta\in\Omega^{+}(8,q)$ such that $\eta$ has elementary divisors
$(t^{2}+1)^{2},(t^{2}+1)^{2}$, which exists by [1, Propositions 16.10, 16.30],
and we may define an element $g_{1}\in\Omega^{-}(8l+6,q)$ as a block-diagonal
direct sum as

(3.1) $g_{1}=h\oplus\bigoplus_{i=1}^{l}\eta,$

and we show that $g_{1}Z$ is weakly real in $\mathrm{P}\Omega^{-}(8l+6,q)$.
First, we have $\eta_{l}=\bigoplus_{i=1}^{l}\eta\in\Omega^{+}(8l,q)$, and
since $\Omega^{+}(8,q)$ is strongly real by Theorem 3.4, then $\eta_{l}$ is
strongly real in $\Omega^{+}(8l,q)$. It follows from [1, Proposition 16.34]
that the centralizer of $\eta_{l}$ in $\mathrm{O}^{+}(8l,q)$ is contained
completely in $\Omega^{+}(8l,q)$. Thus any element in $\mathrm{O}^{+}(8l,q)$
which conjugates $\eta_{l}$ to its inverse must be in $\Omega^{+}(8l,q)$. By
[21, Case (C), (iv) of 2.6], since the eigenvalues of $h$ are distinct from
those of $\eta_{l}$, we have

$C_{\mathrm{O}^{-}(8l+6,q)}(g_{1})\cong C_{\mathrm{O}^{-}(6,q)}(h)\times
C_{\mathrm{O}^{+}(8l,q)}(\eta_{l})=C_{\mathrm{O}^{-}(6,q)}(h)\times
C_{\Omega^{+}(8l,q)}(\eta_{l}).$

We therefore have

(3.2) $C_{\Omega^{-}(8l+6,q)}(g_{1})\cong C_{\Omega^{-}(6,q)}(h)\times
C_{\Omega^{+}(8l,q)}(\eta_{l}).$

Next notice that $g_{1}$ is real in $\Omega^{-}(8l+6,q)$ since $h$ is real in
$\Omega^{-}(6,q)$ and $\eta_{l}$ is real in $\Omega^{+}(8l,q)$, and so
$g_{1}Z$ is real in $\mathrm{P}\Omega^{-}(8l+6,q)$. In particular, there is an
element $x_{1}=(x,x_{l})\in\Omega^{-}(6,q)\times\Omega^{+}(8l,q)$ such that
$xhx^{-1}=h^{-1}$ and $x_{l}\eta_{l}x_{l}^{-1}=\eta_{l}^{-1}$, and so
$x_{1}g_{1}x_{1}^{-1}=g_{1}^{-1}$ in $\Omega^{-}(8l+6,q)$. Since any two
elements which conjugate $g_{1}$ to its inverse differ by an element of the
centralizer, then from (3.2) we have every element of $\Omega^{-}(8l+6,q)$
which conjugates $g_{1}$ to its inverse must be of the form
$(x,x_{l})\in\Omega^{-}(6,q)\times\Omega^{+}(8l,q)$. If $g_{1}Z$ is strongly
real in $\mathrm{P}\Omega^{-}(8l+6,q)$, and since the elementary divisors of
$g_{1}$ prevent the possibility that $g_{1}$ is conjugate to $-g_{1}^{-1}$, we
suppose that $x_{1}g_{1}x_{1}^{-1}=g_{1}^{-1}$ with $x_{1}^{2}=\pm I$. Then
$x_{1}=(x,x_{l})$ with $x^{2}=\pm I$ and $xhx^{-1}=h^{-1}$ for
$x\in\Omega^{-}(6,q)$. This contradicts the fact that $hZ$ is weakly real in
$\Omega^{-}(6,q)$, and thus $g_{1}Z$ must be weakly real in
$\mathrm{P}\Omega^{-}(8l+6,q)$.

In Lemma A.3, we construct an element $h_{0}\in\Omega^{-}(10,q)$ such that
$h_{0}Z$ is weakly real in $\mathrm{P}\Omega^{-}(10,q)$, where $h_{0}$ has
elementary divisors

$(t-1)^{3},(t-1)^{3},(t+1)^{2},(t+1)^{2}.$

Then we consider the element
$g_{0}=(h_{0},\eta_{l})\in\Omega^{-}(10,q)\times\Omega^{+}(8l,q)$, so we
repeat the construction in (3.1) with $h$ replaced by $h_{0}$. By repeating
the same argument, we obtain $g_{0}Z$ is weakly real in
$\mathrm{P}\Omega^{-}(8l+10,q)$. This gives that every
$\mathrm{P}\Omega^{-}(4m+2,q)$, $q\equiv 3(\mathrm{mod}\;4)$, $m\geq 1$, has
weakly real elements. ∎

Remark. In the above construction, it would of course be much simpler to just
add a single elementary divisor of the form $(t^{2}+1)^{2}$, so that only one
base case is needed. However, this does not work because such an element is
not real in $\Omega^{+}(4,q)$ when $q\equiv 3($mod $4)$.

##  4\. Characteristic two

We now assume that $q$ is a power of 2, and consider the vector space
$V=\mathbb{F}_{q}^{2n}$ with $Q$ a non-defective quadratic form on $V$, and
$B$ the associated non-degenerate symplectic form on $V$. All non-degenerate
symplectic forms on $V$ are equivalent, and so we may always embed the
orthogonal group $\mathrm{O}^{\pm}(2n,q)$ associated with $Q$ in the
symplectic group $\mathrm{Sp}(2n,q)$ associated with $B$, and there is only
one such symplectic group up to isomorphism. We also recall that given
$g\in\mathrm{O}^{\pm}(2n,q)$, we have $g\in\Omega^{\pm}(2n,q)$ if and only if
$\mathrm{rank}(g+I)$ is even [14, Proposition 3.2].

Given any $g\in\mathrm{Sp}(2n,q)$ (and so any $g\in\Omega^{\pm}(2n,q)$), we
now describe an orthogonal decomposition (with respect to $B$) of
$V=\mathbb{F}_{q}^{2n}$ which is somewhat similar to the case when $q$ is odd.
We follow the description given in [6, Section 1], and these results are due
to Huppert [9]. We say a subspace $W$ of $V$ is symplectically indecomposable
with respect to $g$ if $W$ has no $B$-orthogonal decomposition into non-
trivial subspaces which are $g$-invariant. Then $V$ can be orthogonally
decomposed, as $V=\oplus V_{i}$, into symplectically indecomposable
$g$-invariant subspaces $V_{i}$ which are one of the following forms:

  1. (1)

$V_{i}$ is a (non-orthogonal) direct sum, $V_{i}=U_{i}\oplus W_{i}$, of
degenerate totally isotropic $g$-invariant cyclic subspaces (with
$\dim_{\mathbb{F}_{q}}(U_{i})=\dim_{\mathbb{F}_{q}}(W_{i})$), and $g$ has a
single elementary divisor $(t-1)^{e}$ on both $U_{i}$ and $W_{i}$;

  2. (2)

$V_{i}$ is $g$-cyclic, such that $g$ has a single elementary divisor
$f(t)^{e}$, where $f(t)$ is irreducible and self-reciprocal, and if $f(t)=t-1$
then $e$ is even;

  3. (3)

$V_{i}$ is $g$-cyclic, and $V_{i}$ a (non-orthogonal) direct sum
$V_{i}=U_{i}\oplus W_{i}$ of totally isotropic $g$-invariant subspaces (with
$\dim_{\mathbb{F}_{q}}(U_{i})=\dim_{\mathbb{F}_{q}}(W_{i})$), such that $g$
has a single elementary divisor $f(t)^{e}$ on $U_{i}$, and a single elementary
divisor $f^{*}(t)^{e}$ on $W_{i}$, such that $f(t)$ is irreducible and
$f(t)\neq f^{*}(t)$.

We continue to say in case (1) that $V_{i}$ is _bicyclic_ with respect to $g$.
Note that $e$ can be even or odd in case (1), while $e$ must be even in case
(2) (different from the case when $q$ is odd). Given the decomposition
$V=\oplus V_{i}$, we will also continue to write $\mathrm{O}(V_{i})$ and
$\Omega(V_{i})$ for the groups corresponding to the quadratic form $Q$
restricted to the subspace $V_{i}$.

Rämö proved that every element of $\Omega^{\pm}(4m,q)$ is strongly real by
considering the decomposition $V=\oplus V_{i}$. By applying the work done
there, we obtain the following statement for the group $\Omega^{\pm}(4m+2,q)$.

######  Proposition 4.1.

Let $q$ be a power of $2$, and let $g\in\Omega^{\pm}(4m+2,q)$. Then there
exists an element $h\in\mathrm{O}^{\pm}(4m+2,q)\setminus\Omega^{\pm}(4m+2,q)$
such that $h^{2}=1$ and $hgh^{-1}=g^{-1}$.

###### Proof.

Write $V=\oplus V_{i}$, where each $V_{i}$ is either cyclic or bicyclic,
$\dim(V)=4m+2$, and $g=\oplus g_{i}$, where $g_{i}=g_{V_{i}}$. Note that we
must have each $\dim(V_{i})$ even, since otherwise $V_{i}$ is cyclic of odd
dimension which is not possible. We consider each possibility for $V_{i}$.

First suppose $4|\dim(V_{i})$. If $V_{i}$ is cyclic, then by [14, Proposition
3.3] there is an involution $h_{i}\in\Omega(V_{i})$ which inverts $g_{i}$. If
$V_{i}$ is bicyclic, then by [14, Proposition 3.5] there is an involution
$h_{i}\in\Omega(V_{i})$ which inverts $g_{i}$.

Now suppose $\dim(V_{i})\equiv 2($mod $4)$. If $V_{i}$ is cyclic, then by [14,
Proposition 3.3], there is an involution
$h_{i}\in\mathrm{O}(V_{i})\setminus\Omega(V_{i})$ which inverts $g_{i}$. If
$V_{i}$ is bicyclic, then by [14, Propositions 3.16 and 3.17], there exists an
involution $h_{i}\in\Omega(V_{i})$ and an involution
$h_{i}^{\prime}\in\mathrm{O}(V_{i})\setminus\Omega(V_{i})$, each of which
inverts $g_{i}$.

Since $\dim(V)=4m+2$, then there are an odd number of $V_{i}$’s such that
$\dim(V_{i})\equiv 2($mod $4)$. So, when taking $h=\oplus h_{i}$, we may
choose an odd number of the $h_{i}$’s such that
$\mathrm{rank}(h_{i}+I_{V_{i}})$ is odd. Thus $\mathrm{rank}(h+I)$ is odd, and
so $h$ inverts $g$, $h^{2}=1$, and
$h\in\mathrm{O}^{\pm}(4m+2,q)\setminus\Omega^{\pm}(4m+2,q)$. ∎

We will also need the following statement.

######  Lemma 4.1.

If $q$ is a power of $2$ and $u\in\Omega^{\pm}(4m,q)$ has a single elementary
divisor corresponding to a cyclic space, then $u$ has an inverting involution
$h\in\mathrm{O}^{\pm}(4m,q)\setminus\Omega^{\pm}(4m,q)$.

###### Proof.

This follows directly from the proof of [14, Proposition 3.4], where we get
$a_{n/2}=a_{2m}=\binom{4m}{2m}=0$ in characteristic 2. The rank of the
resulting inverting involution is then odd, and so is in
$\mathrm{O}^{\pm}(4m,q)\setminus\Omega^{\pm}(4m,q)$. ∎

Rämö also proves that for a unipotent cyclic subspace $V_{i}$, if
$\dim(V_{i})\equiv 2($mod $4)$, then there is an inverting involution
$h_{i}^{\prime}\in\Omega(V_{i})$ [14, Proposition 3.4]. This means that the
only case where there may not be an inverting involution in $\Omega(V_{i})$ is
in the case $V_{i}$ is cyclic, non-unipotent, $\dim(V_{i})\equiv 2($mod $4)$,
and $V_{i}$ corresponds to a self-dual elementary divisor $f(t)^{e}$. We must
have $\deg(f(t))$ even since $f(t)$ is self-dual, and so $e$ must be odd. We
are concerned with the case that these have odd multiplicity, that is, when
there are an odd number of such elementary divisors counting multiplicity
(otherwise, we have an even number of
$h_{i}\in\mathrm{O}(V_{i})\setminus\Omega(V_{i})$, and we will get $h=\oplus
h_{i}\in\Omega^{\pm}(n,q)$). These observations motivate the main result of
this section, which we now prove.

######  Theorem 4.1.

Let $q$ be a power of $2$, $V=\mathbb{F}_{q}^{4m+2}$, and
$g\in\Omega^{\pm}(4m+2,q)$. Then $g$ is strongly real in
$\Omega^{\pm}(4m+2,q)$ if and only if either of the following holds:

  1. (i)

There are an even number of elementary divisors (counting multiplicity) of $g$
of the form $f(t)^{e}$ with $e$ odd, $f(t)$ self-reciprocal, and
$\deg(f(t))=4r+2$, or

  2. (ii)

In the orthogonal decomposition $V=\oplus V_{i}$, there exists a $V_{i}$ on
which $g$ has only eigenvalues equal to $1$, other than bicyclic $V_{i}$ with
dimension divisible by $4$.

Moreover, all real classes in $\Omega^{\pm}(4m+2,q)$ are strongly real.

###### Proof.

We first suppose (i) or (ii) holds. Write $V=\oplus V_{i}$ as before, and
consider the cases for $V_{i}$ as in the proof of Proposition 4.1. If (i) is
true, then since $\dim(V)=4m+2$, there is some other $V_{i}$ in the orthogonal
decomposition such that $\dim(V_{i})\equiv 2($mod $4)$. Such a space is either
bicyclic or cyclic and unipotent, in which case we can choose an inverting
involution in the correct coset. If (ii) is true, then we can again choose an
inverting involution in the correct coset, by Lemma 4.1 and [14, Propositions
3.4, 3.16, and 3.17].

For the converse, we assume $g$ does not satisfy (i) or (ii), and we prove
that $g$ is not real in $\Omega^{\pm}(4m+2,q)$, and this also proves the
second claim. First, by Proposition 4.1, there is an inverting involution
$h\in\mathrm{O}^{\pm}(4m+2,q)\setminus\Omega^{\pm}(4m+2,q)$. If
$k\in\mathrm{O}^{\pm}(4m+2,q)$ is any other inverting involution for $h$, then
we must have $k=ha$ where $a\in C_{G}(g)$ and $G=\mathrm{O}^{\pm}(4m+2,q)$.
Now write $V=V_{y}\oplus V_{u}$, where $V_{y}$ is the direct sum of all
$V_{i}$ on which $g$ has eigenvalues different from $1$, and $V_{u}$ is the
direct sum of those on which $g$ has eigenvalues only equal to $1$. Write
$g=g_{y}\oplus g_{u}$, where $g_{y}=g_{V_{y}}$ and $g_{u}=g_{V_{u}}$. From the
description of centralizers given by Wall [21, Section 3.7], it follows that
we have the direct product

(4.1) $C_{G}(g)=C_{\mathrm{O}(V_{y})}(g_{y})\times
C_{\mathrm{O}(V_{u})}(g_{u}).$

Since $g$ does not satisfy condition (ii), then neither does $g_{u}$. This
implies that $g_{u}$ is an _exceptional_ unipotent element of
$\mathrm{O}(V_{u})$, as defined in [4, pg. 2547]. It then follows from [4,
Theorem 2.5(ii)(a)] that $C_{\mathrm{O}(V_{u})}(g_{u})\subset\Omega(V_{u})$,
and so the conjugacy class of $g_{u}$ in $\mathrm{O}(V_{u})$ splits into two
conjugacy classes in $\Omega(V_{u})$. From (4.1), the conjugacy class of
$g=g_{y}\oplus g_{u}$ in $G$ must also split into two conjugacy classes in
$\Omega^{\pm}(4m+2,q)$, and so

$C_{G}(g)\subset\Omega^{\pm}(4m+2,q).$

Since $h\in G\setminus\Omega^{\pm}(4m+2,q),$ and
$C_{G}(g)\subset\Omega^{\pm}(4m+2,q)$, then every inverting element for $g$ in
$\Omega^{\pm}(4m+2,q)$ must be in the outer coset $h\Omega^{\pm}(4m+2,q)$.
Thus $g$ is not real in $\Omega^{\pm}(4m+2,q)$, giving the claim. ∎

##  5\. Real-valued characters

In this section, we give some results on real-valued irreducible complex
characters of finite simple orthogonal groups, and compare them to the known
reality results of their conjugacy classes.

Given a finite group $G$, we let $\mathrm{Irr}(G)$ denote the collection of
complex irreducible characters of $G$, so that each $\chi\in\mathrm{Irr}(G)$
is afforded by a complex representation $(\pi,W)$ of $G$. Given a basis
$\mathcal{B}$ of $W$, let $[\pi]=[\pi]_{\mathcal{B}}$ denote the associated
matrix representation. We say that $(\pi,W)$ is a real representation if there
is a basis $\mathcal{B}$ of $W$ such that $[\pi(g)]$ has all real entries for
all $g\in G$. The Frobenius-Schur indicator of $\chi\in\mathrm{Irr}(G)$, which
we denote by $\varepsilon(\chi)$, is given by the formula

$\varepsilon(\chi)=\frac{1}{|G|}\sum_{g\in G}\chi(g^{2}),$

and takes the values $\varepsilon(\chi)=0$ if $\chi$ is not real-valued,
$\varepsilon(\chi)=-1$ if $\chi$ is real-valued but $(\pi,W)$ is not a real
representation, and $\varepsilon(\chi)=1$ if $(\pi,W)$ is a real
representation (see [10, Chapter 4]). In particular, because the number of
real conjugacy classes of $G$ is equal to the number of real-valued
irreducible complex characters of $G$, then $\varepsilon(\chi)=\pm 1$ for all
$\chi\in\mathrm{Irr}(G)$ if and only if $G$ is a real group.

There is a twisted variation of the Frobenius-Schur indicator, due to N.
Kawanaka and H. Matsuyama [11], which we will need here. Let $\iota$ be an
automorphism of the group $G$, such that $\iota^{2}=1$, and let
${{}^{\iota}\chi}$ denote the character defined by
${{}^{\iota}\chi}(g)=\chi(\iota(g))$. If we define $\varepsilon_{\iota}(\chi)$
by

$\varepsilon_{\iota}(\chi)=\frac{1}{|G|}\sum_{g\in G}\chi(g\cdot\iota(g)),$

then $\varepsilon_{\iota}(\chi)$ takes only the values $1,-1,0$ for
$\chi\in\mathrm{Irr}(G)$, and $\varepsilon_{\iota}(\chi)=\pm 1$ if and only if
${{}^{\iota}\chi}=\overline{\chi}$. This twisted variant generalizes the
Frobenius-Schur indicator, and has many other interesting properties, but we
do not need them here. We have the following, which is only a slight variation
of [16, Lemma 2.3(i)].

######  Lemma 5.1.

Suppose $H$ and $G$ are finite groups with $H$ an index two subgroup of $G$
such that $G=\langle H,s\rangle$ with $s^{2}=1$. If $\varepsilon(\chi)=1$ for
all $\chi\in\mathrm{Irr}(G)$, then $\varepsilon(\psi)\geq 0$ for all
$\psi\in\mathrm{Irr}(H)$. If $H$ is a real group, then $\varepsilon(\psi)=1$
for all $\psi\in\mathrm{Irr}(H)$.

###### Proof.

We define $\iota$ on $H$ by ${{}^{\iota}g}=sgs^{-1}$. Given any
$\psi\in\mathrm{Irr}(H)$, the induced character $\psi^{G}$ is either
irreducible or the sum of two distinct irreducibles of $G$. As in the proof of
[16, Lemma 2.3], if $\psi^{H}=\chi$ irreducible, we have
$\varepsilon(\chi)=\varepsilon(\psi)+\varepsilon_{\iota}(\psi)$, while if
$\psi^{H}=\chi_{1}+\chi_{2}$ with $\chi_{1},\chi_{2}\in\mathrm{Irr}(G)$, we
have

$\varepsilon(\chi_{1})+\varepsilon(\chi_{2})=\varepsilon(\psi)+\varepsilon_{\iota}(\psi)\;\text{
with }\;\varepsilon(\chi_{1})=\varepsilon(\chi_{2}).$

Since $\varepsilon(\chi)=1$ for all $\chi\in\mathrm{Irr}(G)$, and
$\varepsilon(\psi)$, $\varepsilon_{\iota}(\psi)$ can only take the values $0$
or $\pm 1$, then in the first case we must either have $\varepsilon(\psi)=1$
and $\varepsilon_{\iota}(\psi)=0$, or $\varepsilon(\psi)=0$ and
$\varepsilon_{\iota}(\psi)=1$. In the second case, we must have
$\varepsilon(\psi)=\varepsilon_{\iota}(\psi)=1$. Thus $\varepsilon(\psi)\geq
0$ for all $\psi\in\mathrm{Irr}(H)$ as claimed. Further, if $H$ is a real
group, then $\varepsilon(\psi)=\pm 1$ for all $\psi$, and so we must have
$\varepsilon(\psi)=1$ for all $\psi\in\mathrm{Irr}(H)$. ∎

We may now obtain the following result on the Frobenius-Schur indicators of
the finite simple orthogonal groups.

######  Theorem 5.1.

Let $q$ be any prime power, $n\geq 1$, and $H=\Omega^{\pm}(n,q)$, other than
the case $H=\Omega^{-}(4m+2,q)$ with $q\equiv 3($mod $4)$. Then for any
$\psi\in\mathrm{Irr}(H)$, we have $\varepsilon(\psi)\geq 0$. That is, if
$\psi$ is a real-valued irreducible character of $H$, then $\psi$ is afforded
by a real representation. The same statement holds for the groups
$\mathrm{P}\Omega^{\pm}(n,q)$, other than $\mathrm{P}\Omega^{-}(4m+2,q)$ with
$q\equiv 3($mod $4)$.

###### Proof.

If $q$ is a power of $2$, then this statement is proven in [20, Theorems 8.3
and 8.6] and in [19, Theorem 4.2] (noting that
$\mathrm{SO}(2m+1,q)\cong\mathrm{Sp}(2m,q)$ when $q$ is a power of $2$). Thus
we now assume that $q$ is the power of an odd prime.

For any $\theta\in\mathrm{Irr}(\mathrm{O}^{\pm}(n,q))$, we have
$\varepsilon(\theta)=1$ by a result of Gow [7, Theorem 1]. For each case, we
consider an index 2 subgroup $G$ of $\mathrm{O}^{\pm}(n,q)$ such that $G$ is
strongly real. If $n\neq 4m+2$, then we can take $G=\mathrm{SO}^{\pm}(n,q)$ by
Threorem 3.2. When $n=4m+2$, then we may take $G=K^{\pm}(4m+2,q)$ by parts (i)
and (ii) of Theorem 3.3. In particular, for each of these cases $G$ is a real
group and $\mathrm{O}^{\pm}(n,q)=\langle G,t\rangle$ with $t$ any involution
in $\mathrm{O}^{\pm}(n,q)\setminus G$. It follows from the second statement in
Lemma 5.1 that $\varepsilon(\chi)=1$ for all $\chi\in\mathrm{Irr}(G)$. Now
$[G:H]=2$, and $G=\langle H,s\rangle$ with $s$ any involution in $G\setminus
H$. It now follows from the first statement in Lemma 5.1 that
$\varepsilon(\psi)\geq 0$ for all $\psi\in\mathrm{Irr}(H)$. Note that in the
case that $H=\Omega^{\pm}(n,q)$ is itself a real group, then we have
$\varepsilon(\psi)=\pm 1$ for all $\psi\in\mathrm{Irr}(H)$, and so it follows
that $\varepsilon(\psi)=1$ for all $\psi\in\mathrm{Irr}(H)$. This is the case
for the groups given in Theorem 3.4.

Now consider the group $\tilde{H}=P\Omega^{\pm}(n,q)=H/Z$, where $Z=Z(H)$ is
the center of $H$. Given any $\omega\in\mathrm{Irr}(\tilde{H})$, we may define
$\psi$ of $H$ by $\psi(h)=\omega(hZ)$. It follows from a direct computation
that $\langle\psi,\psi\rangle=\langle\omega,\omega\rangle=1$, and so
$\psi\in\mathrm{Irr}(H)$. Then we may compute

$\varepsilon(\omega)=\frac{|Z|}{|H|}\sum_{hZ\in\tilde{H}}\omega(h^{2}Z)=\frac{|Z|}{|H|}\sum_{hZ\in\tilde{H}}\psi(h^{2})=\frac{1}{|H|}\sum_{h\in
H}\psi(h^{2})=\varepsilon(\psi).$

The result now follows, since $\varepsilon(\psi)\geq 0$. ∎

We cannot make the same conclusion for the group $H=\Omega^{-}(4m+2,q)$ with
$q\equiv 3($mod $4)$, where to do so it would be enough to show that $H$ is
contained in an index 2 subgroup of $\mathrm{O}^{-}(4m+2,q)$ which is real
(but not necessarily strongly real). By Theorem 3.2, such an index 2 subgroup
cannot be $\mathrm{SO}^{-}(4m+2,q)$. In fact, one can show that both of the
groups $\mathrm{K}^{-}(4m+2,q)$ and $\mathrm{T}^{-}(4m+2,q)$, with $q\equiv
3($mod $4)$, contain elements which are not real. That is, in the exceptional
case $H=\Omega^{-}(4m+2,q)$ with $q\equiv 3($mod $4)$, $H$ is not contained in
any real index two subgroup of $\mathrm{O}^{-}(4m+2,q)$. We could also make
this conclusion from the above if we construct some
$\chi\in\mathrm{Irr}(\Omega^{-}(4m+2,q))$, $q\equiv 3($mod $4)$, such that
$\varepsilon(\chi)=-1$. By [12, Proposition 2.9.1(vii)], we have
$\mathrm{P}\Omega^{-}(6,q)\cong\mathrm{PSU}(4,q)$, and the character table for
$\mathrm{PSU}(4,3)$ reveals the existence of characters of
$\mathrm{P}\Omega^{-}(6,3)$ which indeed have Frobenius-Schur indicator $-1$
[2, pg. 54].

Given the results obtained here, we expect that for any values of $n$ and $q$,
all real classes of $\Omega^{\pm}(n,q)$ or $\mathrm{P}\Omega^{\pm}(n,q)$ are
strongly real if and only if all real-valued irreducible complex characters
have Frobenius-Schur indicator equal to 1. We hope to complete the proof of
this statement in the sequel, by constructing irreducible complex characters
with Frobenius-Schur indicator $-1$ of $\Omega^{-}(4m+2,q)$ and
$\mathrm{P}\Omega^{-}(4m+2,q)$ with $q\equiv 3($mod $4)$, and showing all real
classes are strongly real in the groups $\Omega(2n+1,q)$, $\Omega^{+}(4m,q)$,
and $\mathrm{P}\Omega^{+}(4m,q)$ with $q\equiv 3($mod $4)$.

##  Appendix A Weakly real elements

Here we construct elements $h\in\Omega^{-}(6,q)$ and
$h_{0}\in\Omega^{-}(10,q)$, with $q\equiv 3(\mathrm{mod}\;4)$, such that $hZ$
is weakly real in $\mathrm{P}\Omega^{-}(6,q)$ and $h_{0}Z$ is weakly real in
$\mathrm{P}\Omega^{-}(10,q)$.

We define the matrix $J_{n}$ to have 1’s on the main antidiagonal and 0’s
elsewhere, so

$J_{n}=\left(\begin{array}[]{rrr}&&1\\\ &\mathinner{\mkern 2.0mu\raise
0.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 1.0mu\raise
7.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 1.0mu}&\\\ 1&&\end{array}\right).$

We first consider an element with elementary divisors $(t-1)^{2},(t-1)^{2}$,
given by

$u=\left(\begin{array}[]{rrrr}1&-1&&\\\ &1&&\\\ &&1&1\\\
&&&1\end{array}\right).$

If we define a symmetric form on $\mathbb{F}_{q}^{4}$ using the Gram matrix
$J_{4}$, then this is a split form, and by [1, Proposition 16.30] we have
$u\in\Omega^{+}(4,q)$. It follows from [1, Proposition 16.34] that the
centralizer of $u$ in $\mathrm{O}^{+}(4,q)$ is contained completely in
$\Omega^{+}(4,q)$. Now note that the element

$s_{0}=\left(\begin{array}[]{cc}0&I\\\ I&0\end{array}\right)$

satisfies $s_{0}\in\mathrm{SO}^{+}(4,q)$ and $sus^{-1}=u^{-1}$. The
$-1$-eigenspace of $s_{0}$ is spanned by $(1,0,-1,0)^{\top}$ and
$(0,1,0,-1)^{\top}$, and restricting our symmetric form defined by $J_{4}$ to
this subspace, with these vectors as a basis, gives that the symmetric form on
this subspace has matrix $\left(\begin{array}[]{rr}0&-2\\\
-2&0\end{array}\right)$. Since this form has discriminant a non-square (since
$-4$ is not a square when $q\equiv 3(\mathrm{mod}\;4)$), then this is a split
form on the $-1$-eigenspace. By [1, Proposition 16.30], we have
$s_{0}\not\in\Omega^{+}(4,q)$. Since one element in $\mathrm{SO}^{+}(4,q)$
which conjugates $u$ to its inverse is not in $\Omega^{+}(4,q)$ and the
centralizer of $u$ in $\mathrm{SO}^{+}(4,q)$ is contained in
$\Omega^{+}(4,q)$, then every element of $\mathrm{SO}^{+}(4,q)$ which
conjugates $u$ to its inverse must be in
$\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q)$.

We also claim that $u$ is not conjugated to its inverse by any element of
$\mathrm{O}^{+}(4,q)$ which squares to $-I$. By direct computation, such an
element would have to be of the form (just from it conjugating $u$ to its
inverse):

$\left(\begin{array}[]{rrrr}a&b&c&d\\\ 0&-a&0&c\\\ c_{1}&d_{1}&a_{1}&b_{1}\\\
0&c_{1}&0&-a_{1}\end{array}\right),$

where $a^{2}+cc_{1}=-1$ and $ac+ca_{1}=0$, from the assumption that this
squares to $-I$. We cannot have $c=0$, since otherwise $a^{2}=-1$ with
$q\equiv 3(\mathrm{mod}\;4)$, and so we have $a_{1}=-a$. Using the fact that
this matrix must preserve the form defined by $J_{4}$, we obtain
$cc_{1}-aa_{1}=1$, or $a^{2}+cc_{1}=1$. This is a contradiction to
$a^{2}+cc_{1}=-1$, and so such an element cannot exist.

We now consider the symmetric form defined by the Gram matrix

$\left(\begin{array}[]{rr}J_{4}&\\\ &I_{2}\end{array}\right),$

which defines a non-split form on $\mathbb{F}_{q}^{6}$ when $q\equiv
3(\mathrm{mod}\;4)$, and we consider the element

$h=\left(\begin{array}[]{rr}u&\\\ &-I_{2}\end{array}\right).$

Then $h$ has elementary divisors $(t-1)^{2},(t-1)^{2},t+1,t+1$, and
$h\in\Omega^{-}(6,q)$ by [1, Proposition 16.30].

######  Lemma A.1.

The element $h\in\Omega^{-}(6,q)$ is such that $hZ$ is weakly real in
$\mathrm{P}\Omega^{-}(6,q)$ when $q\equiv 3(\mathrm{mod}\;4)$.

###### Proof.

First, take any element
$x_{0}\in\mathrm{SO}^{-}(2,q)\setminus\Omega^{-}(2,q)$, and take the element
$s_{0}\in\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q)$ defined above. Then we
have $\left(\begin{array}[]{ll}s_{0}&\\\
&x_{0}\end{array}\right)\in\Omega^{-}(6,q)$, and this element conjugates $h$
to its inverse. Thus $h$ is real in $\Omega^{-}(6,q)$ and $hZ$ is real in
$\mathrm{P}\Omega^{-}(6,q)$.

To prove that $hZ$ is not strongly real in $\mathrm{P}\Omega^{-}(6,q)$, we
first note that $h$ cannot be conjugate to $-h^{-1}$ since these two elements
have different sets of elementary divisors. So we consider only the
possibility that $shs^{-1}=h^{-1}$ with $s^{2}=\pm I$. Since
$u\in\Omega^{+}(4,q)$ and $-I\in\Omega^{-}(2,q)$ have distinct eigenvalues,
then it follows from [21, Case (C), (iv) of 2.6] that

$C_{\mathrm{O}^{-}(6,q)}(h)=C_{\mathrm{O}^{+}(4,q)}(u)\times
C_{\mathrm{O}^{-}(2,q)}(-I)=C_{\Omega^{+}(4,q)}\times\mathrm{O}^{-}(2,q),$

since the centralizer of $u$ in $\mathrm{O}^{+}(4,q)$ is completely contained
in $\Omega^{+}(4,q)$ (mentioned above). That is, we have

$C_{\Omega^{-}(6,q)}(h)=C_{\Omega^{+}(4,q)}(u)\times\Omega^{-}(2,q).$

We now have the element

$(s_{0},x_{0})\in(\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q))\times(\mathrm{SO}^{-}(2,q)\setminus\Omega^{-}(2,q)),$

which conjugates $h$ to its inverse, and every such element in
$\Omega^{-}(6,q)$ must be a product of this with an element of the centralizer
of $h$ in $\Omega^{-}(6,q)$. So any element $s\in\Omega^{-}(6,q)$ such that
$shs^{-1}=h^{-1}$ and $s^{2}=\pm I$ must be of the form $s=(\sigma,\tau)$
where $\sigma\in\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q)$,
$\tau\in\mathrm{SO}^{-}(2,q)\setminus\Omega^{-}(2,q)$, $\sigma^{2}=\pm I$, and
$\tau^{2}=\pm I$. As proven above, there is no element of
$\mathrm{O}^{+}(4,q)$ which conjugates $u$ to its inverse which squares to
$-I$, and so we must assume $\sigma^{2}=I$ and $\tau^{2}=I$. However,
$\mathrm{SO}^{-}(2,q)$ is cyclic of order $q+1$, and so the only elements
$\tau$ in $\mathrm{SO}^{-}(2,q)$ which square to $I$ are $\pm I$. But $\pm I$
are both elements of $\Omega^{-}(2,q)$, contradicting that such an element
would have to come from the other coset in $\mathrm{SO}^{-}(2,q)$. Thus no
such $s\in\Omega^{-}(6,q)$ exists, and we have $hZ$ is weakly real in
$\mathrm{P}\Omega^{-}(6,q)$. ∎

We next consider the symmetric form on $\mathbb{F}_{q}^{6}$ defined by the
Gram matrix

$\left(\begin{array}[]{ll}J_{3}&\\\ &J_{3}\end{array}\right),$

which is a non-split form when $q\equiv 3(\mathrm{mod}\;4)$, and we consider
the element

$u_{1}=\left(\begin{array}[]{rrrrrr}1&-1&\gamma&&&\\\ &1&1&&&\\\ &&1&&&\\\
&&&1&-1&\gamma\\\ &&&&1&1\\\ &&&&&1\end{array}\right),$

where $\gamma\in\mathbb{F}_{q}$ such that $2\gamma+1=0$ (so if $q$ is a power
of $p$, we can take $\gamma=(p-1)/2$). Then $u_{1}$ preserves the non-split
form above, has elementary divisors $(t-1)^{3},(t-1)^{3}$, and since $u_{1}$
is unipotent then $u_{1}\in\Omega^{-}(6,q)$. We need the following.

######  Lemma A.2.

Any involution in $\mathrm{SO}^{-}(6,q)$ which conjugates the element $u_{1}$
to its inverse, must be an element of $\Omega^{-}(6,q)$.

###### Proof.

A somewhat tedious but straightforward calculation reveals that any orthogonal
involution which conjugates $u_{1}$ to its inverse, using the above Gram
matrix, must be of the form

$\left(\begin{array}[]{rrrrrr}a&a_{1}&a_{2}&c&c_{1}&c_{2}\\\
&-a&a_{1}&&-c&c_{1}\\\ &&a&&&c\\\ c&c_{1}&c_{2}&b&b_{1}&b_{2}\\\
&-c&c_{1}&&-b&b_{1}\\\ &&c&&&b\end{array}\right),$

such that $a^{2}=b^{2}$, $a^{2}+c^{2}=1$, and $c(a+b)=0$, among other required
relations. Thus either $c=0$ or $a=-b$. The determinant of this matrix is
$(ab-c^{2})^{3}$, and so for this to be in $\mathrm{SO}^{-}(6,q)$ we must have
$(ab-c^{2})^{3}=1$. If $c\neq 0$, then $a=-b$, which gives
$(ab-c^{2})^{3}=(-a^{2}-c^{2})^{3}=-1$. Thus we must have $c=0$ and
$a^{3}b^{3}=1$, and since then $a^{2}=b^{2}=1$, we must have $a=b=\pm 1$. It
follows that an involution in $\mathrm{SO}^{-}(6,q)$ which conjugates $u_{1}$
to its inverse must be either of the form

$\left(\begin{array}[]{rrrrrr}1&a_{1}&a_{2}&&c_{1}&c_{2}\\\
&-1&a_{1}&&&c_{1}\\\ &&1&&&\\\ &c_{1}&c_{2}&1&b_{1}&b_{2}\\\
&&c_{1}&&-1&b_{1}\\\ &&&&&1\end{array}\right),$

such that
$a_{1}^{2}+c_{1}^{2}+2a_{2}=b_{1}^{2}+c_{1}^{2}+2b_{2}=c_{1}(a_{1}+b_{1})+2c_{2}=0$,
or of the form

$\left(\begin{array}[]{rrrrrr}-1&a_{1}&a_{2}&&c_{1}&c_{2}\\\
&1&a_{1}&&&c_{1}\\\ &&-1&&&\\\ &c_{1}&c_{2}&-1&b_{1}&b_{2}\\\
&&c_{1}&&1&b_{1}\\\ &&&&&-1\end{array}\right),$

such that
$a_{1}^{2}+c_{1}^{2}-2a_{2}=b_{1}^{2}+c_{1}^{2}-2b_{2}=c_{1}(a_{1}+b_{1})-2c_{2}=0$.

In order to determine whether these involutions are elements of
$\Omega^{-}(6,q)$, we must calculate whether the restriction of the symmetric
form to the $-1$-eigenspace of the involution is split or non-split. In the
first case above, the $-1$-eigenspace is 2-dimensional and spanned by the
vectors

$(-a_{1}/2,1,0,-c_{1}/2,0,0)^{\top},(-c_{1}/2,0,0,-b_{1}/2,1,0)^{\top}.$

In terms of this ordered eigenbasis, the matrix for the symmetric form
restricted to this eigenspace is $I_{2}$, which is a non-split form by [1, pg.
222], and so this involution is an element of $\Omega^{-}(6,q)$ by [1,
Proposition 16.30]. In the second case above, the $-1$-eigenspace of the
involution is spanned by the vectors $(1,0,0,0,0,0)^{\top}$,
$(0,-a_{1}/2,1,0,-c_{1}/2,0)^{\top}$, $(0,0,0,1,0,0)^{\top}$, and
$(0,-c_{1}/2,0,0,-b_{1}/2,1)^{\top}$. When restricting our symmetric form to
this eigenspace, then with respect to this ordered basis, the restricted
symmetric form has Gram matrix

$\left(\begin{array}[]{cccc}0&1&0&0\\\
1&(a_{1}^{2}+c_{1}^{2})/4&0&c_{1}(a_{1}+b_{1})/4\\\ 0&0&0&1\\\
0&c_{1}(a_{1}+b_{1})/4&1&(b_{1}^{2}+c_{1}^{2})/4\end{array}\right),$

which has determinant 1. It follows this is a split form by [1, pg. 222], and
the involution is again an element of $\Omega^{-}(6,q)$ by [1, Proposition
16.30], giving the result. ∎

Remark.  The centralizer of $u_{1}$ in $\mathrm{SO}^{-}(6,q)$ contains
elements both in $\Omega^{-}(6,q)$ and its other coset, and so while there are
elements outside of $\Omega^{-}(6,q)$ which conjugate $u_{1}$ to its inverse,
there are no such involutions.

Finally, we consider the element $h_{0}=\left(\begin{array}[]{rr}u_{1}&\\\
&-u\end{array}\right)$, where $u$ is the element of $\Omega^{+}(4,q)$
described above. Then $h_{0}$ is an element of $\Omega^{-}(10,q)$ with respect
to the symmetric form defined by the Gram matrix

$\left(\begin{array}[]{ccc}J_{3}&&\\\ &J_{3}&\\\ &&J_{4}\end{array}\right),$

and $h_{0}$ has elementary divisors $(t-1)^{3},(t-1)^{3},(t+1)^{2},(t+1)^{2}$.

######  Lemma A.3.

The element $h_{0}\in\Omega^{-}(10,q)$ is such that $h_{0}Z$ is weakly real in
$\mathrm{P}\Omega^{-}(10,q)$ when $q\equiv 3(\mathrm{mod}\;4)$.

###### Proof.

It was shown above that the element $u$ is conjugated to its inverse by
elements in $\mathrm{SO}^{+}(4,q)$ which are not elements of
$\Omega^{+}(4,q)$, and that $u$ is not conjugated to its inverse by any
element in $\mathrm{O}^{+}(4,q)$ which squares to $-I$. Thus both of these
statements hold for $-u$ as well.

As mentioned above, the centralizer of $u_{1}$ contains elements in
$\mathrm{SO}^{-}(6,q)$ which are not in $\Omega^{-}(6,q)$, which follows from
[1, Proposition 16.34]. Taking any element of $\Omega^{-}(6,q)$ which
conjugates $u_{1}$ to its inverse (for example, any of the involutions
described above), and multiplying it by an element of the centralizer of
$u_{1}$ from $\mathrm{SO}^{-}(6,q)\setminus\Omega^{-}(6,q)$, yields an element
$y_{1}$ from that outer coset which conjugates $u_{1}$ to its inverse. We have
already found an involution $s_{0}$ from
$\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q)$ which conjugates $-u$ to its
inverse, and so now the block diagonal element
$(y_{1},s_{0})\in\Omega^{-}(10,q)$ conjugates $h_{0}=(u_{1},-u)$ to its
inverse. Thus $h_{0}Z$ is real in $\mathrm{P}\Omega^{-}(10,q)$. Next we show
that $h_{0}Z$ is not strongly real in $\mathrm{P}\Omega^{-}(10,q)$, and note
that we cannot have $h_{0}$ conjugate to $-h_{0}^{-1}$ since these elements
have different sets of elementary divisors. So we must show there is no
$s\in\Omega^{-}(10,q)$ such that $sh_{0}s^{-1}=h^{-1}$ and $s^{2}=\pm I$.

Since $u_{1}$ and $-u$ have distinct eigenvalues, and since the centralizer of
$-u$ in $\mathrm{O}^{+}(4,q)$ is completely contained in $\Omega^{+}(4,q)$ by
[1, Proposition 16.34], then it follows from [21, Case (C), (iv) of 2.6] that
we have

$C_{\Omega^{-}(10,q)}(h_{0})=C_{\Omega^{-}(6,q)}(u_{1})\times
C_{\Omega^{+}(4,q)}(-u).$

Since there is an element

$(y_{1},s_{0})\in(\mathrm{SO}^{-}(6,q)\setminus\Omega^{-}(6,q))\times(\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q)),$

such that $y_{1}$ conjugates $u_{1}$ to its inverse and $s_{0}$ conjugates
$-u$ to its inverse, then from the structure of the centralizer of $h_{0}$,
any element of $\Omega^{-}(10,q)$ which conjugates $h_{0}$ to its inverse must
take this form. However, there is no element in
$\mathrm{SO}^{+}(4,q)\setminus\Omega^{+}(4,q)$ which conjugates $-u$ to its
inverse and which squares to $-I$, so there is no such conjugating element for
$h_{0}$ in $\Omega^{-}(10,q)$. Since there is no involution in
$\mathrm{SO}^{-}(6,q)\setminus\Omega^{-}(6,q)$ which conjugates $u_{1}$ to its
inverse by Lemma A.2, then there can be no involution in $\Omega^{-}(10,q)$
which conjugates $h_{0}$ to its inverse. Thus $h_{0}Z$ is not strongly real in
$\mathrm{P}\Omega^{-}(10,q)$. ∎

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