# A note on adjoint reality in simple complex Lie algebras

Krishnendu Gongopadhyay and Chandan Maity  Indian Institute of Science
Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar
140306, Punjab, India<EMAIL_ADDRESS><EMAIL_ADDRESS>Indian Institute of Science Education and Research (IISER) Berhampur, Odisha,
India<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. In [GM], an
infinitesimal version of the notion of classical reality, namely adjoint
reality, has been introduced. An element $X\in\mathfrak{g}$ is adjoint real if
$-X$ belongs to the adjoint orbit of $X$ in $\mathfrak{g}$. In this paper, we
investigate the adjoint real and the strongly adjoint real semisimple elements
in complex simple classical Lie algebras. We also prove that every element in
a complex symplectic Lie algebra is adjoint real.

###### Key words and phrases:

Adjoint orbits, real element, Strongly adjoint real element, semisimple
element

###### 2020 Mathematics Subject Classification:

Primary 20E45; Secondary: 22E60, 17B08

ORCID ID: K. Gongopadhyay (0000-0003-4327-0660); C. Maity
(0000-0003-0673-3953)

##  1\. Introduction

In the group theoretical set-up an element $g$ in a group $G$ is called _real_
or _reversible_ if it is conjugate to $g^{-1}$ in $G$. An element $g$ is
_strongly real_ or _strongly reversible_ in $G$ if it is conjugate to $g^{-1}$
by an involution. Classification of real and strongly real elements in a group
is a problem of wide interest; see [OS], [ST].

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Consider the natural
Ad$(G)$-representation of $G$ on its Lie algebra $\mathfrak{g}$

${\rm Ad}\colon G\longrightarrow{\rm GL}(\mathfrak{g})\,.$

For $X\in\mathfrak{g}$, the adjoint orbit of $X$ in $\mathfrak{g}$ is defined
as $\mathcal{O}_{X}:=\\{{\rm Ad}(g)X\mid g\in G\\}$. If $X\in\mathfrak{g}$ is
semisimple, then $\mathcal{O}_{X}$ is called semisimple orbit. Understanding
the adjoint orbits in a semisimple Lie group has been an intense area of
research, cf. [CM], [Mc]. For various results related to semisimple orbits,
see [CM, Chapter 2].

In [GM], the authors introduced the notion of adjoint reality which we recall
now. Consider the Ad$(G)$-representation of $G$ on $\mathfrak{g}$. For a
linear Lie group $G$, ${\rm Ad}(g)X=gXg^{-1}$.

######  Definition 1.1 ([GM, Definition 1.1]).

An element $X\in\mathfrak{g}$ is called AdG-real  if $-X\in\mathcal{O}_{X}$.
An AdG-real element is called strongly AdG-real if $-X={\rm Ad}(\tau)X$ for
some $\tau\in G$ so that $\tau^{2}={\rm Id}$.

This is an infinitesimal analogue of the reality in Lie groups. It was shown
in [GM] that the reality of the unipotent elements in a Lie group and the
${\rm Ad}_{G}$-reality of the nilpotent elements in the corresponding Lie
algebras are equivalent via the exponential map. This correspondence was used
to classify unipotent real elements in classical Lie groups in [GM]. However,
this correspondence does not necessarily hold in general. Nevertheless,
classifying the adjoint reality in Lie agebras is a problem of independent
algebraic interest, and it also helps to understand the real and strongly real
elements in the image of the exponential map, thus providing understanding of
reality in the Lie group to a large extent. The above notion of adjoint
reality has turned useful in classifying the strongly real elements in ${\rm
GL}_{n}(\mathbb{H})$, ${\rm GL}_{n}(\mathbb{D})\ltimes\mathbb{D}^{n}$ for
$\mathbb{D}=\mathbb{R},~{}\mathbb{C}$ or $\mathbb{H}$, respectively, cf.
[GLM1], [GLM2].

With this motivation, it is a natural problem to investigate the adjoint
reality for semisimple orbits. The aim of this note is to classify the adjoint
real and strongly adjoint real semisimple elements in the complex simple
classical Lie algebras; see Theorem 3.5, Theorem 3.9, Theorem 3.12. The
adjoint real nilpotent elements in these Lie algebras are classified in [GM].

By the Jordan decomposition, every element in a semisimple Lie algebra
decomposes as a unique sum of a semisimple and a unipotent element. Thus,
classifying an arbitrary adjoint real element in a Lie algebra is intimately
related to such classification of semisimple and nilpotent elements. We
demonstrate this for the symplectic Lie algebras, i.e. semisimple Lie algebras
of type $C_{n}$. Recall that symplectic group plays an vital role in many
branch of Mathematics. Thus the characterisation of adjoint real elements in
the symplectic Lie algebra are fundamentally important which is done in
Theorem 4.2 by using description of the centralizers. We note here that
classifying strongly real elements using this idea might require further
technicalities as we have seen for the type $A_{n}$ Lie algebras in [GLM3].
Other than Lie algebras of type $A_{n}$ and $C_{n}$, adjoint reality for
arbitrary elements in other semisimple Lie algebras are yet to be fully
understood.

Given $X\in\mathfrak{g}$ one defines the following subsets of $G$. The
centralizer and the reverser  of an element $X$ in $G$ are respectively
defined as

${Z}_{G}(X):=\\{s\in G\mid sXs^{-1}=X\\},\hbox{ and \ }{R}_{G}(X):=\\{r\in
G\mid rXr^{-1}=-X\\}.$

Note that ${Z}_{G}(X)$ is a subgroup but the set ${R}_{G}(X)$ is a right coset
of the centralizer ${Z}_{G}(X)$. Thus the reversing symmetry group or the
extended centralizer ${E}_{G}(X):={Z}_{G}(X)\cup{R}_{G}(X)$ is a subgroup of
$G$ in which ${Z}_{G}(X)$ has index $1$ or $2$. The group ${E}_{G}(X)$ is an
extension of ${Z}_{G}(X)$ of degree at most two. In the group theoretical set-
up we refer to [OS, §2.1.4], [BR] on reversing symmetries for groups.

To find the reversing symmetric group ${E}_{G}(X)$, it is enough to construct
one reversing element which is not in the centralizer. We have explicitly
constructed an element in $R_{G}(X)$ for each adjoint real semisimple element.
Recall that for a simply connected complex semisimple Lie group $G$, the
centralizer $Z_{G}(X)$ of a semisimple element $X$ is connected; see [CM,
Theorem 2.3.3, p.28]. Thus the centralizer is determined by its Lie algebra
which has a nice description in terms of Cartan subalgebra and certain root
vectors; see [CM, Lemma 2.1.2, p. 20]. Therefore, our construction also
classifies the reverser of adjoint real semisimple elements in simple Lie
groups.

##  2\. Notation and background

The Lie groups will be denoted by the capital letters, while the Lie algebra
of a Lie group will be denoted by the corresponding lower case German letter.
For a subgroup $H$ of $G$ and a subset $S$ of $\mathfrak{g}$, the subgroup
$Z_{H}(S)$ of $H$ that fixes $S$ pointwise under the adjoint action is called
the centralizer of $S$ in $H$. Similarly, for a Lie subalgebra
$\mathfrak{h}\,\subset\,\mathfrak{g}$ and a subset $S\,\subset\,\mathfrak{g}$,
by $\mathfrak{z}_{\mathfrak{h}}(S)$ we will denote the subalgebra of
$\mathfrak{h}$ consisting of all the elements that commute with every element
of $S$. For $A\in{\rm M}_{n}(\mathbb{C})$, $A^{t}$ denotes the transpose of
the matrix $A$. Let ${\rm I}_{n}$ denote the $n\times n$ identity matrix, and

${\rm J}_{n}\,:=\,\begin{pmatrix}&-{\rm I}_{n}\\\ {\rm
I}_{n}&\end{pmatrix}\,.$ (2.1)

Here we will work with the following classical simple Lie groups and Lie
algebras over $\mathbb{C}$:

$\displaystyle{\rm SL}_{n}(\mathbb{C})$ $\displaystyle:=\\{g\,\in\,{\rm
GL}_{n}(\mathbb{C})\,\mid\,\det(g)\,=\,1\\},$
$\displaystyle{\mathfrak{s}\mathfrak{l}}_{n}(\mathbb{C}):=\\{z\,\in\,{\rm
M}_{n}(\mathbb{C})\,\mid\,\text{tr}(z)=0\\};$ $\displaystyle{\rm
SO}(n,\mathbb{C})$ $\displaystyle:=\\{g\,\in\,{\rm
SL}_{n}(\mathbb{C})\,\mid\,g^{t}g\,=\,{\rm I}_{n}\\},$
$\displaystyle{\mathfrak{s}\mathfrak{o}}(n,\mathbb{C}):=\\{z\,\in\,\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})\,\mid\,z^{t}{\rm
I}_{n}+{\rm I}_{n}z=0\\};$ $\displaystyle{\rm Sp}(n,\mathbb{C})$
$\displaystyle:=\\{g\,\in\,{\rm SL}_{2n}(\mathbb{C})\,\mid\,g^{t}{\rm
J}_{n}g\,={\rm J}_{n}\\},$
$\displaystyle{\mathfrak{s}\mathfrak{p}}(n,\mathbb{C}):=\\{z\,\in\,\mathfrak{s}\mathfrak{l}_{2n}(\mathbb{C})\,\mid\,z^{t}{\rm
J}_{n}+{\rm J}_{n}z=0\\}.$

For any group $H$, let $H^{n}_{\Delta}$ denote the diagonally embedded copy of
$H$ in the $n$-fold direct product $H^{n}$. Similarly, for a matrix $A\in{\rm
M}_{n}(\mathbb{C})$, let $A^{n}_{\Delta}$ denote the diagonally embedded copy
of $A$ in the $n$-fold direct sum $A\oplus\dots\oplus A$. For a pair of
disjoint ordered sets $(v_{1},\,\dots,\,v_{n})$ and $(w_{1},\,\dots,\,w_{m})$,
the ordered set $(v_{1},\,\dots,\,v_{n},\,w_{1},\,\dots,\,w_{m})$ will be
denoted by

$(v_{1},\,\dots,\,v_{n})\vee(w_{1},\,\dots,\,w_{m})\,.$

For a Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, a subset
$\\{X,H,Y\\}\,\subset\,\mathfrak{g}$ is said to be a
$\mathfrak{s}\mathfrak{l}_{2}$-triple if $X\,\neq\,0$, $[H,\,X]=2X$,
$[H,\,Y]=-2Y$ and $[X,\,Y]=H$. Note that for a
$\mathfrak{s}\mathfrak{l}_{2}$-triple $\\{X,H,Y\\}$ in $\mathfrak{g}$,
$\text{Span}_{\mathbb{C}}\\{X,H,Y\\}$ is isomorphic to
$\mathfrak{s}\mathfrak{l}_{2}(\mathbb{C})$. We now recall a well-known result
due to Jacobson and Morozov.

######  Theorem 2.1 (Jacobson-Morozov, cf. [CM, Theorem 9.2.1]).

Let $X\,\in\,\mathfrak{g}$ be a non-zero nilpotent element in a semisimple Lie
algebra $\mathfrak{g}$ over $\mathbb{C}$. Then there exist
$H,\,Y\,\in\,\mathfrak{g}$ such that $\\{X,H,Y\\}$ is a
$\mathfrak{s}\mathfrak{l}_{2}$-triple.

##  3\. Adjoint reality for semisimple elements

Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak{g}$. Let
$H\in\mathfrak{g}$ be a semisimple element, and $\mathfrak{h}$ be a Cartan
subalgebra in $\mathfrak{g}$. Further we may assume $H\in\mathfrak{h}$ as
${\rm Ad}(g)(H)\in\mathfrak{h}$ for some $g\in G$.

###  3.1. Semi-simple elements in $\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$

Let $\mathfrak{g}:=\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$ and
$\mathfrak{h}$ be the subalgebra consisting of all diagonal matrices in
$\mathfrak{g}$. Then $\mathfrak{h}$ is a Cartan subalgebra in $\mathfrak{g}$.

######  Lemma 3.1.

Let $H$ be a semisimple element in $\mathfrak{g}\mathfrak{l}_{n}(\mathbb{C})$.
Then $H$ is ${\rm Ad}_{{\rm GL}_{n}(\mathbb{C})}$-real in
$\mathfrak{g}\mathfrak{l}_{n}(\mathbb{C})$ if and only if whenever $\lambda$
is an eigenvalue of $H$, $-\lambda$ is also an eigenvalue of $H$ with the same
multiplicity.

######  Remark 3.2.

A semisimple element $H\in\mathfrak{g}\mathfrak{l}_{n}(\mathbb{C})$ is ${\rm
Ad}_{{\rm GL}_{n}(\mathbb{C})}$-real if and only if $H$ is ${\rm Ad}_{{\rm
SL}_{n}(\mathbb{C})}$-real in $\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$. The
analogous statement is also true in the group theoretic sense; see [OS, p.
77].

######  Theorem 3.3.

Every real semisimple element in ${\rm Lie(\,PSL}_{n}(\mathbb{C}))$ is
strongly ${\rm Ad}_{{\rm PSL}_{n}(\mathbb{C})}$-real.

Proof. Let $H\in{\rm Lie(\,PSL}_{n}(\mathbb{C}))$ be a ${\rm Ad}_{{\rm
PSL}_{n}(\mathbb{C})}$-real semisimple element. We may assume $H={\rm
diag}(h_{1},\ldots,h_{m},-h_{1},\ldots,-h_{m},0,\ldots,0)$. Then $\sigma:={\rm
diag}({\rm J}_{m},\,\sqrt{-1}\,{\rm I}_{s})$ will conjugate $H$ and $-H$,
where $2m+s=n$, and ${\rm J}_{n}$ is as in (2.1). ∎

######  Example 3.4.

Consider the semi-simple element $H={\rm diag}(x_{1},-x_{1})$. Let $g\in{\rm
GL}_{2}(\mathbb{C})$ so that $gH=-Hg$. Then $g$ is of the form
$\left(\begin{smallmatrix}0&b\\\ c&0\end{smallmatrix}\right)$. Hence $H$ is a
strongly ${\rm Ad}_{{\rm GL}_{2}(\mathbb{C})}$-real element in
$\mathfrak{g}\mathfrak{l}_{2}(\mathbb{C})$, and is an ${\rm Ad}_{{\rm
SL}_{2}(\mathbb{C})}$-real element but not strongly ${\rm Ad}_{{\rm
SL}_{2}(\mathbb{C})}$-real in $\mathfrak{s}\mathfrak{l}_{2}(\mathbb{C})$. Note
that $g\in{\rm Sp}(1,\mathbb{C})$ if $bc=-1$. Similarly, $H$ is a ${\rm
Ad}_{{\rm Sp}(1,\mathbb{C})}$-real element but not strongly ${\rm Ad}_{{\rm
Sp}(1,\mathbb{C})}$-real element in $\mathfrak{s}\mathfrak{p}(1,\mathbb{C})$.
∎

The next result classifies strong ${\rm Ad}_{{\rm
SL}_{n}(\mathbb{C})}$-reality in the Lie algebra
$\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$; see [GLM3, Proposition 2.5]. Here
we provide a detailed proof.

######  Theorem 3.5.

An ${\rm Ad}_{{\rm SL}_{n}(\mathbb{C})}$-real semisimple element in
$\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$ is strongly ${\rm Ad}_{{\rm
SL}_{n}(\mathbb{C})}$-real if and only if either $0$ is an eigenvalue or
$n\not\equiv 2\pmod{4}$.

Proof. Let $H\in\mathfrak{s}\mathfrak{l}_{n}(\mathbb{C})$ be a ${\rm Ad}_{{\rm
SL}_{n}(\mathbb{C})}$-real semisimple element. If $0$ is an eigenvalue of $H$,
then using Example 3.4 it follows that $H$ is strongly ${\rm Ad}_{{\rm
SL}_{n}(\mathbb{C})}$-real. Suppose $0$ is not an eigenvalue of $H$, and
$n\not\equiv 2\pmod{4}$, then $n=4m$ for $m\in\mathbb{N}$ and we can assume
$H={\rm diag}(h_{1},\ldots,h_{2m},-h_{1},\ldots,-h_{2m})$. Then, $H$ and $-H$
will be conjugated by $g=\left(\begin{smallmatrix}&{\rm I}_{2m}\\\ {\rm
I}_{2m}&\\\ \end{smallmatrix}\right)$.

Next assume that $H$ is strongly ${\rm Ad}_{{\rm SL}_{n}(\mathbb{C})}$-real
and $0$ is not an eigenvalue of $H$. We will show $n\in 4\mathbb{N}$. Without
loss of generality, we can assume $H={\rm
diag}(h_{1},\ldots,h_{m},-h_{1},\ldots,-h_{m})$ and $gH=-Hg$ for some
involution. Let $e_{j}$ be the standard column vector in $\mathbb{C}^{n}$ with
$1$ in $j^{\rm th}$ place and $0$ elsewhere. For $1\leq j\leq m$,

$Hge_{j}\,=\,-gHe_{j}\,=\,-gh_{j}e_{j}\,=\,-h_{j}ge_{j}.$

Let $V_{j}:=\mathbb{C}e_{j}\oplus\mathbb{C}ge_{j}$ and
$\mathcal{C}_{j}:=\\{e_{j},ge_{j}\\}$. Since $g^{2}={\rm I}_{2}$,
$g(V_{j})\subset V_{j}$. Then $\\{e_{j},ge_{j}\mid 1\leq j\leq m\\}$ forms a
basis of $\mathbb{C}^{2m}$. Set
$\mathcal{C}:=\mathcal{C}_{1}\vee\cdots\vee\mathcal{C}_{m}$. Then the matrix
$[g]_{\mathcal{C}}$ is a $2\times 2$ block-diagonal matrix and
$\det\,[g]_{\mathcal{C}}=(-1)^{m}$. As $\det g=1$, it follows that $m\in
2\mathbb{N}$. This completes the proof. ∎

###  3.2. Semi-simple elements in $\mathfrak{o}(n,\mathbb{C})$ and
$\mathfrak{s}\mathfrak{o}(n,\mathbb{C})$

Up to conjugacy, any semisimple element in $\mathfrak{o}(n,\mathbb{C})$ or
$\mathfrak{s}\mathfrak{o}(n,\mathbb{C})$ belongs to the following Cartan
subalgebra $\mathfrak{h}$, see [Kn, p. 127]:

$\displaystyle\mathfrak{h}:=\begin{cases}{\rm diag}(H_{1},\ldots,H_{m},0)&{\rm
if}\,n=2m+1\\\ \ {\rm diag}(H_{1},\ldots,H_{m})&{\rm
if}\,n=2m\end{cases},{\rm\,where\,}H_{j}=\begin{pmatrix}0&x_{j}\\\
-x_{j}&0\end{pmatrix},x_{j}\in\mathbb{C}.$ (3.1)

######  Example 3.6.

Consider $H=\left(\begin{smallmatrix}0&x\\\
-x&0\end{smallmatrix}\right)\in\mathfrak{s}\mathfrak{o}(2,\mathbb{C})=\mathfrak{o}(2,\mathbb{C})$,
where $x\in\mathbb{C}$. Let $g\in{\rm GL}_{2}(\mathbb{C})$ so that
$gHg^{-1}=-H$. Then $g$ is of the form $\left(\begin{smallmatrix}a&b\\\
b&-a\end{smallmatrix}\right)$. Hence, $\det g=-a^{2}-b^{2}$, and
$gg^{t}=g^{2}=(-\det g){\rm I}_{2}$. Thus, one can choose $g\in{\rm
O}(2,\mathbb{C})$ with $\det g=-1$. This shows that $H$ is a ${\rm Ad}_{{\rm
O}(n,\mathbb{C})}$-real, as well as strongly ${\rm Ad}_{{\rm
O}(n,\mathbb{C})}$-real, element in ${\mathfrak{o}}(2,\mathbb{C})$ but not
${\rm Ad}_{{\rm SO}(n,\mathbb{C})}$-real in
${\mathfrak{s}\mathfrak{o}}(2,\mathbb{C})$. ∎

The following result follows from the construction done in the above example.

######  Lemma 3.7.

Every semisimple element in $\mathfrak{o}(n,\mathbb{C})$ is ${\rm Ad}_{{\rm
O}(n,\mathbb{C})}$-real.

Next we investigate strongly ${\rm Ad}_{{\rm O}(n,\mathbb{C})}$-real elements
in $\mathfrak{o}(n,\mathbb{C})$.

######  Proposition 3.8.

Every semisimple element in $\mathfrak{o}(n,\mathbb{C})$ is strongly ${\rm
Ad}_{{\rm O}(n,\mathbb{C})}$-real.

Proof. Enough to consider the elements of $\mathfrak{h}$. For the elements in
$\mathfrak{h}$ one can easily construct the required involution using the
Example 3.6. ∎

Now we classify the strongly AdSO(n,C)-real semisimple element in
$\mathfrak{s}\mathfrak{o}(n,\mathbb{C})$.

######  Theorem 3.9.

Let $H\in\mathfrak{s}\mathfrak{o}(n,\mathbb{C})$ be a semisimple element. Then
$H$ is a strongly AdSO(n,C)-real if and only if either $0$ is an eigenvalue of
$H$ or $n\not\equiv 2\pmod{4}$.

Proof. Since $H$ is semisimple, there exists $\sigma\in{\rm SO}(n,\mathbb{C})$
such that ${\rm Ad}(\sigma)H={\rm
diag}(H_{1},\dots,H_{m},\underset{r\text{-many}}{\underbrace{0,\dots,0}})$,
$r\geq 1$ where $H_{j}$ is as in (3.1).

First assume that $0$ is an eigenvalue of $H$. Let $g:={\rm
diag}(\underset{m\text{-many}}{\underbrace{{\rm I}_{1,1},\dots,{\rm
I}_{1,1}}},{\rm I}_{r})$, where ${\rm I}_{1,1}:={\rm diag}(1,-1)$. Then ${\rm
Ad}(\sigma^{-1}g\sigma)H=-H$. Note that $g^{2}={\rm I}_{n}$. If $\det g=1$,
then we are done. Otherwise replace $g$ by ${\rm diag}({\rm
I}_{1,1},\dots,{\rm I}_{1,1},{\rm I}_{r-1},-1)$ to get the required involution
in ${\rm SO}(n,\mathbb{C})$. Next assume that $0$ is not an eigenvalue, and
$n\not\equiv 2\pmod{4}$. Thus $n\equiv 0\pmod{4}$, and hence up to conjugacy
$H$ is of the form ${\rm diag}(H_{1},\dots,H_{2k})$, $4k=n$. By choosing an
involution $g$ as above with $m=2k,r=0$, we have that $H$ is strongly ${\rm
Ad}_{{\rm SO}(n,\mathbb{C})}$-real.

Next suppose that $H$ is strongly AdSO(n,C)-real. Further assume that $0$ is
not an eigenvalue of $H$, and $H={\rm diag}(H_{1},\dots,H_{m})$ where $H_{j}$
is as in (3.1). Let $e_{j}$ be the standard column vector in $\mathbb{C}^{n}$.
Then $e_{2j-1}+\sqrt{-1}e_{2j}$ and $g(e_{2j-1}+\sqrt{-1}e_{2j})$ are
eigenvector of $H$ corresponding to the eigenvalues $\sqrt{-1}x_{j}$ and
$-\sqrt{-1}x_{j}$, respectively. Let
$V_{j}:=\mathbb{C}(e_{2j-1}+\sqrt{-1}e_{2j})\oplus\mathbb{C}g(e_{2j-1}+\sqrt{-1}e_{2j})$,
and
$\mathcal{C}_{j}:=\\{e_{2j-1}+\sqrt{-1}e_{2j},\,g(e_{2j-1}+\sqrt{-1}e_{2j})\\}$.
Since $g^{2}={\rm I_{2}}$, $g(V_{j})\subset V_{j}$. Then
$\\{e_{2j-1}+\sqrt{-1}e_{2j},\,g(e_{2j-1}+\sqrt{-1}e_{2j})\,\mid\,1\leq j\leq
m\\}$ forms a basis of $\mathbb{C}^{n}$. Set
$\mathcal{C}:=\mathcal{C}_{1}\vee\dots\vee\mathcal{C}_{m}$. Then the matrix
$[g]_{\mathcal{C}}$ is a $2\times 2$ block-diagonal matrix and
$\det\,[g]_{\mathcal{C}}=(-1)^{m}$. As $\det g=1$, it follows that $m\in
2\mathbb{N}$. This completes the proof. ∎

###  3.3. Semi-simple elements in $\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$

Recall that in Example 3.4, ${\rm diag}(x,-x)$ is not strongly AdSp(1,C)-real
in but we can choose a conjugating involution $g$ from ${\rm
PSp}(1,\mathbb{C})$. Thus, we will first consider the semisimple element in
the Lie algebra of ${\rm PSp}(n,\mathbb{C})$. Let $\mathfrak{h}:=\\{{\rm
diag}(h_{1},\ldots,h_{m},-h_{1},\ldots,-h_{m})\mid h_{j}\in\mathbb{C}\\}$.

######  Theorem 3.10.

Every semisimple element in ${\rm Lie\,(PSp}(n,\mathbb{C}))$ is strongly
AdPSp(n,C)-real.

Proof. Let $X\in{\rm Lie\,(PSp}(n,\mathbb{C}))$. Then Ad$(g)X\in\mathfrak{h}$
for some $g\in{\rm PSp}(n,\mathbb{C})$. Thus we define required involution
using Example 3.4. ∎

The following corollary is immediate.

######  Corollary 3.11.

Every semisimple element in $\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$ is
AdSp(n,C)-real.

######  Theorem 3.12.

A semisimple element in $\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$ is strongly
AdSp(n,C)-real if and only if the multiplicity of each non-zero eigenvalue is
even.

Proof. Let $H$ be a strongly real semisimple element in
$\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$. We can assume $H={\rm
diag}(h_{1},\ldots,h_{n},-h_{1},\ldots,-h_{n})$, and $gH=-Hg$ for some
involution $g\in{\rm Sp}(n,\mathbb{C})$. Suppose $h_{j}\neq 0$ and
multiplicity of $h_{j}$ is $m$. Let
$\mathcal{C}_{j}:=\\{e_{j_{1}},\ldots,e_{j_{m}}\\}$ be an ordered basis of the
eigenspace of $H$ corresponding to the eigenvalue $h_{j}$. Then
$\mathcal{C}_{n+j}:=\\{e_{n+j_{1}},\ldots,e_{n+j_{m}}\\}$ is an ordered basis
of the eigenspace corresponding to the eigenvalue $-h_{j}$. Let $V_{j}$ be the
$\mathbb{C}$-span of $\mathcal{C}_{j}\vee\mathcal{C}_{n+j}$. Then the
involution $g$ keeps $V_{j}$ invariant and $(g|_{V_{j}})^{t}{\rm
J}_{m}(g|_{V_{j}})\,={\rm J}_{m}$, where ${\rm J}_{m}$ is as in (2.1). As
$H(ge_{j_{l}})=-h_{j}ge_{j_{l}}$ for $1\leq l\leq m$, we can write

$\displaystyle[g]_{\mathcal{C}_{j}\vee\mathcal{C}_{n+j}}:=\big{(}\begin{smallmatrix}0&B\\\
C&0\end{smallmatrix}\big{)}\,,\quad{\rm for\,some\,}B,C\in{\rm
GL}_{m}(\mathbb{C})\,.$ (3.2)

Since $g|_{V_{j}}$ is an involution and $(g|_{V_{j}})^{t}{\rm
J}_{m}(g|_{V_{j}})\,={\rm J}_{m}$, it follows that $C^{-1}=B=-B^{t}$. Hence,
$m$ (the multiplicity of $h_{j}$) has to be even.

Next assume that multiplicity of each non-zero eigenvalue is even. Can assume
any semisimple element $H={\rm
diag}(h_{1},\ldots,h_{n},-h_{1},\ldots,-h_{n})$, where $h_{2j-1}=h_{2j}$ for
$j=1,\ldots,n/2$. In this case to define a required involution $g$, set
$B:=\begin{pmatrix}&&{\rm J}_{1}\\\ &\reflectbox{$\ddots$}\\\ {\rm
J}_{1}\end{pmatrix}$ in (3.2). This completes the proof. ∎

##  4\. ${\rm Ad}_{{\rm Sp}(n,\mathbb{C})}$-reality in
$\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$

The aim of this section is to prove that every element in
$\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$ is adjoint real. For this, we need to
recall some known construction and results, cf. [BCM1]. The structure of the
centralizer of nilpotent elements play an important role here.

Let $0\neq X\in\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$ be a nilpotent element.
Let $\mathfrak{a}$ be a $\mathfrak{s}\mathfrak{l}_{2}$-triple in
$\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$ containing $X$; see Theorem 2.1. Note
that $\mathbb{C}^{2n}$ is $\mathbb{C}$-module over $\mathfrak{a}$. By
decomposing $\mathbb{C}^{2n}$ as direct sum of irreducible
$\mathfrak{a}$-module, let $\mathbb{N}_{\mathbf{d}}:=\\{d_{1},\dots,d_{s}\\}$
be the dimensions of irreducible $\mathfrak{a}$-modules. Let $M(d-1)$ denote
the sum of all $\mathbb{C}$-subspaces of $\mathbb{C}^{2n}$ which are
irreducible $\mathfrak{a}$-submodule of dimension $d$. Then $M(d-1)$ is the
isotypical component of $\mathbb{C}^{2n}$ containing all the irreducible
submodules of $\mathbb{C}^{2n}$ with highest weight $d-1$. Let

$L(d-1)\,:=\,\text{Span}_{\mathbb{C}}\\{v\in M(d-1)\,\mid\,\,{\rm
weight\,of}\,v\,{\rm is}\,1-d\,\\}\,.$

Then it follows that $M(d-1)\,=\,L(d-1)\oplus XL(d-1)\oplus\dots\oplus
X^{d-1}L(d-1)$; see [BCM1, Lemma A.1]. Let
$t_{d}\,:=\,\dim_{\mathbb{C}}L(d-1).$ Then
$\sum_{d\in\mathbb{N}_{\mathbf{d}}}dt_{d}\,=\,2n$, and let
$\\{X^{l}v^{d}_{j}\,\mid\,0\leq l<d,\,1\leq j\leq
t_{d},d\in\mathbb{N}_{\mathbf{d}}\\}$ be a basis of $\mathbb{C}^{2n}$ as
constructed in [BCM1, Lemma A.6]. We need to fix an ordering of the above
basis. Let $(v^{d}_{1},\,\dotsc,\,v^{d}_{t_{d}})$ be an ordered
$\mathbb{C}$-basis of $L(d-1)$ for $d\,\in\,\mathbb{N}_{\mathbf{d}}$. Then it
follows that

$\mathcal{B}^{l}(d)\,:=\,(X^{l}v^{d}_{1},\,\dotsc,\,X^{l}v^{d}_{t_{d}})$

is an ordered $\mathbb{C}$-basis of $X^{l}L(d-1)$ for $0\,\leq\,l\,\leq\,d-1$
with $d\,\in\,\mathbb{N}_{\mathbf{d}}$. Define

$\mathcal{B}(d)\,:=\,\mathcal{B}^{0}(d)\vee\cdots\vee\mathcal{B}^{d-1}(d)\
\forall\ d\,\in\,\mathbb{N}_{\mathbf{d}}\,,\ \text{ and }\
\mathcal{B}\,:=\,\mathcal{B}(d_{1})\vee\cdots\vee\mathcal{B}(d_{s})\,.$

Let
$\langle\cdot,\,\cdot\rangle\,:\,\mathbb{C}^{2n}\times\mathbb{C}^{2n}\,\longrightarrow\,\mathbb{C}\
$ be the symplectic form given by $\langle x,\,y\rangle\,:=x^{t}{\rm J}_{n}y$.
Define another form on $L(d-1)$ below, as in [CM, p. 139],

$(\cdot,\,\cdot)_{d}\,:\,L(d-1)\times
L(d-1)\,\longrightarrow\,\mathbb{C}\quad;\ \ \ (v,\,u)_{d}\,:=\,\langle
v\,,\,X^{d-1}u\rangle\,.$ (4.1)

The following result is due to Springer-Steinberg, which describes the
structure of the centralizer of the $\mathfrak{s}\mathfrak{l}_{2}$-triple in
${\rm Sp}(n,\mathbb{C})$ and $\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$.

######  Lemma 4.1 (cf. [CM, Theorem 6.1.3], [BCM1, Lemma 4.4]).

The following isomorphisms hold:

  1. (1)

If $\mathfrak{a}$ is a $\mathfrak{s}\mathfrak{l}_{2}$-triple in
$\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$, then

$\mathcal{Z}_{{\rm Sp}(n,\mathbb{C})}(\mathfrak{a})=\left\\{g\in{\rm
SL}(\mathbb{C}^{2n})\middle|\begin{array}[]{cccc}g(X^{l}L(d-1))\,\subset\,X^{l}L(d-1),\vspace{.14cm}\\\
\\!\big{[}g|_{X^{l}L(d-1)}\big{]}_{{\mathcal{B}}^{l}(d)}=\big{[}g|_{L(d-1)}\big{]}_{{\mathcal{B}}^{0}(d)},(gx,gy)_{d}=(x,y)_{d}\vspace{.14cm}\\\
\text{ for all }d\in\mathbb{N}_{\mathbf{d}},~{}0\leq l\leq d-1,\text{ and
}x,y\in L(d-1)\end{array}\right\\};$

here $(\cdot,\,\cdot)_{d}$ is as in (4.1).

  2. (2)

In particular,

$\displaystyle\mathcal{Z}_{{\rm Sp}(n,\mathbb{C})}(\mathfrak{a})\,$
$\displaystyle\simeq\,\Big{\\{}g\in\prod_{d\,\in\,\mathbb{N}_{\mathbf{d}}}{\rm
GL}(L(d-1))\,\bigm{|}\,\,(gx,gy)_{d}=(x,y)_{d},\,\mbox{\Large$\chi$}_{\mathbf{d}}(g)\,=\,1\Big{\\}}$
$\displaystyle\simeq\,\Big{\\{}\prod_{d\in\mathbb{O}_{\mathbf{d}}}{\rm
Sp}(t_{\mathbf{d}}/2,\,\mathbb{C})_{\Delta}^{d}\,\times\,\prod_{d\in\mathbb{E}_{\mathbf{d}}}{\rm
O}(t_{\mathbf{d}},\,\mathbb{C})_{\Delta}^{d}\Big{\\}}\,.$ (4.2)

  3. (3)

$\mathfrak{z}_{{\mathfrak{s}\mathfrak{p}}(n,\mathbb{C})}(\mathfrak{a})=\left\\{A\in{\mathfrak{s}\mathfrak{l}}(\mathbb{C}^{2n})\middle|\begin{array}[]{cccc}\\!A(X^{l}L(d-1))\,\subset\,X^{l}L(d-1),\vspace{.14cm}\\\
\\!\\!\big{[}A|_{X^{l}L(d-1)}\big{]}_{{\mathcal{B}}^{l}(d)}=\big{[}A|_{L(d-1)}\big{]}_{{\mathcal{B}}^{0}(d)},(Ax,y)_{d}+(x,Ay)_{d}=0\\!\\!\vspace{.14cm}\\\
\text{ for all }d\in\mathbb{N}_{\mathbf{d}},~{}0\leq l\leq d-1,\text{ and
}x,y\in L(d-1)\end{array}\right\\}\\!.$

  4. (4)

In particular,

$\mathfrak{z}_{{\mathfrak{s}\mathfrak{p}}(n,\mathbb{C})}(\mathfrak{a})\,\simeq\,\Big{\\{}\Big{(}\bigoplus_{d\in\mathbb{O}_{\mathbf{d}}}{\mathfrak{s}\mathfrak{p}}(t_{\mathbf{d}}/2,\,\mathbb{C})_{\Delta}^{d}\Big{)}\,\,\,{{\bigoplus}}\,\,\Big{(}\bigoplus_{d\in\mathbb{E}_{\mathbf{d}}}{\mathfrak{o}}(t_{\mathbf{d}},\,\mathbb{C})_{\Delta}^{d}\Big{)}\Big{\\}}\,.$

Now we will characterize the ${\rm Ad}_{{\rm Sp}(n,\mathbb{C})}$-real elements
in $\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$.

######  Theorem 4.2.

Every element of $\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$ is ${\rm Ad}_{{\rm
Sp}(n,\mathbb{C})}$-real.

Proof. Let $X\in\mathfrak{s}\mathfrak{p}(n,\mathbb{C})$, and
$X=X_{s}\,+\,X_{n}$ be the Jordan decomposition of $X$ where $X_{s}$ and
$X_{n}$ are the semisimple and nilpotent part of $X$, respectively. In view of
Corollary 3.11 and [GM, Lemma 4.2], we may further assume that $X_{s}\neq
0,\,X_{n}\neq 0$. We will construct below two elements $\sigma$ and $\tau$ in
${\rm Sp}(n,\mathbb{C})$ so that

  * •

$\sigma X_{n}\sigma^{-1}=-X_{n}$ and $\sigma X_{s}\sigma^{-1}=X_{s}$,

  * •

$\tau X_{s}\tau^{-1}=-X_{s}$ and $\tau X_{n}\tau^{-1}=X_{n}$.

Then $\sigma\tau$ will do the job, i.e., $\sigma\tau X(\sigma\tau)^{-1}=-X$.

Construction of $\sigma$. Define $\sigma\in{\rm GL}\,(\mathbb{C}^{2n})$ below,
as done in [GM, (4.3)] :

$\displaystyle\sigma(X^{l}v^{d}_{j}):=\begin{cases}(-1)^{l}X^{l}v^{d}_{j}&\qquad\text{\
if\ }d\in\mathbb{O}_{\mathbf{d}}\,,\\\
(-1)^{l}\sqrt{-1}X^{l}v^{d}_{j}&\qquad\text{\ if\
$d\in\mathbb{E}_{\mathbf{d}}$}\,.\end{cases}$ (4.3)

Note that $\sigma X_{n}=-X_{n}\sigma$, and $\langle\sigma x,\sigma
y\rangle=\langle x,y\rangle$ for all $x,y\in\mathbb{C}^{2n}$. This shows that
$\sigma\in{\rm Sp}(n,\mathbb{C})$; cf. [GM, Section 4.3].

The element $X_{s}$ is a semisimple element and commutes with $X_{n}$, and
$\mathfrak{z}_{{\mathfrak{s}\mathfrak{p}}(n,\mathbb{C})}(\mathfrak{a})$ is
reductive part (Levi part) of
$\mathfrak{z}_{{\mathfrak{s}\mathfrak{p}}(n,\mathbb{C})}(X_{n})$; see [CM,
Lemma 3.7.3]. Thus,
$X_{s}\in\mathfrak{z}_{{\mathfrak{s}\mathfrak{p}}(n,\mathbb{C})}(\mathfrak{a})$.
Write $[X_{s}|_{L(d-1)}]_{\mathcal{B}^{0}_{d}}:=X_{sd}$. Then using Lemma
4.1(3), $X_{s}=\oplus_{d\in\mathbb{N}_{\mathbf{d}}}(X_{sd})_{\Delta}^{d}$.
Also $[\sigma]_{\mathcal{B}^{0}(d)}$ is a scalar matrix; see $\eqref{real-
nilpotent-sp-n-c}$. Thus from Lemma 4.1(3), it follows that $\sigma
X_{s}\sigma^{-1}=X_{s}$.

Construction of $\tau$. Note that

$X_{sd}\,\in\,\begin{cases}{\mathfrak{s}\mathfrak{p}}(t_{d}/2,\,\mathbb{C})&\
{\rm for}\ \theta\in\mathbb{O}_{\mathbf{d}}\\\
{\mathfrak{o}}(t_{d},\,\mathbb{C})&\ {\rm for}\
\theta\in\mathbb{E}_{\mathbf{d}}\,.\end{cases}$

In view of Corollary 3.11 and Proposition 3.8, there exists
$\tau_{d}\in\begin{cases}{\rm Sp}(t_{d}/2,\,\mathbb{C})&{\rm for}\
\theta\in\mathbb{O}_{\mathbf{d}}\\\ {\rm O}(t_{d},\,\mathbb{C})&{\rm for}\
\theta\in\mathbb{E}_{\mathbf{d}}\end{cases}$ such that
$\tau_{d}X_{sd}\tau_{d}^{-1}\,=\,-X_{sd}$ for all
$d\in\mathbb{N}_{\mathbf{d}}$. Finally set
$\tau:=\mbox{\Large$\chi$}(\tau_{d})$, where $\chi$ is an isomorphism in (2);
see [BCM2, Section 3.4] for such isomorphism. Then $\tau
X_{s}\tau^{-1}=-X_{s}$ and $\tau X_{n}\tau^{-1}=X_{n}$. This completes the
proof. ∎

### Acknowledgment

Gongopadhyay acknowledges SERB grant CRG/2022/003680. The authors thank Tejbir
for his comments.

## References

  * [BR] M. Baake and J. A. G. Roberts, The structure of reversing symmetry groups, Bull. Austral. Math. Soc., 73(2006), 445–459.
  * [BCM1] I. Biswas, P. Chatterjee, C. Maity, The Second cohomology of nilpotent orbits in classical Lie algebra, Kyoto J. Math. , vol. 60, No. 2, 2020, pp 717-799.
  * [BCM2] I. Biswas, P. Chatterjee, C. Maity, Homotopy types of the nilpotent orbits in classical Lie algebra, Kyoto J. Math., vol. 63, no. 4, 851–891, 2023.
  * [CM] D. H. Collingwood, W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.
  * [GLM1] K. Gongopadhyay, T. Lohan, C. Maity, Reversibility and real adjoint orbits of linear maps, Essays in Geometry, dedicated to Norbert A’Campo (ed. A. Papadopoulos), European Mathematical Society Press, Berlin, 2023, DOI 10.4171/IRMA/34/15. .
  * [GLM2] K. Gongopadhyay, T. Lohan, C. Maity, Reversibility of affine transformations, Proceedings of the Edinburgh Mathematical Society, 66 (2023), no. 4, 1217–1228.
  * [GLM3] K. Gongopadhyay, T. Lohan, C. Maity, Real adjoint orbits of special linear groups, to appear in Illinois Journal of Mathematics, arXiv:2204.03624.
  * [GM] K. Gongopadhyay, C. Maity, Reality of unipotent elements in simple Lie groups, Bull. Sci. Math., 185, 2023, 103261.
  * [Kn] A. W. Knapp, Lie groups beyond an introduction, Second edition, Progress in Mathematics, Vol. 140, Birkhäusar, Basel, 2002.
  * [Mc] W. M. McGovern, The adjoint representation and the adjoint action in Algebraic quotients. Torus actions and cohomology. Encyclopaedia Math. Sci., 131, Invariant Theory Algebr. Transform. Groups, II, Springer, Berlin, 2002, 159–238.
  * [OS] A. G. O’Farrell, I. Short, Reversibility in Dynamics and Group Theory, London Mathematical Society Lecture Note Series, vol.416, Cambridge University Press, Cambridge, 2015.
  * [ST] A. Singh, M. Thakur, Reality properties of conjugacy classes in algebraic groups. Israel J. Math., 165 (2008), 1–27.


all}\;m,\text{if}\;q_{\perp}\neq 0,\\\ \frac{1}{2}(i\mathcal{D}\cdot
V)^{2}&\text{if}\;m=0,\text{and}\;q_{\perp}=0,\\\ i\mathcal{D}\cdot
V&\text{if}\;m=1,\text{and}\;q_{\perp}=0,\\\
1&\text{if}\;m=2,\text{and}\;q_{\perp}=0,\\\ 0&\text{if}\;m\geqslant
3,\text{and}\;q_{\perp}=0.\\\ \end{dcases}$ (119)

The function, for $q_{\perp}=0$, is equal to the term number $2-m$, from the
Taylor expansion.

With the help of this function, the contribution to the type (10)-integral,
Section XVI, from a pair from plane 1, i.e. $(V_{1},\hat{\mu})$, is

$\begin{split}e^{i(\mathcal{D}\cdot
V_{1})_{m_{x}^{1}+m_{y}^{1}}}(-i\mathcal{D}_{x})^{m_{x}^{1}}(-i\mathcal{D}_{y})^{m_{y}^{1}}\\\
\frac{W\cdot\hat{\lambda}}{W\cdot\mathcal{D}}(-\Delta(f_{1}f_{2})_{\hat{\mu}})\frac{1}{\mathcal{D}\cdot\hat{\mu}},\end{split}$
(120)

where $\mathcal{D}$ is arbitrary, subject to the condition that
$\mathcal{D}\cdot\hat{\mu}\neq 0$ for all $\hat{\mu}$ in plane 1. The
contribution to the type (01)-integral from all real points from plane 2, i.e.
$V_{2}$, is a mirror image.

The contribution to the type (11)-integral from a virtual point, $V_{12}$,
under constrained translation has a different aspect, being

$\begin{split}\sum_{\text{all
four}\;\hat{\mu}^{\text{'}}\text{s}}e^{i(\mathcal{D}\cdot
V_{12})_{m_{x}^{1}+m_{y}^{1}+m_{x}^{2}+m_{y}^{2}}}&\\\
(\lambda_{1x})^{m_{x}^{1}}(\lambda_{1y})^{m_{y}^{1}}(\lambda_{2x})^{m_{x}^{2}}(\lambda_{2y})^{m_{y}^{2}}&\\\
\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{2}}{\hat{\mu}_{1}\cdot\hat{\lambda}_{2}}\right)^{m_{x}^{1}+m_{y}^{1}}\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{1}}{\hat{\mu}_{2}\cdot\hat{\lambda}_{1}}\right)^{m_{x}^{2}+m_{y}^{2}}&\\\
\frac{W\cdot\hat{\lambda}}{W\cdot\mathcal{D}}(-\Delta(f_{1}f_{2})_{\hat{\mu}})\frac{1}{\mathcal{D}\cdot\hat{\mu}}\;.\end{split}$
(121)

To avoid carrying over a minus sign and to arrive at the above formula, we use
$\hat{\mu}_{2}\cdot\hat{\lambda}_{1}=-\hat{\mu}_{1}\cdot\hat{\lambda}_{2}$.
The sum, for a fixed virtual point $V_{12}$, is over all four
$\hat{\mu}^{\text{'}}\text{s}$, which have the tail located at $V_{12}$.

The above sum, if needed, can be place in a form which is independent of the
gauge vector $W$. For that, first notice that

$\begin{split}(\lambda_{1x})^{m_{x}^{1}}(\lambda_{1y})^{m_{y}^{1}}(\lambda_{2x})^{m_{x}^{2}}(\lambda_{2y})^{m_{y}^{2}}\\\
\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{2}}{\hat{\mu}_{1}\cdot\hat{\lambda}_{2}}\right)^{m_{x}^{1}+m_{y}^{1}}\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{1}}{\hat{\mu}_{2}\cdot\hat{\lambda}_{1}}\right)^{m_{x}^{2}+m_{y}^{2}}\end{split}$
(122)

is the same for all four $\hat{\mu}^{\text{'}}\text{s}$, which have the tail
located at $V_{12}$.

Second, use the relation

$\begin{split}\sum_{\text{all
four}\;\hat{\mu}^{\text{'}}\text{s}}\frac{W\cdot\hat{\lambda}}{W\cdot\mathcal{D}}(-\Delta(f_{1}f_{2})_{\hat{\mu}})\frac{1}{\mathcal{D}\cdot\hat{\mu}}=\\\
\frac{\absolutevalue{\hat{\lambda}_{1}\cdot\hat{\mu}_{2}}}{(\mathcal{D}\cdot\hat{\mu}_{1})(\mathcal{D}\cdot\hat{\mu}_{2})}(\Delta
f)_{\hat{\mu}_{1}}(\Delta g)_{\hat{\mu}_{2}}\end{split}$ (123)

valid for any two choices of $\hat{\mu}_{1}$ and $\hat{\mu}_{2}$, which have
the tail located at $V_{12}$ and the tip on line 1 and line 2, respectively.

The contribution of a virtual point can be thus brought into a W-independent
form

$\begin{split}\sum_{V_{12},\text{two}\;\hat{\mu}^{\text{'}}\text{s}}e^{i(\mathcal{D}\cdot
V_{12})_{m_{x}^{1}+m_{y}^{1}+m_{x}^{2}+m_{y}^{2}}}&\\\
\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{2}}{\hat{\mu}_{1}\cdot\hat{\lambda}_{2}}\lambda_{1x}\right)^{m_{x}^{1}}\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{2}}{\hat{\mu}_{1}\cdot\hat{\lambda}_{2}}\lambda_{1y}\right)^{m_{y}^{1}}&\\\
\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{1}}{\hat{\mu}_{2}\cdot\hat{\lambda}_{1}}\lambda_{2x}\right)^{m_{x}^{2}}\left(i\frac{\mathcal{D}\cdot\hat{\mu}_{1}}{\hat{\mu}_{2}\cdot\hat{\lambda}_{1}}\lambda_{2y}\right)^{m_{y}^{2}}&\\\
\frac{\absolutevalue{\hat{\lambda}_{1}\cdot\hat{\mu}_{2}}}{(\mathcal{D}\cdot\hat{\mu}_{1})(\mathcal{D}\cdot\hat{\mu}_{2})}(\Delta
f)_{\hat{\mu}_{1}}(\Delta g)_{\hat{\mu}_{2}}\;.\end{split}$ (124)

To show (123), choose two out of four vectors that have the tail on the
virtual point and direction on the line in plane 1 and plane 2, and call them
$\hat{\mu}_{1}$ and $\hat{\mu}_{2}$, respectively. The other two vectors are
$-\hat{\mu}_{1}$ and $-\hat{\mu}_{2}$. Use

$\begin{split}-\Delta(f_{1}f_{2})_{\hat{\mu}_{1}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{1}}=\\\
-(-\Delta(f_{1}f_{2})_{\hat{\mu}_{2}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{2}})\end{split}$
(125)

and get

$\begin{split}\sum_{\text{all
four}\;\hat{\mu}^{\text{'}}\text{s}}\frac{W\cdot\hat{\lambda}}{W\cdot\mathcal{D}}(-\Delta(f_{1}f_{2})_{\hat{\mu}})\frac{1}{\mathcal{D}\cdot\hat{\mu}}=&\\\
\left(\frac{W\cdot\hat{\lambda}_{1}}{W\cdot\mathcal{D}}\frac{1}{\mathcal{D}\cdot\hat{\mu}_{1}}-\frac{W\cdot\hat{\lambda}_{2}}{W\cdot\mathcal{D}}\frac{1}{\mathcal{D}\cdot\hat{\mu}_{2}}\right)&\\\
(-\Delta(f_{1}f_{2})_{\hat{\mu}_{1}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{1}}).\end{split}$
(126)

A vector identity, valid for any $W$, $\mathcal{D}$ and two pairs
$(\hat{\lambda}_{1},\hat{\mu}_{1})$ and $(\hat{\lambda}_{2},\hat{\mu}_{2})$,

$({W\cdot\hat{\lambda}_{1}})(\mathcal{D}\cdot\hat{\mu}_{2})-({W\cdot\hat{\lambda}_{2}})(\mathcal{D}\cdot\hat{\mu}_{1})=(W\cdot\mathcal{D})(\hat{\lambda}_{1}\cdot\hat{\mu}_{2})$
(127)

brings the right-hand side of (126) into the form

$\begin{split}\left(\frac{\hat{\lambda}_{1}\cdot\hat{\mu}_{2}}{(\mathcal{D}\cdot\hat{\mu}_{1})(\mathcal{D}\cdot\hat{\mu}_{2})}\right)(-\Delta(f_{1}f_{2})_{\hat{\mu}_{1}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{1}}).\end{split}$
(128)

The final step is to prove that

$\begin{split}(-\Delta(f_{1}f_{2})_{\hat{\mu}_{1}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{1}})(\hat{\lambda}_{1}\cdot\hat{\mu}_{2})=\\\
(\Delta f_{1})_{\hat{\mu}_{1}}(\Delta
f_{2})_{\hat{\mu}_{2}}\absolutevalue{\hat{\lambda}_{1}\cdot\hat{\mu}_{2}}\;.\end{split}$
(129)

This last equality can be established by going through every of the four cases
of pairs $(\hat{\lambda},\hat{\mu})$. Another way is to notice that
$-\Delta(f_{1}f_{2})_{\hat{\mu}_{1}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{1}}$ is
equal to $(\Delta f)_{\hat{\mu}_{1}}(\Delta g)_{\hat{\mu}_{2}}$ up to a sign,
i.e.
$-\Delta(f_{1}f_{2})_{\hat{\mu}_{1}}-\Delta(f_{1}f_{2})_{-\hat{\mu}_{1}}=\pm(\Delta
f_{1})_{\hat{\mu}_{1}}(\Delta f_{2})_{\hat{\mu}_{2}}$. This is so, because the
left-hand side is invariant under the change in sign
$\hat{\mu}_{1}\rightarrow-\hat{\mu}_{1}$, whereas the right-hand side changes
its sign. The absolute value eliminates this change in sign and so the
equality is valid.

Appendix I: The optimization procedure for selecting the dielectric constants

Table 12: Mode ordering and Rayleigh frequency for $\theta=17.5^{\circ}$ and
$\varphi=0^{\circ}$

$(M_{x},M_{y})$ | m | Pol.,m |  $f_{R}$[GHz]
---|---|---|---
(0,0) | 1 | S1=S(0,0), P1=P(0,0) | 0.0
(-1,0) | 2 | S2=S(-1,0), P2=P(-1,0) | 36.6827
(0,-1) | 3 | S3=S(0,-1), P3=P(0,-1) | 50.0289
(0,1) | 4 | S4=S(0,1), P4=P(0,1) | 50.0289
(-1,-1) | 5 | S5=S(-1,-1), P5=P(-1,-1) | 56.7146
(-1,1) | 6 | S6=S(-1,1), P6=P(-1,1) | 56.7146
(1,0) | 7 | S7=S(1,0), P7=P(1,0) | 68.23087
(-2,0) | 8 | S8=S(-2,0), P8=P(-2,0) | 73.36548
(-2,-1) | 9 | S9=S(-2,-1), P9=P(-2,-1) | 84.68336
(-2,1) | 10 | S10=S(-2,1), P10=P(-2,1) | 84.68336

00footnotetext: The odd and even Polarization-Bloch-Floquet No., in column 3,
corresponds to the S and P polarization, respectively. It is easier to use the
index that runs from 1 to 20 in columns 3, when working with a $20\times 20$
matrix. However, to associate a meaning to the Polarizatio-Bloch-Floquet No.,
the other two styles shown in column 3 may be better.

Construct, out of (52), a $8\times 5$ sub-matrix composed of transfer matrix
elements that couples only the S-polarization of the four modes selected,
$(0,0),(-1,0),(0,-1),(0,1)$. The equations to solve are obtained from formula
(49). The OUT set (DIR,POL,BF) is composed of ($\pm$,S,m), with $m=1,\dots,4$
from Table 12. Given the absence of any source at $z=+\infty$, the
coefficients $C_{\text{POL,BF}}^{\text{DIR}}$ are all zero for ($-$,S,m),
$m=1,\dots,4$. At IN, the set (dir,pol,bf) contains the incoming ($+$,S,1) and
the reflected ($-$,S,m), with $m=1,\dots,4$. There are 8 unknown coefficients,
4 at OUT and another 4 at IN.

The $8\times 5$ matrix transfers the input column vector
$(\bm{\mathcal{C}}^{+}_{S1},\bm{\mathcal{C}}^{-}_{S1},\bm{\mathcal{C}}^{-}_{S2},\bm{\mathcal{C}}^{-}_{S3},\bm{\mathcal{C}}^{-}_{S4})$
into the output $(C^{+}_{S1},C^{+}_{S2},C^{+}_{S3},C^{+}_{S4},0,0,0,0)$. For
ease of reading, the IN and OUT coefficients are marked using distinct fonts.
Within this approximation, we can solve $8$ equations for $8$ unknowns
coefficients, by fixing the input $\bm{\mathcal{C}}^{+}_{S1}=1$. The
coefficients thus found are the transmission and reflection coefficients for
the $4$ modes under consideration. To construct an objective function for the
optimization process we use a series of properties. The first property is that
in the vicinity of the resonant frequency the absolute value of the
determinant of the $4\times 4$ matrix $(T^{-,-}_{Sm,Sn})$ attains a minimum.
The second one is that the absolute value of the transmission and reflection
coefficients of the evanescent modes need to attain high values. The third
property is based on the absorptance factor defined as
$1-\absolutevalue{C^{+}_{S1}}^{2}-\absolutevalue{C^{+}_{S2}}^{2}-\absolutevalue{\bm{\mathcal{C}}^{-}_{S1}}^{2}-\absolutevalue{\bm{\mathcal{C}}^{-}_{S2}}^{2}$,
similarly to the definition from [44]. The absorptance factor goes through a
maximum around the resonant frequency. The objective function we use is

$\displaystyle\absolutevalue{Det(T^{-,-}_{Sm,Sn})}+$ (130)
$\displaystyle\absolutevalue{\absolutevalue{C^{+}_{S1}}^{2}+\absolutevalue{C^{+}_{S2}}^{2}+\absolutevalue{\bm{\mathcal{C}}^{-}_{S1}}^{2}+\absolutevalue{\bm{\mathcal{C}}^{-}_{S2}}^{2}-0.5}$
$\displaystyle+\absolutevalue{\absolutevalue{C^{+}_{S3}}-10}+\absolutevalue{\absolutevalue{C^{+}_{S4}}-10}\;.$

Only the transmission coefficients for the evanescent waves are present in
(130) because we expect that the reflection coefficients be very close, in
absolute value, to the transmission values. We choose $10$ as a target number
for this values. For the absorptance factor the target values is chosen at
$0.5$.

For a fixed frequency in the range $37$ GHz to $50$ GHz, we found a local
minimum for this objective function in the 6-dimensional space of the
dielectric constants, each constant constrained to be between 1 and 12. After
scanning 100 frequencies, the local minimum we selected has $f=41.29$ GHz,
$Det(T^{-,-}_{Sm,Sn})=2.5\times 10^{-5}-3.7\times 10^{-5}i$,
$C^{+}_{S1}=0.59-0.0022i$, $\bm{\mathcal{C}}^{-}_{S1}=-0.21-0.59i$,
$C^{+}_{S2}=0.041+0.31i$, $\bm{\mathcal{C}}^{-}_{S2}=0.04+0.34i$,
$C^{+}_{S3}=-10.03-0.15i$, $\bm{\mathcal{C}}^{-}_{S3}=-10.61-0.18i$,
$C^{+}_{S4}=-3.92+9.20i$ and $\bm{\mathcal{C}}^{-}_{S4}=-4.13+9.74i$. We see
that the obtained determinant is a small number and that the reflection and
the transmissions coefficients are close, in absolute value, to the value 10
we imposed. The absorptance factor came out as $0.04$, far from the value of
$0.5$. This is because in the vicinity of the resonant frequency the
transmission and reflection coefficients of the propagative modes attain a
minimum at frequencies slightly different than the resonance. The dielectric
permittivity set for the local minimum is

$\displaystyle\epsilon_{\text{1B}}=4.98655,\epsilon_{\text{2B}}=2.95855,\epsilon_{\text{1L}}=7.59549,$
(131)
$\displaystyle\epsilon_{\text{2L}}=5.663,\epsilon_{\text{1R}}=5.92965,\epsilon_{\text{2R}}=1.63434$

Appendix J: Complex frequency plane

The real resonant frequency is located at the minimum of the determinant on
the real $\Omega$-axis. The zero of the cofactor is located in the upper
complex plane, $\text{Im}(f_{zero})>0$

Figure 26:  The domain used for the cofactor surrounds the zero $f_{zero}$.
The domain for determinant does not contain $f_{pole}$, but it contains the
resonant frequency. Both domains are above the real $\Omega$ axis

. A search for the minimum of the absolute value of the
$\text{Cofactor}_{S(0,0),S(0,1)}(T^{-,-})$ gives a complex frequency
$\Omega_{zero}=0.213999052599+0.00014322547392i$, for which the Cofactor
attains the value of $-4.17186\times 10^{-7}+5.51181\times 10^{-7}i$.
Translated in frequency, using $c=0.25\text{mm}$, we get
$f_{zero}=40.8425+0.0273351i$ in GHz. Once this zero is localized, the
parameters $a$ and $b$ are estimated through a linear regression, [38].

The linear approximation of the cofactor in the vicinity of $\Omega_{zero}$,

$\text{Cofactor}_{S(0,0),S(0,1)}(T^{-,-})\cong(a+ib)(\Omega-\Omega_{zero}),$
(132)

contains two unknown parameters, $a$ and $b$. The numerical data consists of
pairs $(\Omega,\text{Cofactor}_{S(0,0),S(0,1)}(T^{-,-}))$ obtained from
sampling the upper complex-frequency plane using the analytical formulas for
the transfer matrix.

The linear approximation for the determinant starts with localizing
$\text{Re}(\Omega_{resonance})=0.21632449245970709$ on the real axis at the
point where the absolute value of the determinant becomes minimum, attaining
the value $1.15869$. In terms of the frequency, the resonance is at
$\text{Re}(f_{resonance})=41.28635279688047$ GHz. The linear approximation of
the determinant in the vicinity of
$\Omega_{pole}=0.21632449245970709+iy_{pole}$,

$Det(T^{-,-})\cong(a^{{}^{\prime}}+ib^{{}^{\prime}})(\Omega-\Omega_{pole}),$
(133)

contains three unknown parameters, $a^{{}^{\prime}}$, $b^{{}^{\prime}}$ and
$y_{pole}$. The numerical data consists of pairs $(\Omega,Det(T^{-,-}))$
obtained from sampling the upper complex-frequency plane around
$Re(\Omega_{resonance})$ using the analytical formulas for the transfer
matrix. The fit this time is nonlinear because of the product
$(a^{{}^{\prime}}+ib^{{}^{\prime}})y_{pole}$.

Figure 27:  The real part of the scattering matrix element. The dashed curve
(orange) is the zero-pole approximation, whereas the continuous curve (black)
is derived from the 20 by 20 transfer matrix using the first 10 Rayleigh-
ordered modes. Figure 28:  The imaginary part of the scattering matrix
element. The dashed curve (orange) is the zero-pole approximation, whereas the
continuous curve (black) is derived from the 20 by 20 transfer matrix using
the first 10 Rayleigh-ordered modes.

## References

  * Xiong  _et al._ [2021] J. Xiong, E.-L. Hsiang, Z. He, T. Zhan, and S.-T. Wu, Augmented reality and virtual reality displays: emerging technologies and future perspectives, Light: Science & Applications 10, 1 (2021).
  * Kress and Chatterjee [2021] B. C. Kress and I. Chatterjee, Waveguide combiners for mixed reality headsets: a nanophotonics design perspective, Nanophotonics 10, 41 (2021).
  * Shlezinger  _et al._ [2021] N. Shlezinger, G. C. Alexandropoulos, M. F. Imani, Y. C. Eldar, and D. R. Smith, Dynamic metasurface antennas for 6g extreme massive mimo communications, IEEE Wireless Communications 28, 106 (2021).
  * Lorenzo  _et al._ [2016] J. Lorenzo, A. Lázaro, R. Villarino, and D. Girbau, Modulated frequency selective surfaces for wearable rfid and sensor applications, IEEE Transactions on Antennas and Propagation 64, 4447 (2016).
  * Descrovi  _et al._ [2010] E. Descrovi, T. Sfez, M. Quaglio, D. Brunazzo, L. Dominici, F. Michelotti, H. P. Herzig, O. J. Martin, and F. Giorgis, Guided bloch surface waves on ultrathin polymeric ridges, Nano letters 10, 2087 (2010).
  * Lova  _et al._ [2018] P. Lova, G. Manfredi, and D. Comoretto, Advances in functional solution processed planar 1d photonic crystals, Advanced Optical Materials 6, 1800730 (2018).
  * Feynman [2018] R. P. Feynman, Simulating physics with computers, in  _Feynman and computation_ (CRC Press, 2018) pp. 133–153.
  * Hagar [2014] A. Hagar,  _Discrete or continuous?: the quest for fundamental length in modern physics_ (Cambridge University Press, 2014).
  * Nörlund [1924] N. E. Nörlund,  _Vorlesungen über differenzenrechnung_ , Vol. 13 (J. Springer, 1924).
  * Nijhoff and Capel [1995] F. Nijhoff and H. Capel, The discrete korteweg-de vries equation, Acta Applicandae Mathematica 39, 133 (1995).
  * Feigenbaum [1978] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, Journal of statistical physics 19, 25 (1978).
  * Cvitanović [2017] P. Cvitanović, Universality in chaos, in  _Universality in chaos_ (Routledge, 2017) pp. 3–34.
  * Mandelbrot and Mandelbrot [1982] B. B. Mandelbrot and B. B. Mandelbrot,  _The fractal geometry of nature_ , Vol. 1 (WH freeman New York, 1982).
  * Korteweg and De Vries [1895] D. J. Korteweg and G. De Vries, Xli. on the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39, 422 (1895).
  * Hirota [1977] R. Hirota, Nonlinear partial difference equations iii; discrete sine-gordon equation, Journal of the Physical Society of Japan 43, 2079 (1977).
  * Hirota [1981] R. Hirota, Discrete analogue of a generalized toda equation, Journal of the Physical Society of Japan 50, 3785 (1981).
  * Krichever  _et al._ [1997] I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, Quantum integrable models and discrete classical hirota equations, Communications in Mathematical Physics 188, 267 (1997).
  * Lipan  _et al._ [1997] O. Lipan, P. B. Wiegmann, and A. Zabrodin, Fusion rules for quantum transfer matrices as a dynamical system on grassmann manifolds, Modern Physics Letters A 12, 1369 (1997).
  * Pendry and MacKinnon [1992] J. Pendry and A. MacKinnon, Calculation of photon dispersion relations, Physical Review Letters 69, 2772 (1992).
  * Pendry [1994] J. Pendry, Photonic band structures, Journal of Modern Optics 41, 209 (1994).
  * Pendry and Bell [1996] J. Pendry and P. Bell, Transfer matrix techniques for electromagnetic waves, in  _Photonic Band Gap Materials_ (Springer, 1996) pp. 203–228.
  * Li and Lin [2003] Z.-Y. Li and L.-L. Lin, Photonic band structures solved by a plane-wave-based transfer-matrix method, Physical Review E 67, 046607 (2003).
  * Sánchez-Soto  _et al._ [2012] L. L. Sánchez-Soto, J. J. Monzón, A. G. Barriuso, and J. F. Cariñena, The transfer matrix: A geometrical perspective, Physics Reports 513, 191 (2012).
  * Rhazi and Atalla [2010] D. Rhazi and N. Atalla, A simple method to account for size effects in the transfer matrix method, The journal of the acoustical society of america 127, EL30 (2010).
  * Yeh  _et al._ [1977] P. Yeh, A. Yariv, and C.-S. Hong, Electromagnetic propagation in periodic stratified media. i. general theory, JOSA 67, 423 (1977).
  * Fan [2002] S. Fan, Sharp asymmetric line shapes in side-coupled waveguide-cavity systems, Applied Physics Letters 80, 908 (2002).
  * Koufidis and McCall [2022] S. F. Koufidis and M. W. McCall, Möbius transformation and coupled-wave theory: Complete identification of the transfer matrix, Physical Review A 106, 062213 (2022).
  * Faddeev [1996] L. Faddeev, How algebraic bethe ansatz works for integrable model, arXiv preprint hep-th/9605187 (1996).
  * Grammaticos  _et al._ [2004] B. Grammaticos, T. Tamizhmani, and Y. Kosmann-Schwarzbach, Discrete integrable systems, Discrete Integrable Systems 644 (2004).
  * Pendry [1974] J. B. Pendry,  _Low energy electron diffraction: the theory and its application to determination of surface structure_ , 2 (Academic Press, 1974).
  * Png and Akimov [2017] C. E. J. Png and Y. Akimov, Nanophotonics and plasmonics: An integrated view, (2017).
  * Fano [1935] U. Fano, Sullo spettro di assorbimento dei gas nobili presso il limite dello spettro d’arco, Il Nuovo Cimento (1924-1942) 12, 154 (1935).
  * Avrutsky  _et al._ [2013] I. Avrutsky, R. Gibson, J. Sears, G. Khitrova, H. Gibbs, and J. Hendrickson, Linear systems approach to describing and classifying fano resonances, Physical Review B 87, 125118 (2013).
  * Schwinger  _et al._ [1998] J. Schwinger, L. L. DeRaad Jr, K. Milton, and W.-y. Tsai,  _Classical electrodynamics_ (Westview Press, 1998).
  * Lee and Mittra [1983] S.-W. Lee and R. Mittra, Fourier transform of a polygonal shape function and its application in electromagnetics, IEEE Transactions on Antennas and Propagation 31, 99 (1983).
  * Wuttke [2017] J. Wuttke, Form factor (fourier shape transform) of polygon and polyhedron, arXiv preprint arXiv:1703.00255 (2017).
  * [37] full-wave CST Microwave studio .
  * [38] W. R. Inc., Mathematica, Version 11.3, champaign, IL, 2018\.
  * Barbuto  _et al._ [2021] M. Barbuto, Z. Hamzavi-Zarghani, M. Longhi, A. Monti, D. Ramaccia, S. Vellucci, A. Toscano, and F. Bilotti, Metasurfaces 3.0: a new paradigm for enabling smart electromagnetic environments, IEEE Transactions on Antennas and Propagation (2021).
  * Zhou  _et al._ [2021] J. Zhou, P. Zhang, J. Han, L. Li, and Y. Huang, Metamaterials and metasurfaces for wireless power transfer and energy harvesting, Proceedings of the IEEE (2021).
  * Atwater and Polman [2010] H. A. Atwater and A. Polman, Plasmonics for improved photovoltaic devices, Nature materials 9, 205 (2010).
  * Eteng  _et al._ [2021] A. A. Eteng, H. H. Goh, S. K. A. Rahim, and A. Alomainy, A review of metasurfaces for microwave energy transmission and harvesting in wireless powered networks, IEEE Access 9, 27518 (2021).
  * Ferreira  _et al._ [2018] D. Ferreira, R. F. Caldeirinha, I. Cuinas, and T. R. Fernandes, A review of manufacturing materials and production methods for frequency-selective structures [wireless corner], IEEE Antennas and Propagation Magazine 60, 110 (2018).
  * Polo  _et al._ [2013] J. Polo, T. Mackay, and A. Lakhtakia,  _Electromagnetic surface waves: a modern perspective_ (Newnes, 2013).