# Dirac series for complex $E_{7}$

Chao-Ping Dong  and  Kayue Daniel Wong  School of Mathematical Sciences,
Soochow University, Suzhou 215006, P. R. China<EMAIL_ADDRESS>School of
Science and Engineering, The Chinese University of Hong Kong, Shenzhen,
Guangdong 518172, P. R. China<EMAIL_ADDRESS>
###### Abstract.

This paper classifies the Dirac series for complex $E_{7}$. As applications,
we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional
Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić
[BP], our classification should be helpful for understanding the unitary dual
of complex $E_{7}$.

###### Key words and phrases:

Dirac cohomology, spin norm, unitary representation.

###### 2010 Mathematics Subject Classification:

Primary 22E46.

##  1\. Introduction

Let $G$ be a connected simple Lie group with finite center. Let $\theta$ be
the Cartan involution. Then $K:=G^{\theta}$ is a maximal compact subgroup of
$G$. Let $\mathfrak{g}_{0}$ (resp., $\mathfrak{k}_{0}$) be the Lie algebra of
$G$ (resp., $K$). Let $T$ be a maximal torus of $K$. Let
$\mathfrak{a}_{0}=\sqrt{-1}\mathfrak{t}_{0}$. Put $A=\exp(\mathfrak{a}_{0})$.
Then $H:=TA$ is the fundamental Cartan subgroup of $G$. Let

$\mathfrak{g}_{0}=\mathfrak{k}_{0}+\mathfrak{p}_{0}$

be the Cartan decomposition on the Lie algebra level. As usual, we drop the
subscripts to denote the complexifications. Let $\langle\cdot,\cdot\rangle$ be
the Killing form on $\mathfrak{g}$. Let $U(\mathfrak{g})$ be the universal
enveloping algebra of $\mathfrak{g}$ and let $C(\mathfrak{p})$ be the Clifford
algebra of $\mathfrak{p}$.

To give geometric construction of the discrete series, Parthasarathy
introduced the Dirac operator $D\in U(\mathfrak{g})\otimes C(\mathfrak{p})$
whose square is a natural Laplacian on $G$ in 1972 [P1]. See (5). Let $\pi$ be
an irreducible $(\mathfrak{g},K)$ module. Choose a spin module $S$ of
$C(\mathfrak{p})$. Let $\widetilde{K}$ be the subgroup of $K\times{\rm
Spin\,\mathfrak{p}_{0}}$ consisting of the pairs $(k,s)$ such that ${\rm
Ad}(k)=p(s)$, where ${\rm Ad}:K\to SO(\mathfrak{p}_{0})$ is the adjoint action
and $p:{\rm Spin}\,\mathfrak{p}_{0}\to SO(\mathfrak{p}_{0})$ is the spin
double covering map.

The Dirac operator $D$ acts on $X\otimes S$. Whenever $\pi$ is unitary, the
operator $D$ is self-adjoint on $X\otimes S$; moreover, the eigenvalue of
$D^{2}$ on any $\widetilde{K}$-type of $X\otimes S$ is non-negative. This
leads to Parthasarathy’s Dirac operator inequality [P2]. See (7). It is very
effective in detecting non-unitary modules and thus a valuable tool for
studying the unitary dual $\widehat{G}$.

To sharpen the Dirac inequality, and to obtain a better understanding of
$\widehat{G}$, Vogan introduced Dirac cohomology [V2] as the following
$\widetilde{K}$-module:

(1) $H_{D}(\pi):=\mathop{\hbox{Ker}}\nolimits D/\mathop{\hbox{Ker}}\nolimits
D\cap\mathop{\hbox{Im}}\nolimits D.$

Vogan conjectured that $H_{D}(\pi)$ (whenever nonzero) should reveal the
infinitesimal character of $\pi$, suggesting that Dirac cohomology should be a
finer invariant of $\pi$. This conjecture was proven by Huang and Pandžić
[HP1] in 2002. See Theorem 2.2. Since then, classifying
$\widehat{G}^{\mathrm{d}}$—the set of all the equivalence classes of
irreducible unitarizable $(\mathfrak{g},K)$ modules with non-vanishing Dirac
cohomology—became an interesting open problem. As suggested by Huang, we call
$\widehat{G}^{\mathrm{d}}$ the _Dirac series_ of $G$.

It is worth noting that when $\pi$ is unitary,
$H_{D}(\pi)=\mathop{\hbox{Ker}}\nolimits D=\mathop{\hbox{Ker}}\nolimits
D^{2}$. Thus $\pi$ is a Dirac series of $G$ if and only if zero is an
eigenvalue of $D^{2}$ on $\pi\otimes S$. Therefore, Dirac series are exactly
the members of $\widehat{G}$ on which the Dirac inequality is attained.

This paper aims to classify $\widehat{E}_{7}^{\mathrm{d}}$—the Dirac series of
the connected and simply connected complex Lie group such that
$\mathfrak{g}_{0}$ is of type $E_{7}$. Although many results quoted below hold
in a much wider setting, for convenience, we specialize $G$ to be _complex_
from now on. Let us build up necessary notation to state our main result. We
identify

(2)
$\mathfrak{g}\cong\mathfrak{g}_{0}\oplus\mathfrak{g}_{0},\quad\mathfrak{h}\cong\mathfrak{h}_{0}\oplus\mathfrak{h}_{0},\quad\mathfrak{t}\cong\\{(x,-x):x\in\mathfrak{h}_{0}\\},\quad\mathfrak{a}\cong\\{(x,x):x\in\mathfrak{h}_{0}\\}.$

Denote the Weyl group of $\Delta(\mathfrak{g}_{0},\mathfrak{h}_{0})$ by $W$,
which has identity element $e$ and longest element $w_{0}$. Then
$W(\Delta(\mathfrak{g},\mathfrak{h}))$ is isomorphic to $W\times W$. By
Theorem 2.1, every irreducible admissible $(\mathfrak{g},K)$ module is
isomorphic to certain $J(\lambda_{L},\lambda_{R})$, see Section 2.1.

Figure 1. Dynkin diagram for $E_{7}$

Let us fix the simple roots of complex $E_{7}$ as in Fig. 1. We enumerate the
corresponding fundamental weights as $\varpi_{1},\dots,\varpi_{7}$, and use
them as a basis to express weights. That is,
$\lambda=[\lambda_{1},\cdots,\lambda_{7}]$ stands for the weight
$\sum_{i=1}^{7}\lambda_{i}\varpi_{i}$. We say that $\lambda$ is _integral_
(resp., _half-integral_) if each coordinate $\lambda_{i}$ (resp.,
$2\lambda_{i}$) is an integer. In this terminology, half sum of the positive
roots is expressed as $\rho(E_{7})=[1,1,1,1,1,1,1]$ and it is integral
regular. There are $10208$ involutions in the Weyl group $W(E_{7})$ in total,
among which $8479$ have the property that

(3) $I(s):=\\{1\leq i\leq 7\mid s(\varpi_{i})=\varpi_{i}\\}$

is empty.

Write $\\{1,2,3,\cdots\\}$ as $\mathbb{P}$ and $\\{0,1,2,\cdots\\}$ as
$\mathbb{N}$. For any fixed involution $s\in W(E_{7})$, we collect all these
$\lambda=[\lambda_{1},\dots,\lambda_{l}]$ such that

  * $\bullet$

$2\lambda_{i}\in\mathbb{P}$;

  * $\bullet$

$\lambda+s\lambda$ is integral;

  * $\bullet$

$\lambda-s\lambda$ is a $\mathbb{N}$-combination of simple roots

as $\Lambda(s)$. By page 5 of [BP] and Section 2.6 of [D2], any representation
in $\widehat{E}_{7}^{\mathrm{d}}$ must bear the form

(4) $J(\lambda,-s\lambda),$

where $s\in W(E_{7})$ is an involution and $\lambda\in\Lambda(s)$. This module
has lowest $K$-type $\\{\lambda+s\lambda\\}$, the unique dominant weight to
which $\lambda+s\lambda$ is conjugate under the action of $W(E_{7})$.
_Scattered representations_ (resp., _string representations_) of complex
$E_{7}$ consist of those $J(\lambda,-s\lambda)$ in
$\widehat{E}_{7}^{\mathrm{d}}$ such that $I(s)$ is empty (resp., non-empty).

By Theorem A of [DD], the set $\widehat{E}_{7}^{\mathrm{d}}$ consists of
finitely many scattered representations and finitely many strings; moreover,
to figure out $\widehat{E}_{7}^{\mathrm{d}}$, it suffices to pin down the
scattered representations in $\widehat{L}_{\mathrm{ss}}^{\mathrm{d}}$, where
$L$ runs over the Levi factors of all the $\theta$-stable parabolic
subalgebras of $G$ and $L_{\mathrm{ss}}:=[L,L]$ is the semisimple factor of
$L$. An algorithm for computing these scattered representations is provided in
[D2, Section 4]. It will be improved slightly in Section 4.

Our main result is the following.

Theorem A. _The set $\widehat{E}_{7}^{\mathrm{d}}$ consists of $66$ scattered
representations (see Tables 3 and 4) whose spin-lowest $K$-types are all
unitarily small, and $578$ strings of representations (see Section 3).
Moreover, each representation $\pi\in\widehat{E}_{7}^{\mathrm{d}}$ has a
unique spin-lowest $K$-type occurring with multiplicity one._

In Theorem A, the notion unitarily small (_u-small_ for short henceforth) was
introduced by Salamanca-Riba and Vogan [SV]. The above result confirms the
following conjectures for the group complex $E_{7}$:

  * $\bullet$

Conjecture 1.1 of [BP] in 2011 (see Corollary 6.1);

  * $\bullet$

Conjecture 3.4 of [BP] in 2011;

  * $\bullet$

Conjecture B of [D2] in 2019;

  * $\bullet$

Conjecture C of [DD] in 2020 (via the aid of [DW1, Lemma 3.4]).

The paper is organized as follows: Section 2 collects necessary preliminaries.
Section 3 (resp., Section 4) descries the string part (resp., the scattered
part) of $\widehat{E}_{7}^{\mathrm{d}}$. We realize 63 scattered members in
$\widehat{E}_{7}^{\mathrm{d}}$ as string limits in Section 5, thus providing
interesting links between the string part and the scattered part. The three
scattered members which can _not_ be realized as string limits are unipotent
representations (not necessarily special). They are discussed in Section 6.
The last section aims to develop effective ways to determine the spin LKTs.
For string limits, the branching result from [BDW] and a technique suggested
by Vogan are helpful. For unipotent representations, we use McGovern [Mc],
Losev, Mason-Brown and Matvieievskyi [LMM].

Among all the exceptional Lie groups, complex $E_{8}$ is the hardest one from
the computational point of view. It is conceivable that our journey here will
help people to classify the Dirac series of complex $E_{8}$ in the future.

##  2\. Preliminaries

Let us adopt the setting of the introduction, and assume that $G$ is a
connected complex Lie group (viewed as a real Lie group).

###  2.1. Zhelobenko classification

Fix a Borel subgroup $B$ of $G$ containing $H$. Put

$\Delta^{+}(\mathfrak{g}_{0},\mathfrak{h}_{0})=\Delta(\mathfrak{b}_{0},\mathfrak{h}_{0}).$

Let $\rho$ be the half sum of roots in
$\Delta^{+}(\mathfrak{g}_{0},\mathfrak{h}_{0})$. Set

$\Delta^{+}(\mathfrak{g},\mathfrak{h})=\Delta^{+}(\mathfrak{g}_{0},\mathfrak{h}_{0})\times\\{0\\}\cup\\{0\\}\times(-\Delta^{+}(\mathfrak{g}_{0},\mathfrak{h}_{0})).$

Let $\rho_{\mathfrak{g}}$ be the half sum of roots in
$\Delta^{+}(\mathfrak{g},\mathfrak{h})$. When restricted to $\mathfrak{t}$, we
get $\Delta^{+}(\mathfrak{g},\mathfrak{t})$,
$\Delta^{+}(\mathfrak{k},\mathfrak{t})$ and
$\Delta^{+}(\mathfrak{p},\mathfrak{t})$. Denote by $\rho_{c}$ the half-sum of
roots in $\Delta^{+}(\mathfrak{k},\mathfrak{t})$. We may and we will identify
a $K$-type ($\widetilde{K}$-type, $\mathfrak{k}$-type, etc) with its highest
weight, which is necessarily dominant integral with respect to
$\Delta^{+}(\mathfrak{k},\mathfrak{t})$.

Let
$(\lambda_{L},\lambda_{R})\in\mathfrak{h}_{0}^{*}\times\mathfrak{h}_{0}^{*}$
be such that $\lambda_{L}-\lambda_{R}$ is a weight of a finite dimensional
holomorphic representation of $G$. We view $(\lambda_{L},\lambda_{R})$ as a
real-linear functional on $\mathfrak{h}$ by (2), and write
$\mathbb{C}_{(\lambda_{L},\lambda_{R})}$ as the character of $H$ with
differential $(\lambda_{L},\lambda_{R})$ (which exists). By (2) again, we have

$\mathbb{C}_{(\lambda_{L},\lambda_{R})}|_{T}=\mathbb{C}_{\lambda_{L}-\lambda_{R}},\quad\mathbb{C}_{(\lambda_{L},\lambda_{R})}|_{A}=\mathbb{C}_{\lambda_{L}+\lambda_{R}}.$

Extend $\mathbb{C}_{(\lambda_{L},\lambda_{R})}$ to a character of $B$, and put

$X(\lambda_{L},\lambda_{R}):=K\mbox{-finite part of
Ind}_{B}^{G}(\mathbb{C}_{(\lambda_{L},\lambda_{R})}).$

Using Frobenius reciprocity, the $K$-type with extremal weight
$\lambda_{L}-\lambda_{R}$ occurs with multiplicity one in
$X(\lambda_{L},\lambda_{R})$. Let $J(\lambda_{L},\lambda_{R})$ be the unique
subquotient of $X(\lambda_{L},\lambda_{R})$ containing the $K$-type
$\\{\lambda_{L}-\lambda_{R}\\}$ (recall that this notation stands for the
unique dominant weight to which $\lambda_{L}-\lambda_{R}$ is conjugate under
the Weyl group action).

######  Theorem 2.1.

(Zhelobenko [Zh]) In the above setting, we have that

  * a)

Every irreducible admissible ($\mathfrak{g}$, $K$)-module is of the form
$J(\lambda_{L},\lambda_{R})$.

  * b)

Two such modules $J(\lambda_{L},\lambda_{R})$ and
$J(\lambda_{L}^{\prime},\lambda_{R}^{\prime})$ are equivalent if and only if
there exists $w\in W$ such that $w\lambda_{L}=\lambda_{L}^{\prime}$ and
$w\lambda_{R}=\lambda_{R}^{\prime}$.

  * c)

$J(\lambda_{L},\lambda_{R})$ admits a nondegenerate Hermitian form if and only
if there exists $w\in W$ such that
$w(\lambda_{L}-\lambda_{R})=\lambda_{L}-\lambda_{R},w(\lambda_{L}+\lambda_{R})=-\overline{(\lambda_{L}+\lambda_{R})}$.

  * d)

The representation $X(\lambda_{L},\lambda_{R})$ is tempered if and only if
$\lambda_{L}+\lambda_{R}\in i\mathfrak{h}_{0}^{*}$. In this case,
$X(\lambda_{L},\lambda_{R})=J(\lambda_{L},\lambda_{R})$.

Note that $J(\lambda_{L},\lambda_{R})$ has lowest $K$-type
$\\{\lambda_{L}-\lambda_{R}\\}$ and infinitesimal character the $W\times W$
orbit of $(\lambda_{L},\lambda_{R})$.

###  2.2. Dirac cohomology

Let $Z_{1},\dots,Z_{n}$ be an orthonormal basis of $\mathfrak{p}_{0}$ with
respect to the inner product induced by the Killing form
$\langle\cdot,\cdot\rangle$. The Dirac operator is defined as

(5) $D=\sum_{i=1}^{n}Z_{i}\otimes Z_{i}\in U(\mathfrak{g})\otimes
C(\mathfrak{p}).$

It is easy to check that $D$ does not depend on the choice of the orthonormal
basis $\\{Z_{i}\\}_{i=1}^{n}$ and it is $K$-invariant for the diagonal action
of $K$ given by adjoint actions on both factors. By construction, $D^{2}$ is a
natural Laplacian on $G$. Indeed,

(6) $D^{2}=\Omega_{\mathfrak{k}_{\Delta}}-\Omega_{\mathfrak{g}}\otimes
1+(\|\rho_{c}\|^{2}-\|\rho_{\mathfrak{g}}\|^{2})1\otimes 1.$

Here $\Omega_{\mathfrak{g}}$ (resp., $\Omega_{\mathfrak{k}}$) is the Casimir
operator for $\mathfrak{g}$ (resp., $\mathfrak{k}$), and
$\Omega_{\mathfrak{k}_{\Delta}}$ is the image of $\Omega_{\mathfrak{k}}$ under
a diagonal embedding $\Delta$ of $\mathfrak{k}$ into $U(\mathfrak{g})\otimes
C(\mathfrak{p})$. See Section 3.1 of [HP2] for more.

Let $\gamma$ be the highest weight of any $\widetilde{K}$-type of $\pi\otimes
S$, where $\pi$ is an irreducible _unitary_ $(\mathfrak{g},K)$ module which
has infinitesimal character $\Lambda$. Then the eigenvalue of $D^{2}$ on
$(X\otimes S)(\gamma)$ is $\|\gamma+\rho_{c}\|^{2}-\|\Lambda\|^{2}$. This
leads to Parthasarathy’s Dirac operator inequality

(7) $\|\gamma+\rho_{c}\|\geq\|\Lambda\|.$

As been sharpened in Theorem 3.5.2 of [HP2], equality holds in (7) if and only
if $\gamma+\rho_{c}$ is conjugate to $\Lambda$ under the action of
$W(\mathfrak{g},\mathfrak{h})$. Indeed, if (7) becomes equality on certain
$\widetilde{K}$-type $\gamma$ of $\pi\otimes S$, then

$(\pi\otimes S)(\gamma)\subseteq\mathop{\hbox{Ker}}\nolimits
D^{2}=H_{D}(\pi).$

Thus by Theorem 2.2, $\gamma+\rho_{c}$ is conjugate to $\Lambda$ under the
action of $W(\mathfrak{g},\mathfrak{h})$.

The following result is the Vogan conjecture proven by Huang and Pandžić
[HP1]. It is foundational for the study of Dirac cohomology. For instance,
although $\pi$ is usually infinite-dimensional, it allows one to focus on
finitely many $K$-types to compute $H_{D}(\pi)$.

######  Theorem 2.2.

(Huang and Pandžić) Let $\pi$ be an irreducible ($\mathfrak{g}$, $K$) module.
Assume that the Dirac cohomology of $\pi$ is nonzero, and that it contains the
$\widetilde{K}$-type $E_{\gamma}$ with highest weight
$\gamma\in\mathfrak{t}^{*}\subset\mathfrak{h}^{*}$. Then the infinitesimal
character of $\pi$ is conjugate to $\gamma+\rho_{c}$ under
$W(\mathfrak{g},\mathfrak{h})$.

###  2.3. Spin norm and spin-lowest $K$-type

Inspired by the ideas of Parthasarathy, Vogan, Huang and Pandžić, the first
named author introduced spin norm and spin-lowest $K$-type in his 2011 HKUST
thesis (cf. [D1]). Given a $K$-type $\mu$, its _spin norm_ is defined as

(8) $\|\mu\|_{\rm spin}:=\|\\{\mu-\rho\\}+\rho\|.$

The spin norm $\|\pi\|_{\rm spin}$ of an irreducible $(\mathfrak{g},K)$ module
$\pi$ is defined as the minimum of the spin norm of all the $K$-types showing
up in $\pi$. We call $\mu$ a _spin-lowest $K$-type_ of $\pi$ if $\mu$ occurs
in $\pi$ and $\|\mu\|_{\rm spin}=\|\pi\|_{\rm spin}$.

Now assume that $\pi$ is unitary and has infinitesimal character $\Lambda$.
Let $\mu$ be any $K$-type of $\pi$. Note that the spin module $S$ is some
copies of the $\mathfrak{k}$-type $\rho$ (see Lemma 2.2 of [BP]). Thus using
the knowledge of PRV-component [PRV], one sees that the Dirac inequality (7)
can be reformulated as

(9) $\|\mu\|_{\rm spin}\geq\|\Lambda\|.$

Moreover, results from the previous section guarantee that:

  * $\bullet$

$\|\pi\|_{\rm spin}\geq\|\Lambda\|$, and equality holds if and only if
$H_{D}(\pi)$ is non-zero;

  * $\bullet$

$\|\mu\|_{\rm spin}\geq\|\Lambda\|$, and equality holds if and only if $\mu$
contributes to $H_{D}(\pi)$;

  * $\bullet$

If $H_{D}(\pi)\neq 0$, it is exactly the spin-lowest $K$-types of $\pi$ that
contribute to $H_{D}(\pi)$.

These properties motivate us to pay close attention to spin norm and spin-
lowest $K$-types when classifying Dirac series.

###  2.4. Vogan pencil

Let $\beta$ be the highest root of
$\Delta^{+}(\mathfrak{g}_{0},\mathfrak{h}_{0})$. As a special case of Lemma
3.4 and Corollary 3.5 of [V1], for any infinite-dimensional irreducible
($\mathfrak{g}$, $K$)-module $\pi$, there is a unique set

$\\{\mu_{\mathfrak{i}}\,|\,\mathfrak{i}\in I\\}\subseteq
i\mathfrak{t}_{0}^{*}$

of dominant integral weights such that all the $K$-types of $\pi$ are
precisely

$\\{\mu_{\mathfrak{i}}+n\beta\,|\,\mathfrak{i}\in I,n\in\mathbb{N}\\}.$

We call a set of $K$-types

(10) $P(\delta):=\\{\delta+n\beta\,|\,n\in\mathbb{N}\\}$

a _Vogan pencil_. For instance, $P(0)$ denotes the pencil starting from the
trivial $K$-type. Put

(11)
$P_{\delta}:=\min\\{\|\delta+n\beta\|_{\mathrm{spin}}\,|\,n\in\mathbb{N}\\}.$

That is, the minimum spin norm of the $K$-types on $P(\delta)$ is denoted as
$P_{\delta}$.

##  3\. The string part of $\widehat{E}_{7}^{\mathrm{d}}$

To understand the string part of $\widehat{E}_{7}^{\mathrm{d}}$, we should
obtain the scattered parts of $\widehat{A}_{i}^{\mathrm{d}}$ for $1\leq i\leq
6$ and $\widehat{D}_{j}^{\mathrm{d}}$ for $4\leq j\leq 6$. Here by $D_{j}$ we
actually mean the group $Spin(2j,\mathbb{C})$. Let us present the scattered
parts of $\widehat{A}_{6}^{\mathrm{d}}$ and $\widehat{D}_{6}^{\mathrm{d}}$ in
Tables 1 and 2, respectively. Here we present the folded version as in [D2,
Section 3]. Note that $N_{A_{6}}=32$ and $N_{D_{6}}=34$. By Lemma 3.4 of
[DW1], we conclude that each spin-lowest $K$-type of any scattered
representation in Table 1 or Table 2 is u-small. Note that scattered
representations of $\widehat{A_{i}}^{d}$ for $1\leq i\leq 5$ and
$\widehat{D_{j}}^{d}$ for $4\leq j\leq 5$ have been recorded in [D2].

We remark that the scattered representations of complex classical Lie groups
have been studied in [DW1] and [DW2]. Be careful that the group of type
$D_{j}$ in [DW2] actually means $SO(2j,\mathbb{C})$. In particular, a
representation in Table 2 passes to a representation of $SO(12,\mathbb{C})$ if
and only if $\lambda_{5}=\lambda_{6}$ holds there; otherwise, it is a genuine
representation of $Spin(12,\mathbb{C})$. In a subsequent work, we will
classify genuine unitary representations of $Spin(n,\mathbb{C})$ with non-zero
Dirac cohomology.

Table 1. The scattered part of $\widehat{A}_{6}^{\mathrm{d}}$ (folded version) $s\rho$ | $\lambda$ | spin LKT | mult
---|---|---|---
$[3,-5,3,3,-5,3]$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[-3,1,4,-6,5,-2]^{*}$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,1,1,2,0,1]$ | $1$
$[-2,-1,5,-6,5,-2]^{*}$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,0,0,4,0,1]$ | $1$
$[-4,2,-1,4,-6,4]^{*}$ | $[\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,1,0,1,2,1]$ | $1$
$[-2,-1,-1,5,-6,4]^{*}$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,0,2,1,1]$ | $1$
$[-2,-1,-1,5,-6,4]^{*}$ | $[1,1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,0,0,0,4,1]$ | $1$
$[-5,3,-1,-1,3,-5]$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[-5,3,-1,-1,3,-5]$ | $[\frac{1}{2},\frac{1}{2},1,1,\frac{1}{2},\frac{1}{2}]$ | $[3,1,0,0,1,3]$ | $1$
$[-3,1,-3,1,3,-5]^{*}$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[-2,3,-5,3,-4,2]^{*}$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,0,2,1,1]$ | $1$
$[-2,-2,1,-2,4,-5]^{*}$ | $[1,\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[2,1,0,0,2,2]$ | $1$
$[-2,-2,1,-2,4,-5]^{*}$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,0,0,3,1]$ | $1$
$[-2,-1,-1,-1,4,-5]^{*}$ | $[1,1,1,1,\frac{1}{2},\frac{1}{2}]$ | $[0,0,0,0,0,4]$ | $1$
$[-2,-2,3,-4,2,-3]^{*}$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,0,0,3,1]$ | $1$
$[-1,-3,1,1,-3,-1]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,2,0,0,2,1]$ | $1$
$[-1,-2,-1,2,-3,-1]^{*}$ | $[1,1,1,\frac{1}{2},\frac{1}{2},1]$ | $[2,0,0,0,0,4]$ | $1$
$-\rho$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,2,0,0,2,1]$ | $1$
$-\rho$ | $[1,1,\frac{1}{2},\frac{1}{2},1,1]$ | $[3,0,0,0,0,3]$ | $1$
$-\rho$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$-\rho$ | $\rho$ | $[0,0,0,0,0,0]$ | $1$
Table 2. The scattered part of $\widehat{D}_{6}^{\mathrm{d}}$ (folded version) $s\rho$ | $\lambda$ | spin LKT | mult
---|---|---|---
$[-3,1,5,-7,5,5]$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[-2,-1,6,-7,5,5]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,0,0,3,1,1]$ | $1$
$[-2,7,-8,6,-1,-1]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[1,0,4,1,0,0]$ | $1$
$[4,-6,5,4,-6,-2]^{*}$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,2,0,1,2,1]$ | $1$
$[7,-9,7,-3,1,1]$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[-5,4,-2,7,-8,-2]^{*}$ | $[\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2},1]$ | $[3,0,1,0,4,1]$ | $1$
$[5,-7,5,-7,5,5]$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[7,-9,7,-1,-1,-1]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1,1]$ | $[1,5,1,0,0,0]$ | $1$
$[-1,-3,1,7,-9,-1]^{*}$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[2,0,1,0,3,2]$ | $1$
$[-6,4,5,-6,-2,4]^{*}$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2}]$ | $[2,0,2,0,2,1]$ | $1$
$[-1,-2,-1,8,-9,-1]^{*}$ | $[1,1,1,\frac{1}{2},\frac{1}{2},1]$ | $[0,1,0,0,7,0]$ | $1$
$[-9,7,-5,3,-1,-1]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[1,3,1,1,0,0]$ | $1$
$[-2,-1,6,-8,-1,7]^{*}$ | $[1,1,\frac{1}{2},\frac{1}{2},1,\frac{1}{2}]$ | $[0,2,0,0,5,0]$ | $1$
$[-7,5,-7,5,-1,-1]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[1,3,1,1,0,0]$ | $1$
$[-9,7,-1,-3,1,1]$ | $[\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,1,2,1,0,0]$ | $1$
$[-1,-7,5,-3,1,1]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,1,2,0,0]$ | $1$
$[-9,7,-1,-1,-1,-1]$ | $[\frac{1}{2},\frac{1}{2},1,1,1,1]$ | $[7,1,0,0,0,0]$ | $1$
$[-1,-5,3,-5,3,3]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,1,2,0,0]$ | $1$
$[-2,4,-5,4,-6,-2]^{*}$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[0,2,0,1,0,3]$ | $1$
$[-1,-7,5,-1,-1,-1]$ | $[1,\frac{1}{2},\frac{1}{2},1,1,1]$ | $[5,2,0,0,0,0]$ | $1$
$[-1,-1,-5,3,-1,-1]$ | $[1,1,\frac{1}{2},\frac{1}{2},1,1]$ | $[3,3,0,0,0,0]$ | $1$
$[1,-3,1,-3,-1,1]^{*}$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2}]$ | $[0,2,0,2,0,1]$ | $1$
$[-1,-1,-1,-3,1,1]$ | $\frac{\rho}{2}$ | $\rho$ | $1$
$[-1,-1,-1,-3,1,1]$ | $[1,1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,4,0,0,0,0]$ | $1$
$-\rho$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,1,2,0,0]$ | $1$
$-\rho$ | $\rho$ | [0,0,0,0,0,0] | $1$

It is not hard to equip these scattered representations into strings of
$\widehat{E}_{7}^{\mathrm{d}}$. Let us see some examples.

######  Example 3.1.

Consider one extreme case $L=HA$. Then the corresponding string in
$\widehat{E}_{7}^{\mathrm{d}}$ is:

$[\lambda_{1},\dots,\lambda_{7}]\in\Lambda(e),$

where each $\lambda_{i}\in\frac{1}{2}\mathbb{P}$. All of them are tempered
representations. ∎

######  Example 3.2.

Let us consider another extreme case. Namely, now $L_{\mathrm{ss}}$ is
$D_{6}$. We focus on the first representation in Table 2, where
$s=s_{1}s_{2}s_{4}s_{3}s_{2}s_{1}s_{5}s_{6}s_{4}$ and

$\lambda=[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}].$

Note that $\lambda+s\lambda=[-1,1,3,-3,3,3]$ and
$\lambda-s\lambda=[2,0,-2,4,-2,-2]$. Recall that
$s\rho(D_{6})=[-3,1,5,-7,5,5]$.

There is a unique $\theta$-stable parabolic subalgebra of $G$ whose semisimple
factor is of type $D_{6}$—the one corresponding to
$\\{\alpha_{2},\dots,\alpha_{7}\\}$. Let us denote the counterpart of the
involution $s$ in it by $s^{\prime}$. One can find that

$s^{\prime}=s_{7}s_{6}s_{4}s_{5}s_{6}s_{7}s_{2}s_{3}s_{4},$

and that

$s^{\prime}\rho(E_{7})=[\textbf{3},5,5,-7,5,1,-3].$

Here the non-bolded coordinate comes from those of $s\rho(D_{6})$ via
permutations. Now the representation $J(\lambda,-s\lambda)$ of $D_{6}$ gives
the following string of $\widehat{E}_{7}^{\mathrm{d}}$:

$\lambda^{\prime}:=[\frac{a}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]\in\Lambda(s^{\prime}),$

where $a\in\mathbb{P}$. Note that

$\lambda^{\prime}+s^{\prime}\lambda^{\prime}=[a+1,3,3,-3,3,1,-1],\quad\lambda^{\prime}-s^{\prime}\lambda^{\prime}=[-1,-2,-2,4,-2,0,2].$

∎

We will not explicitly give all the strings of $\widehat{E}_{7}^{\mathrm{d}}$.
Instead, let us count them. For this purpose, we denote by $N_{i}$ the number
of $I$-strings in $\widehat{E}_{7}^{\mathrm{d}}$ such that $|I|=7-i$. Then
$N_{0}=1$ (see Example 3.1), and

$\displaystyle N_{1}$ $\displaystyle=7N_{A_{1}}=7,$ $\displaystyle N_{2}$
$\displaystyle=6N_{A_{2}}+15N_{A_{1}}N_{A_{1}}=27,$ $\displaystyle N_{3}$
$\displaystyle=6N_{A_{3}}+18N_{A_{2}}N_{A_{1}}+11N_{A_{1}}N_{A_{1}}N_{A_{1}}=71,$
$\displaystyle N_{4}$
$\displaystyle=N_{D_{4}}+5N_{A_{4}}+11N_{A_{3}}N_{A_{1}}+12N_{A_{2}}N_{A_{1}}N_{A_{1}}+4N_{A_{2}}N_{A_{2}}+2N_{A_{1}}N_{A_{1}}N_{A_{1}}N_{A_{1}}=135,$
$\displaystyle N_{5}$
$\displaystyle=2N_{D_{5}}+3N_{A_{5}}+N_{D_{4}}N_{A_{1}}+5N_{A_{4}}N_{A_{1}}+3N_{A_{3}}N_{A_{2}}+3N_{A_{3}}N_{A_{1}}N_{A_{1}}+3N_{A_{2}}N_{A_{2}}N_{A_{1}}$
$\displaystyle+N_{A_{2}}N_{A_{1}}N_{A_{1}}N_{A_{1}}=181,$ $\displaystyle
N_{6}$
$\displaystyle=N_{D_{6}}+N_{A_{6}}+N_{A_{5}}N_{A_{1}}+N_{A_{3}}N_{A_{2}}N_{A_{1}}+N_{A_{4}}N_{A_{2}}+N_{D_{5}}N_{A_{1}}+N_{E_{6}}=156.$

Here recall that $N_{G}$ is the cardinality of the scattered part of
$\widehat{G}^{\mathrm{d}}$. Therefore, $\widehat{E}_{7}^{\mathrm{d}}$ contains
$\sum_{i=0}^{6}N_{i}=578$ strings in total.

##  4\. The scattered part of $\widehat{E}_{7}^{\mathrm{d}}$

Similar to Section 4 of [D2], we proceed as follows to sieve out all the _non-
trivial_ scattered representations of complex $E_{7}$:

  * $\bullet$

collect the finitely many $\lambda\in\Lambda(s)$ (see the introduction) such
that $\|\lambda-s\lambda\|^{2}\leq 464$ and that $\|2\lambda\|^{2}\leq
P_{\\{\lambda+s\lambda\\}}^{2}$ (see (11)).

  * $\bullet$

For these $\lambda$ surviving from the above step (if exist), use atlas [ALTV,
At] to study the unitarity and $K$-types of $J(\lambda,-s\lambda)$.

In the first step, note that the earlier upper bound for
$\|\lambda-s\lambda\|^{2}$ is $\|2\rho\|^{2}$, which equals to $798$ for
complex $E_{7}$. This improvement is guaranteed by Theorem 1.1 of [D3].

The relevant files are available from the following link:

    https://www.researchgate.net/publication/353914493_E7-Files

######  Example 4.1.

Consider the involution
$s=s_{1}s_{4}s_{2}s_{3}s_{1}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}$. Note that
$s\rho(E_{7})=[-2,6,7,-8,6,1,-3]$. Carrying out the first step above leaves us
with the following $6$ candidate representations:

$\displaystyle[\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}],\quad[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}],\quad[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{2}],$
$\displaystyle[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2},1],\quad[1,\frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}],\quad[\frac{3}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}].$

Only the
$\lambda=[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$
produces a unitary representation. Note that

$\displaystyle\lambda+s\lambda=[0,4,4,-4,4,1,-1],\quad\lambda-s\lambda=[2,-3,-3,5,-3,0,2].$

It has a unique spin-lowest $K$-type $[1,1,0,2,1,1,1]$ whose spin norm equals
to $\|2\lambda\|$. Moreover, it occurs exactly once. This is the 1st entry of
Table 3. ∎

######  Example 4.2.

Consider the involution

$s=s_{1}s_{2}s_{3}s_{4}s_{2}s_{3}s_{4}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{5}s_{6}s_{7}.$

Note that $s\rho(E_{7})=[-17,-1,15,-1,-1,-1,-1]$. Carrying out the first step
above leaves us with $241$ candidate representations. Only
$\lambda=[\frac{1}{2},1,\frac{1}{2},1,1,1,1]$ produces a unitary
representation. Note that

$\lambda+s\lambda=[-14,0,14,0,0,0,0],\quad\lambda+s\lambda=[15,2,-13,2,2,2,2].$

It has a unique spin-lowest $K$-type $[13,0,1,0,0,0,0]$ whose spin norm equals
to $\|2\lambda\|$. Moreover, it occurs exactly once. This is the 1st entry of
Table 4. ∎

######  Example 4.3.

Consider the longest involution $w_{0}=-1$ in $W(E_{7})$. Note that
$w_{0}\rho(E_{7})=[-1,-1,-1,-1,-1,-1,-1]$. Carrying out the first step above
leaves us with $116$ candidate representations. Only the
$\lambda=[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$
produces a unitary representation. It has a unique spin-lowest $K$-type
$[1,0,1,2,0,2,0]$ whose spin norm equals to $\|2\lambda\|$. Moreover, it
occurs exactly once. This is the 21st entry of Table 4. ∎

##  5\. String limits

This section aims to explain the last columns of Tables 3 and 4.

######  Example 5.1.

The first row of [D2, Table 6] gives a scattered member of
$\widehat{E}_{6}^{\mathrm{d}}$ with
$s=s_{4}s_{5}s_{6}s_{5}s_{1}s_{3}s_{2}s_{4}s_{1}$ and
$\lambda=[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$. Note that
$s\rho(E_{6})=[-2,5,6,-7,6,-2]$. This scattered representation in
$\widehat{E}_{6}^{\mathrm{d}}$ produces the following string in
$\widehat{E}_{7}^{\mathrm{d}}$:

(12)
$J(\lambda^{\prime},-s\lambda^{\prime}),\quad\lambda^{\prime}=[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{g}{2}],$

where $g\in\mathbb{P}$. Note that

$\lambda^{\prime}+s\lambda^{\prime}=[0,4,4,-4,4,0,g+2],\quad\lambda^{\prime}-s\lambda^{\prime}=[2,-3,-3,5,-3,2,-2].$

If we set $g=-1$, the corresponding representation is no longer a member of
the string. However,
$s_{7}([1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,-\frac{1}{2}])=[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$;
moreover, if we set $s^{\prime}=s_{7}ss_{7}^{-1}$, then $s^{\prime}$ is an
involution of $W(E_{7})$ such that $I(s^{\prime})$ is empty and that
$s^{\prime}\rho(E_{7})=[-2,6,7,-8,6,1,-3]$. One sees that $s^{\prime}$ and
$[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$
give the first row of Table 3. Therefore, we may view the first entry of Table
3 (see Example 4.1), a scattered member of $\widehat{E}_{7}^{\mathrm{d}}$, as
a limit of the above string $J(\lambda^{\prime},-s\lambda^{\prime})$, whose
underlying representation is a scattered member of
$\widehat{E}_{6}^{\mathrm{d}}$.

Let us see how to realize the above things in atlas.

    set G=complex(simply_connected (E7))
    set x=KGB(G,2903039)
    set q=parameter(x,[0,4,4,-4,4,1,-1,  0,0,0,0,0,0,0],
                      [2,-3,-3,5,-3,0,2, 0,0,0,0,0,0,0])
    set p=monomials(finalize(q))[0]
    p
    Value: final parameter(x=17376,lambda=[2,-2,-2,4,-2,0,2,2,0,0,2,0,1,1]/1,
                                       nu=[2,-3,-3,5,-3,0,2,2,-3,-3,5,-3,0,2]/2)

The representation p is the first entry of Table 3.

    set q1=parameter(x,[0,4,4,-4,4,0,3, 0,0,0,0,0,0,0],
                       [2,-3,-3,5,-3,2,-2, 0,0,0,0,0,0,0])
    set p1=monomials(finalize(q1))[0]
    p1
    Value: final parameter(x=6951,lambda=[2,-1,-1,3,-2,2,-1,2,0,0,2,0,1,1]/1,
                                      nu=[2,-3,-3,5,-3,2,-2,2,-3,-3,5,-3,2,-2]/2)

The representation p1 is the string member with $g=1$.

    set (Q,pp1)=reduce_good_range(p1)
    Q
    Value: ([0,1,2,3,4,5,7,8,9,10,11,12],KGB element #6951)
    pp1
    Value: final parameter(x=2670,lambda=[2,-1,-1,3,-2,2,-10,2,0,0,2,0,1,-8]/1,
                                      nu=[2,-3,-3,5,-3,2,-2,2,-3,-3,5,-3,2,-2]/2)

Let us move the inducing module from $g=1$ to $g=-1$, and then do
cohomological induction.

    set ppm1=parameter(x(pp1),lambda(pp1)-[0,0,0,0,0,0,2,0,0,0,0,0,0,0],nu(pp1))
    goodness(ppm1,G)
    Value: "Fair"
    set ind=theta_induce_irreducible(ppm1,G)
    ind
    Value:
    1*parameter(x=17376,lambda=[2,-2,-2,4,-2,0,2,2,0,0,2,0,1,1]/1,
                            nu=[2,-3,-3,5,-3,0,2,2,-3,-3,5,-3,0,2]/2) [411]
    monomials(ind)[0]=p
    Value: true

Thus the induced module is irreducible, and it is exactly the representation
p. ∎

For convenience, we simply say that the first entry of Table 3 is a string
limit coming from the first scattered representation of $E_{6}$ with $g=-1$.

######  Example 5.2.

The last row of [D2, Table 6] is the trivial representation of complex
$E_{6}$. Namely, we have that $s\rho(E_{6})=[-1,-1,-1,-1,-1,-1]$ and
$\lambda=[1,1,1,1,1,1]$. It produces the following string in
$\widehat{E}_{7}^{\mathrm{d}}$:

(13)
$J(\lambda^{\prime},-s\lambda^{\prime}),\quad\lambda^{\prime}=[1,1,1,1,1,1,\frac{g}{2}],$

where $g\in\mathbb{P}$. Note that

$\lambda^{\prime}+s\lambda^{\prime}=[0,0,0,0,0,0,g+16],\quad\lambda^{\prime}-s\lambda^{\prime}=[2,2,2,2,2,2,-16].$

This string produces the following eight limits in total:

  * $\bullet$

$g=-1$, the 8th entry of Table 4;

  * $\bullet$

$g=-3$, the 9th entry of Table 4;

  * $\bullet$

$g=-5$, the 11th entry of Table 4;

  * $\bullet$

$g=-7$, the 13th entry of Table 4;

  * $\bullet$

$g=-9$, the 14th entry of Table 4;

  * $\bullet$

$g=-11$, the 16th entry of Table 4;

  * $\bullet$

$g=-13$, the 17th entry of Table 4;

  * $\bullet$

$g=-15$, the 20th entry of Table 4.

Note that in each case, the inducing module is in the fair range. ∎

As recorded in the last columns of Tables 3 and 4, we _exhaust_ all the string
limits coming from the scattered representations of the $E_{6}$ factor, and
the inducing module is always in the fair range. Similar things have been done
for the $A_{6}$ and $D_{6}$ factor. Let us give a few more examples.

######  Example 5.3.

The 13th row of Table 1 gives a scattered member of
$\widehat{A}_{6}^{\mathrm{d}}$ with $s\rho(A_{6})=[-2,-1,-1,-1,4,-5]$ and
$\lambda=[1,1,1,1,\frac{1}{2},\frac{1}{2}]$. It produces the following string
in $\widehat{E}_{7}^{\mathrm{d}}$:

(14)
$J(\lambda^{\prime},-s\lambda^{\prime}),\quad\lambda^{\prime}=[1,\frac{b}{2},1,1,1,\frac{1}{2},\frac{1}{2}],$

where $b\in\mathbb{P}$. Note that

$\lambda^{\prime}+s\lambda^{\prime}=[0,b+9,0,0,0,4,-4],\quad\lambda^{\prime}-s\lambda^{\prime}=[2,-9,2,2,2,-3,5].$

This string gives the following three limits in total:

  * (a)

$b=-1$, the 17th entry of Table 3;

  * (b)

$b=-3$, the 24th entry of Table 3;

  * (c)

$b=-20$, the 18th entry of Table 3.

Note that both (a) and (b) are also string limits coming from the $E_{6}$
factor, while (c) is not. Moreover, the inducing module for (c) is not in the
fair range.

The 13th row of Table 1 is marked with a star, meaning that it has a dual
representation. Indeed, it is a scattered member of $\widehat{A}_{6}^{d}$ such
that $s^{*}\rho(A_{6})=[-5,4,-1,-1,-1,-2]$ and
$\lambda^{*}=[\frac{1}{2},\frac{1}{2},1,1,1,1]$. It produces the following
string in $\widehat{E}_{7}^{\mathrm{d}}$:

(15)
$J(\lambda^{\prime\prime},-s^{*}\lambda^{\prime\prime}),\quad\lambda^{\prime\prime}=[\frac{1}{2},\frac{b}{2},\frac{1}{2},1,1,1,1],$

where $b\in\mathbb{P}$. Note that

$\lambda^{\prime\prime}+s^{*}\lambda^{\prime\prime}=[-4,b+8,4,0,0,0,0],\quad\lambda^{\prime\prime}-s^{*}\lambda^{\prime\prime}=[5,-8,-3,2,2,2,2].$

This string gives the following three limits in total:

  * (a’)

$b=-1$, the representation (c);

  * (b’)

$b=-18$, the representation (b);

  * (c’)

$b=-20$, the representation (a).

Note that the inducing module for (a’) is now in the fair range. ∎

######  Example 5.4.

The 19th row of Table 2 gives a scattered member of
$\widehat{D}_{6}^{\mathrm{d}}$ with $s\rho(D_{6})=[-2,4,-5,4,-6,-2]$ and
$\lambda=[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$. It produces
the following string in $\widehat{E}_{7}^{\mathrm{d}}$:

(16)
$J(\lambda^{\prime},-s\lambda^{\prime}),\quad\lambda^{\prime}=[\frac{a}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1],$

where $a\in\mathbb{P}$. This string gives no limit.

On the other hand, the dual representation has
$s^{*}\rho(D_{6})=[-2,4,-5,4,-2,-6]$ and
$\lambda^{*}=[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2}]$. It
produces the following string in $\widehat{E}_{7}^{\mathrm{d}}$:

(17)
$J(\lambda^{\prime\prime},-s^{*}\lambda^{\prime\prime}),\quad\lambda^{\prime\prime}=[\frac{a}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1],$

where $a\in\mathbb{P}$. Note that

$\lambda^{\prime\prime}+s^{*}\lambda^{\prime\prime}=[a+8,-3,0,3,-3,3,0],\quad\lambda^{\prime\prime}-s^{*}\lambda^{\prime\prime}=[-8,4,2,-2,4,-2,2].$

When $a=-1$, this string gives the 31st entry of of Table 3. ∎

######  Example 5.5.

Consider the maximal Levi obtained by removing the simple root $\alpha_{6}$.
Put trivial representations on both the $D_{5}$ factor and the $A_{1}$ factor.
Correspondingly, we have the string $J(\lambda,-s\lambda)$ in
$\widehat{E}_{7}^{\mathrm{d}}$ such that

(18)
$\lambda+s\lambda=[0,0,0,0,0,2f+11,0],\quad\lambda+s\lambda=[2,2,2,2,2,-11,2]$

where $f\in\mathbb{P}$. This string gives the following three limits in total:

  * (a)

$f=-1$, the 28th entry of Table 3;

  * (b)

$f=-3$, the 36th entry of Table 3;

  * (c)

$f=-5$, the last entry of Table 3.

We may also put the scattered representation of $\widehat{D}_{5}^{d}$ sitting
in the second row of [D2, Table 5], and put the trivial representation on the
$A_{1}$ factor. They produce the string
$J(\lambda^{\prime},-s^{\prime}\lambda^{\prime})$ in
$\widehat{E}_{7}^{\mathrm{d}}$ such that

(19)
$\lambda^{\prime}+s^{\prime}\lambda^{\prime}=[4,0,-4,4,0,2f+4,0],\quad\lambda^{\prime}+s^{\prime}\lambda^{\prime}=[-3,2,5,-3,2,-4,2].$

where $f\in\mathbb{P}$. Setting $f=-1$ gives the third entry of Table 3. ∎

Table 3. The scattered representations in $\widehat{E}_{7}^{\mathrm{d}}$ (part one) $s\rho$ | $\lambda$ | spin LKT | mult | string limits
---|---|---|---|---
$[-2,6,7,-8,6,1,-3]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,0,2,1,1,1]$ | $1$ |  $E_{6}$, 1st, $g=-1$
$[-2,6,7,-8,7,-1,-2]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[1,1,0,4,0,0,1]$ | $1$ |  $A_{2}.A_{1}.A_{3}$, $d=-1$
$[6,-2,-8,6,5,-6,4]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,1,2,1,0,2,1]$ | $1$ |  $D_{5}$ 2nd.$A_{1}$, $f=-1$
$[-5,-7,3,5,5,-7,5]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |  $A_{6}$, 4th, $b=-1$
$[-4,-2,3,7,-9,8,-2]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[2,1,0,1,3,0,1]$ | $1$ |  $D_{6}$, 3rd*, $a=-1$
$[-2,-8,-2,7,6,-7,5]$ | $[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,4,1,0,0,3,1]$ | $1$ |  $A_{6}$, 6th, $b=-1$
$[-1,-1,-2,9,-10,9,-2]$ | $[1,1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[0,0,1,0,6,0,1]$ | $1$ |  $A_{4}.A_{2}$, $e=-1$
$[8,-2,-10,9,-2,4,-5]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[1,1,4,0,1,0,3]$ | $1$ |  $E_{6}$, 4th*, $g=-1$
$[-6,-9,4,8,-2,4,-6]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[2,3,1,0,1,1,2]$ | $1$ |  $A_{6}$, 8th, $b=-1$
$[-8,-1,6,-1,8,-10,8]$ | $[\frac{1}{2},1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[4,0,1,0,1,4,1]$ | $1$ |  $D_{6}$, 8th, $a=-1$
$[-2,5,4,-6,10,-12,10]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,0,1,1,2,1,1]$ | $1$ |  $E_{6}$, 4th, $g=-1$
$[9,-1,-11,9,1,-3,-1]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,2,3,0,1,0,2]$ | $1$ |  $E_{6}$, 4th*, $g=-3$
$[-6,-10,5,8,1,-3,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,2,2,0,1,1,1]$ | $1$ |  $E_{6}$, 5th*, $g=-1$
$[-11,5,9,-7,5,5,-7]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |  $E_{6}$, 7th, $g=-1$
$[9,-1,-11,10,-1,-2,-1]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},1,1,1]$ | $[1,0,7,0,0,1,0]$ | $1$ |  $A_{1}.A_{5}$, $c=-1$
$[-1,-1,-6,4,9,-11,9]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,0,2,0,1,3,1]$ | $1$ |  $D_{6}$, 8th, $a=-3$
$[-2,-12,-1,10,-1,5,-6]$ | $[1,\frac{1}{2},1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[1,6,0,1,0,0,4]$ | $1$ |  $E_{6}$, 8th, $g=-1$
$[-6,-10,5,9,-1,-2,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1,1]$ | $[4,6,0,0,0,1,1]$ | $1$ |  $A_{6}$, 13th*, $b=-1$
$[-9,7,7,-9,7,5,-7]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |  $E_{6}$, 9th, $g=-1$
$[-2,9,8,-10,8,-10,8]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,0,1,1,2,1,1]$ | $1$ |  $A_{4}.A_{2}$, $e=-3$
$[-10,-1,8,4,-6,8,-10]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,0,1,1,1,2,1]$ | $1$ |  $E_{6}$, 6th, $g=-3$
$[-1,-13,-2,11,3,-4,-1]$ | $[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[2,5,0,1,0,0,3]$ | $1$ |  $E_{6}$, 8th, $g=-3$
$[-10,-1,8,6,-8,6,-8]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,0,1,1,1,2,1]$ | $1$ |  $D_{6}$, 14th, $g=-1$
$[-2,-1,9,-11,9,6,-7]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[1,3,0,2,0,0,3]$ | $1$ |  $E_{6}$, 11th, $g=-1$
$[-12,8,11,-9,7,-2,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[1,0,4,0,1,1,1]$ | $1$ |  $D_{6}$, 11th, $a=-3$
$[1,-3,-3,1,11,-13,11]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |  $E_{6}$, 4th, $g=-3$
$[-10,10,9,-11,9,-2,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[1,0,4,0,1,1,1]$ | $1$ |  $A1.A5$, $c=-3$
$[-1,-2,-1,-1,12,-13,11]$ | $[1,1,1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[0,1,0,0,0,9,1]$ | $1$ |  $D_{5}.A_{1}$, $f=-1$
$[-1,-14,-1,13,-2,-1,-1]$ | $[1,\frac{1}{2},1,\frac{1}{2},1,1,1]$ | $[0,10,0,0,1,0,0]$ | $1$ |  $A_{6}$, $b=-1$
$[-12,-2,11,-1,-1,8,-10]$ | $[\frac{1}{2},1,\frac{1}{2},1,1,\frac{1}{2},\frac{1}{2}]$ | $[8,1,0,0,0,1,6]$ | $1$ |  $D_{6}$, 17th, $a=-1$
$[-14,-7,12,5,-6,4,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,0,2,1,1,1,1]$ | $1$ |  $D_{6}$, 19th*, $a=-1$
$[-1,-2,12,-13,12,-2,-1]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[0,7,0,0,2,0,0]$ | $1$ |  $A_{6}$, $b=-3$
$[-1,-1,-11,9,-1,9,-11]$ | $[1,1,\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[6,2,0,0,0,1,5]$ | $1$ |  $D_{6}$, 17th, $a=-3$
$[-1,-2,-1,9,-11,13,-14]$ | $[1,1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[0,1,0,1,0,6,1]$ | $1$ |  $E_{6}$, 13th, $g=-3$
$[-13,-2,12,-1,6,-8,-1]$ | $[\frac{1}{2},1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},1]$ | $[7,1,0,0,0,2,4]$ | $1$ |  $D_{6}$, 20th, $a=-1$
$[-1,-2,-1,11,-13,11,-12]$ | $[1,1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[0,1,0,1,0,6,1]$ | $1$ |  $D_{5}.A_{1}$, $f=-3$
$[-1,-1,-12,10,7,-9,-1]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[5,2,0,0,0,2,3]$ | $1$ |  $D_{6}$, 20th, $a=-3$
$[-14,-2,13,4,-6,-1,-1]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[6,1,0,0,0,3,2]$ | $1$ |  $D_{6}$, 21st, $a=-1$
$[-2,8,-1,-9,7,10,-12]$ | $[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[4,3,0,0,0,1,4]$ | $1$ |  $D_{6}$, 17th, $a=-5$
$[-1,7,7,-9,-1,13,-15]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[0,2,0,1,0,4,2]$ | $1$ |  $E_{6}$, 13th, $g=-5$
$[-17,1,15,-3,1,-3,1]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |  $D_{6}$, 22nd, $a=-1$
$[11,-2,-12,11,-13,11,-2]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[0,4,1,0,2,0,1]$ | $1$ |  $A_{6}$, $b=-5$
$[-1,8,7,-9,11,-13,-2]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[0,1,0,2,0,3,1]$ | $1$ |  $E_{6}$, 15th*, $g=-5$
$[-1,10,9,-11,9,-11,-2]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[0,1,0,2,0,3,1]$ | $1$ |  $D_{5}.A_{1}$, $f=-5$
Table 4. The scattered representations in $\widehat{E}_{7}^{\mathrm{d}}$ (part two) $s\rho$ | $\lambda$ | spin LKT | mult | string limits
---|---|---|---|---
$[-17,-1,15,-1,-1,-1,-1]$ | $[\frac{1}{2},1,\frac{1}{2},1,1,1,1]$ | $[13,0,1,0,0,0,0]$ | $1$ |  $D_{6}$, $a=-1$
$[4,-6,-6,4,-1,14,-16]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,\frac{1}{2},\frac{1}{2}]$ | $[1,2,0,1,0,3,2]$ | $1$ |  $E_{6}$, 13th, $g=-7$
$[-1,-1,-16,14,-1,-1,-1]$ | $[1,1,\frac{1}{2},\frac{1}{2},1,1,1]$ | $[10,0,2,0,0,0,0]$ | $1$ |  $D_{6}$, $a=-3$
$[4,-7,-6,5,12,-14,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,1,0,2,0,2,1]$ | $1$ |  $E_{6}$, 15th*, $g=-7$
$[-11,9,9,-11,9,-11,9]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |  $A_{6}$, $b=-7$
$[-3,-1,1,-3,1,15,-17]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,2,0,1,0,2,2]$ | $1$ |  $E_{6}$, 13th, $g=-9$
$[-1,13,-1,-15,13,-1,-1]$ | $[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},1,1]$ | $[7,0,3,0,0,0,0]$ | $1$ |  $D_{6}$, $a=-5$
$[-2,-1,-1,-1,-1,16,-17]$ | $[1,1,1,1,1,\frac{1}{2},\frac{1}{2}]$ | $[1,0,0,0,0,0,15]$ | $1$ |  $E_{6}$, $g=-1$
$[-1,-1,-2,-1,15,-16,-1]$ | $[1,1,1,1,\frac{1}{2},\frac{1}{2},1]$ | $[2,0,0,0,0,0,13]$ | $1$ |  $E_{6}$, $g=-3$
$[-1,-13,-1,11,-13,11,-1]$ | $[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[5,0,3,0,0,1,0]$ | $1$ |  $D_{6}$, $a=-7$
$[-1,-1,-1,13,-15,-1,-1]$ | $[1,1,1,\frac{1}{2},\frac{1}{2},1,1]$ | $[3,0,0,0,0,0,11]$ | $1$ |  $E_{6}$, $g=-5$
$[-1,-1,9,-11,9,-11,9]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[3,0,3,0,0,2,0]$ | $1$ |  $D_{6}$, $a=-9$
$[-1,11,12,-13,-1,-2,-1]$ | $[1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1,1]$ | $[4,0,0,0,0,0,9]$ | $1$ |  $E_{6}$, $g=-7$
$[10,-10,-11,9,-1,-2,-2]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1,1]$ | $[4,0,0,0,0,1,7]$ | $1$ |  $E_{6}$, $g=-9$
$[6,-1,-8,6,-8,6,-8]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[2,0,2,1,0,2,0]$ | $1$ |  $D_{6}$, $a=-11$
$[-9,-1,7,-8,6,-1,-2]$ | $[\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},1,1]$ | $[4,0,0,0,0,2,5]$ | $1$ |  $E_{6}$, $g=-11$
$[-1,-2,-6,4,-5,4,-2]$ | $[1,1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[4,0,0,0,0,3,3]$ | $1$ |  $E_{6}$, $g=-13$
$[-5,3,3,-5,3,-5,-1]$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,0,1,2,0,2,0]$ | $1$ |  $D_{6}$, $a=-13$
$[-1,1,-1,-3,1,-3,1]$ | $\frac{\rho}{2}$ | $\rho$ | $1$ |
$[-1,1,-1,-3,1,-3,1]$ | $[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ | $[4,0,0,0,0,4,1]$ | $1$ |  $E_{6}$, $g=-15$
$-\rho$ | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$ | $[1,0,1,2,0,2,0]$ | $1$ |
$-\rho$ | $\rho$ | [0,0,0,0,0,0,0] | $1$ |

Note that there are three scattered representations of complex $E_{7}$ which
can not be realized as string limits. Namely, the 19th, 21st and 22nd entries
of Table 4. They are all unipotent representations. We will look at them more
carefully in the next section.

By the way, we note that for complex $E_{6}$, only the two _spherical_
scattered representations can _not_ be realized as string limits: the model
representation [Mc], and the trivial representation.

##  6\. Unipotent representations

The following representations are discussed in McGovern [Mc]:

  * (i)

$J(\frac{\rho}{2},\frac{\rho}{2})$ (the 1st entry of [Mc, Table 10]);

  * (ii)

$J(\frac{\rho}{2},[\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2},-\frac{1}{2},\frac{3}{2},-\frac{1}{2}])$
(cf. $Q^{\prime}$ in the proof of [Mc, Theorem 2.2]);

  * (iii)

$J(\lambda,\lambda)$ with
$\lambda=[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$
(the 2nd entry of [Mc, Table 10]);

  * (iv)

$J(\lambda,\lambda)$ with
$\lambda=[1,\frac{1}{2},1,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$
(the 3rd entry of [Mc, Table 10]);

  * (v)

$J(\lambda,\lambda)$ with $\lambda=[1,1,1,0,1,0,1]$ (the 4th entry of [Mc,
Table 10]);

  * (vi)

$J(\lambda,\lambda)$ with $\lambda=[1,1,1,0,1,1,1]$ (the minimal
representation);

  * (vii)

$J(\rho,\rho)$, the trivial representation.

These representations are all unipotent in the sense of Losev, Mason-Brown and
Matvieievskyi ([LMM], [MM]). In particular, if $\lambda$ is integral, then
they are special unipotent in the sense of [BV]. All of them are unitary
(David Vogan helped us with the representation (v), which used about 290 GB
memory in atlas). However, (i) and (iv) have zero Dirac cohomology since
neither $\rho$ nor $[2,1,2,1,1,1,1]$ lies in the root lattice of $E_{7}$,
while (v) and (vi) also have zero Dirac cohomology since $\lambda$ is
singular. The remaining three representations are all Dirac series. Indeed,
(ii), (iii), (vii) are the 19th, 21st and 22nd entries of Table 4,
respectively.

Let us summarize the discussions as the following.

######  Corollary 6.1.

Any Dirac series of complex $E_{7}$ is either a unipotent representation, or
cohomologically induced from a unipotent representation with non-zero Dirac
cohomology in the fair range.

##  7\. Calculations of Spin Lowest $K$-types

We mention some calculations on the spin lowest $K$-types $\sigma$ in Table 4.
It is easy to check that all of them satisfy
$\|\sigma\|_{\mathrm{spin}}=\|\Lambda\|=\|2\lambda\|$.

Firstly, let us handle the cases that $J(\lambda,-s\lambda)\in\widehat{G}^{d}$
can be realized as a string limit. In such a case, it suffices to check that

(20) $[V(\sigma):J(\lambda,-s\lambda)]>0.$

Indeed, the branching result in [BDW] implies that $\sigma$ must be the unique
the spin lowest and it must occur exactly once. Usually, one can us the atlas
command branch_irr to check (20) directly. However, for a few representations,
such a direct calculation in atlas would be time-consuming. Let us mention
some ways that are more effective.

###  $\bullet$ $E_{6}$, $g=-15$ (the 20th entry of Table 4)

This representation p has LKT $[0,0,0,0,0,0,1]$ whose height is $27$, while
the $K$-type $[4,0,0,0,0,4,1]$ has height $371$. Instead of doing
branch_irr(p, 371), we do branch_irr(p, 235) and find that the $K$-type
$[0,0,0,0,0,4,1]$ shows up in p exactly once. Using Vogan pencil, we conclude
that the $K$-type

$\sigma:=[0,0,0,0,0,4,1]+4\beta=[4,0,0,0,0,4,1]$

shows up in p, as desired.

It is interesting to note that if we look at the representations $E_{6}$,
$g=-1,-3,\dots,-15$ consecutively, the spin LKTs still demonstrate a vague
pattern, whilst the pattern is described explicitly in [DW2] for all classical
groups.

There are cases that we can not move _down_ from the desired spin LKT along
the Vogan pencil. Then the following clever way (suggested by David Vogan)
could be helpful to get the multiplicity of $\sigma$ in
$J(\lambda,-s\lambda)$.

###  $\bullet$ $D_{6}$, $a=-13$ (the 18th entry of Table 4)

Here $J(\lambda,-s\lambda)={\rm
Ind}_{D_{6}\times\mathbb{C}^{*}}^{E_{7}}(\mathrm{triv}\boxtimes\mathop{\hbox{det}}\nolimits^{2})$
with lowest $K$-type $2\beta=[2,0,0,0,0,0,0]$. As mentioned above, it suffices
to check

$[V([1,0,1,2,0,2,0]):{\rm
Ind}_{D_{6}\times\mathbb{C}^{*}}^{E_{7}}(\mathrm{triv}\boxtimes\mathop{\hbox{det}}\nolimits^{2})]>0.$

Using Frobenius reciprocity, it is equivalent to show that

(21)
$[\mathrm{triv}\boxtimes\mathbb{C}_{2\beta}:V([1,0,1,2,0,2,0])|_{D_{6}\times\mathbb{C}^{*}}]>0.$

The branching $V([1,0,1,2,0,2,0])|_{D_{6}\times\mathbb{C}^{*}}$ can be done as
follows: $V([1,0,1,2,0,2,0])$ defines a finite-dimensional
$(\mathfrak{e}_{7},D_{6}\times A_{1})$-module for the quarterionic real form
E7_q, so the $K$-types of this finite-dimensional module are precisely the
branching of $V([1,0,1,2,0,2,0])$ to $D_{6}\times A_{1}$.

The following atlas command gives the finite-dimensional module
$V([1,0,1,2,0,2,0])$ and its dimension.

    set G=E7_q
    set p=parameter(x_open(G),[1,0,1,2,0,2,0]+rho(G),[1,0,1,2,0,2,0]+rho(G))
    dimension(p)
    Value: 2399133156669849600

Branching of the module p into $D_{6}\times A_{1}$-modules could cost a lot of
time. However, it is easy to see that it starts as follows:

    print_branch_irr_long(p,KGB(G,62),40)
    m  x    lambda                            hw                       dim  height
    1  20   [ 0, 0, 0, 0, 0, 0, 1 ]/1         [ 1, 0, 0, 0, 0, 0, 0 ]  3    27
    1  337  [  1,  0, -1,  1, -1,  1,  0 ]/1  [ 2, 0, 0, 0, 0, 0, 0 ]  5    38

with the last 6 coordinates of hw are the highest weights of the
$D_{6}$-module in the decomposition. Therefore,

$V([1,0,1,2,0,2,0])|_{D_{6}\times
A_{1}}=(\mathrm{triv}\boxtimes\mathbb{C}^{3})\oplus(\mathrm{triv}\boxtimes\mathbb{C}^{5})\oplus\cdots.$

Here $\mathbb{C}^{3}$ means the 3-dimensional irreducible representation of
$A_{1}$. Thus the restriction

$V([1,0,1,2,0,2,0])|_{D_{6}\times\mathbb{C}^{*}}$

decomposes as

$\left(\mathrm{triv}\boxtimes(\mathbb{C}_{\beta}\oplus\mathbb{C}_{0}\oplus\mathbb{C}_{-\beta})\right)\oplus\left(\mathrm{triv}\boxtimes(\mathbb{C}_{2\beta}\oplus\mathbb{C}_{\beta}\oplus\mathbb{C}_{0}\oplus\mathbb{C}_{-\beta}\oplus\mathbb{C}_{-2\beta})\right)\oplus\cdots$

and (21) has multiplicity $>0$, as desired.

We have also handled the 12th and 15th entry of Table 4 in this way.

Secondly, let us determine the spin lowest $K$-types of the two unipotent
representations.

###  $\bullet$
$J(\frac{\rho}{2},[\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2},-\frac{1}{2},\frac{3}{2},-\frac{1}{2}])$
(the 19th entry of Table 4)

As mentioned in the previous section, this is the unipotent representation
$Q^{\prime}$ appearing in the proof of [Mc, Theorem 2.2], whose $K$-types have
highest weights in the weight lattice _but not in the root lattice_ , all
appearing with multiplicity one. In particular $\sigma=\rho$ appears in
$Q^{\prime}$, and this is the only $\sigma$ that satisfies
$\\{\sigma-\rho\\}+\rho=2\lambda=\rho$.

###  $\bullet$$J(\lambda,\lambda)$,
$\lambda=[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},1]$
(the 21st entry of Table 4)

According to [Mc, Table 10], it is conjectured that this unipotent
representation would have $K$-types equal to $[a,0,b,c,0,d,0]$ with
multiplicity one, where $a,b,c,d$ are non-negative integers.

This can be verified by [MM], which states that this representation is _the_
quantization model for the birationally rigid orbit
$\mathcal{O}=3A_{1}^{\prime}$, and hence its $K$-spectrum is equal to the ring
of regular functions of $\mathcal{O}$. Then the conjecture follows from the
results of [C]. It is then easy to check that $[1,0,1,2,0,2,0]$ is the
_unique_ spin lowest $K$-type in $J(\lambda,\lambda)$.

It is also worth noting that this unipotent module, along with the module with
string limit $D_{6}$, $a=-13$ (the 18th entry of Table 4), are the only
composition factors of the end point of the complementary series in [BP,
Equation (5.29)], each occurring once. Hence both modules are unitary.

Acknowledgments

We are deeply grateful to the atlas mathematicians. We also thank Dr. Luan for
helping us considering the branching (21).

Funding

Dong is supported by the National Natural Science Foundation of China (grant
12171344). Wong is supported by the National Natural Science Foundation of
China (grant 11901491) and the Presidential Fund of CUHK(SZ).

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