# On the Helgason-Johnson bound

Chao-Ping Dong  School of Mathematical Sciences, Soochow University, Suzhou
215006, P. R. China<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a simple non-compact linear Lie group. Let $\pi$ be any irreducible
unitary representation of $G$ with infinitesimal character $\Lambda$ whose
continuous part is $\nu$. The beautiful Helgason-Jonson bound in 1969 says
that the norm of $\nu$ is upper bounded by the norm of $\rho(G)$, which stands
for the half sum of the positive roots of $G$. The current paper aims to give
a framework to sharpen the Helgason-Johnson bound when $\pi$ is infinite-
dimensional. We have explicit results for exceptional Lie groups. Ingredients
of the proof include Parathasarathy’s Dirac operator inequality, Vogan pencil,
and the unitarily small convex hull introduced by Salamanca-Riba and Vogan.

###### 2010 Mathematics Subject Classification:

Primary 22E46

##  1\. Introduction

Let $G$ be a connected simple non-compact linear Lie group which is in the
Harish-Chandra class [14]. For convenience, we further assume that $G$ has
finite center. Let $\theta$ be the Cartan involution of $G$, and assume that
$K:=G^{\theta}$ is a maximal compact subgroup of $G$. Let $\mathfrak{g}_{0}$
(resp., $\mathfrak{k}_{0}$) be the Lie algebra of $G$ (resp., $K$). Then on
the Lie algebra level, we have the corresponding Cartan decomposition

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

where $\mathfrak{p}_{0}$ is the $-1$ eigenspace of $\theta$ on
$\mathfrak{g}_{0}$. The complexification of $\mathfrak{g}_{0}$ is written as

$\mathfrak{g}:=\mathfrak{g}_{0}\otimes_{\mathbb{R}}\mathbb{C}$

and similar notation will be used for other groups. We adopt the Killing form
on $\mathfrak{g}$, and denote by $\|\cdot\|$ the corresponding norm.

Let $T_{f}$ be a maximal torus of $K$, and let $\mathfrak{a}_{f,0}$ be the
centralizer of $\mathfrak{t}_{f,0}$ in $\mathfrak{p}_{0}$. Let
$A_{f}=\exp(\mathfrak{a}_{f,0})$. Then $H_{f}=T_{f}A_{f}$ is called the
fundamental Cartan subgroup of $G$. We have root systems
$\Delta(\mathfrak{g},\mathfrak{h}_{f})$,
$\Delta(\mathfrak{k},\mathfrak{k}_{f})$ and the restricted root system
$\Delta(\mathfrak{g},\mathfrak{t}_{f})$. Denote the corresponding Weyl groups
by $W(\mathfrak{g},\mathfrak{h}_{f})$, $W(\mathfrak{k},\mathfrak{t}_{f})$ and
$W(\mathfrak{g},\mathfrak{t}_{f})$. We fix a positive root system
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ once for all. Let $\rho_{c}$ be
the half sum of the roots in $\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$.
Choose a positive root system $\Delta^{+}(\mathfrak{g},\mathfrak{t}_{f})$
containing $\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$, and denote the half
sum of roots in $\Delta^{+}(\mathfrak{g},\mathfrak{t}_{f})$ by $\rho(G)$.

We will denote a $K$-type (that is, an irreducible representation of $K$) as
$E_{\eta}$, where $\eta$ is its highest weight. Similar notation will apply to
$\mathfrak{k}$-types. We may just refer to a $K$-type (or $\mathfrak{k}$-type)
by its highest weight. Vogan realized the Langlands classification of the
irreducible admissible representations algebraically, where lambda norm and
lowest $K$-type played key roles, see Chapter 6 of [23]. In this setting, an
irreducible admissible $(\mathfrak{g},K)$ module $\pi$ will have infinitesimal
character

(1)
$\Lambda=(\lambda_{a}(\mu),\nu)\in\mathfrak{h}^{*}=\mathfrak{t}^{*}+\mathfrak{a}^{*}.$

Let us mention the relevant notation: $E_{\mu}$ is a lowest $K$-type of $\pi$;
the vector $\lambda_{a}(\mu)$ will be recalled in (6), its norm is the _lambda
norm_ of $\mu$; $G(\lambda_{a}(\mu))$ is the isotropy group at
$\lambda_{a}(\mu)$ for the $G$ action; $H=TA$ is a maximally split
$\theta$-stable Cartan subgroup of $G(\lambda_{a}(\mu))$. Note that the quasi-
split group $G(\lambda_{a}(\mu))$ is still in the Harish-Chandra class, with
Cartan involution $\theta|_{G(\lambda_{a}(\mu))}$.

Now let us state the following beautiful result.

The Helgason-Johnson bound. Let $\pi$ be any irreducible unitary
$(\mathfrak{g},K)$ module whose infinitesimal character is given by (1). Then
$\|\nu\|\leq\|\rho(G)\|$.

Helgason and Johnson obtained the above result for $\pi$ spherical (that is,
$\pi$ contains the trivial $K$-type) in 1969 [15]. It is not hard to deduce
the general case using the Langlands classification, see Theorem 5.2 of
Chapter IV of [4]. Note that the Helgason-Johnson bound can be tight. For
instance, when $G$ is complex, the $\nu$ parameter of the trivial
representation does have norm $\|\rho(G)\|$.

However, since a typical irreducible unitary representation of $G$ is
infinite-dimensional, it is natural to ask the following question: Can we
improve the Helgason-Johnson bound for infinite-dimensional irreducible
unitary representations?

After giving a general framework in Section 3, we will handle this question
for exceptional Lie groups. It is conceivable that the results reported here
will be helpful for understanding the unitary duals.

To make the things clean, we postpone the Hermitian symmetric case to the last
section. Our result in this case is Theorem 6.1. Unless stated otherwise, we
assume that $\mathfrak{k}$ has no center from now on. Then the
$\mathfrak{k}$-module $\mathfrak{p}$ is irreducible and thus has a highest
weight which will be denoted by $\beta$. Let $\pi$ be any infinite-dimensional
representation of $G$. If the $K$-type $E_{\mu}$ occurs in $\pi$, then by
Section 3 of Vogan [22], the $K$-types $E_{\mu+m\beta}$ will show up in $\pi$
for all $m\in\mathbb{Z}_{\geq 0}$. Let us call the following $K$-types the
Vogan pencil starting from $\mu$

(2) ${\rm Pencil}(\mu):=\\{\mu+m\beta\mid m\in\mathbb{Z}_{\geq 0}\\}.$

Let us also introduce

(3) ${\rm MP}(\mu):=\min\\{\|\mu+m\beta\|_{\rm spin}^{2}\mid
m\in\mathbb{Z}_{\geq 0}\\},$

where $\|\cdot\|_{\rm spin}$ stands for the spin norm [8]. It will be recalled
in Section 2.1.

######  Theorem 1.1.

Let $G$ be a complex exceptional Lie group. Let $\pi$ be any infinite-
dimensional irreducible unitary $(\mathfrak{g},K)$ module whose infinitesimal
character is given by (1). Then we have the following bounds for $\|\nu\|$:

  * $\bullet$

${\rm Lie}(G)$ is of type $G_{2}:$ $\|\nu\|\leq\sqrt{26}$, while
$\|\rho(G)\|=\sqrt{56}$.

  * $\bullet$

${\rm Lie}(G)$ is of type $F_{4}:$ $\|\nu\|\leq\sqrt{78}$, while
$\|\rho(G)\|=\sqrt{156}$.

  * $\bullet$

${\rm Lie}(G)$ is of type $E_{6}:$ $\|\nu\|\leq\sqrt{170}$, while
$\|\rho(G)\|=\sqrt{312}$.

  * $\bullet$

${\rm Lie}(G)$ is of type $E_{7}:$ $\|\nu\|\leq\sqrt{464}$, while
$\|\rho(G)\|=\sqrt{798}$.

  * $\bullet$

${\rm Lie}(G)$ is of type $E_{8}:$ $\|\nu\|\leq\sqrt{1522}$, while
$\|\rho(G)\|=\sqrt{2480}$.

Note that the unitary dual of complex $G_{2}$ has been determined by Duflo
[12] in 1979. Note also that the above bound can be tight. For instance, when
$G$ is complex $F_{4}$, the $\nu$ parameter of the minimal representation does
have norm $\sqrt{78}$.

######  Theorem 1.2.

Let $G$ be a real exceptional Lie group. Let $\pi$ be any infinite-dimensional
irreducible unitary $(\mathfrak{g},K)$ module whose infinitesimal character is
given by (1). Then we have the following bounds for $\|\nu\|$:

  * $\bullet$

${\rm Lie}(G)=G_{2(2)}:$ $\|\nu\|\leq\sqrt{\frac{51}{8}}$, while
$\|\rho(G)\|=\sqrt{14}$.

  * $\bullet$

${\rm Lie}(G)={\rm FI}=F_{4(4)}:$ $\|\nu\|\leq\sqrt{\frac{39}{2}}$, while
$\|\rho(G)\|=\sqrt{39}$.

  * $\bullet$

${\rm Lie}(G)={\rm FII}=F_{4(-20)}:$ $\|\nu\|\leq\frac{9}{2}$, while
$\|\rho(G)\|=\sqrt{39}$.

  * $\bullet$

${\rm Lie}(G)={\rm EI}=E_{6(6)}:$ $\|\nu\|\leq\sqrt{42}$, while
$\|\rho(G)\|=\sqrt{78}$.

  * $\bullet$

${\rm Lie}(G)={\rm EII}=E_{6(2)}:$ $\|\nu\|\leq\sqrt{42}$, while
$\|\rho(G)\|=\sqrt{78}$.

  * $\bullet$

${\rm Lie}(G)={\rm EIV}=E_{6(-26)}:$ $\|\nu\|\leq\sqrt{37}$, while
$\|\rho(G)\|=\sqrt{78}$.

  * $\bullet$

${\rm Lie}(G)={\rm EV}=E_{7(7)}:$ $\|\nu\|\leq\sqrt{\frac{231}{2}}$, while
$\|\rho(G)\|=\sqrt{\frac{399}{2}}$.

  * $\bullet$

${\rm Lie}(G)={\rm EVI}=E_{7(-5)}:$ $\|\nu\|\leq\sqrt{\frac{227}{2}}$, while
$\|\rho(G)\|=\sqrt{\frac{399}{2}}$.

  * $\bullet$

${\rm Lie}(G)={\rm EVIII}=E_{8(8)}:$ $\|\nu\|\leq\sqrt{380}$, while
$\|\rho(G)\|=\sqrt{620}$.

  * $\bullet$

${\rm Lie}(G)={\rm EIX}=E_{8(-24)}:$ $\|\nu\|\leq\sqrt{\frac{723}{2}}$, while
$\|\rho(G)\|=\sqrt{620}$.

Note that the unitary dual of $G$ when ${\rm Lie}(G)$ is $G_{2(2)}$ has been
determined by Vogan [24] in 1994. Note also that the above bound can be tight.
For instance, when ${\rm Lie}(G)$ is EI, the $\nu$ parameter of the
representation in the penultimate row of Table 11 of [7] does have norm
$\sqrt{42}$. This is a minimal representation. It is worth noting that our
bound, among the above cases, is attained at the trivial $K$-type if and only
if $G/K$ is _not_ quaternionic as in Appendix C of Knapp [16], which traces
back to Wolf [28]. See Section 5 for details.

On the $p$-adic side, Dirac operators have been introduced by Barbasch,
Ciubotaru and Trapa for graded affine Hecke algebras [3]. In that setting one
also has Dirac inequality, and an analogue of our results on the $p$-adic side
has been deduced in Corollary 5.4 of [3]. See also Theorem 1.2 (1) of
Ciubotaru [6]. Moreover, Theorem 1.2 (3) of [6] gives the second non-unitarity
gap by understanding the minimal representations. It is conceivable that there
should be an analogue on the real side. We will address this problem in
future.

Throughout this paper, the root systems will be adopted as in Appendix C of
Knapp [16].

##  2\. Preliminaries

Let $\mathfrak{g}_{0}$ be the Lie algebra of $G$. We always fix a Vogan
diagram for $\mathfrak{g}_{0}$ as [16, Appendix C]. By doing this we have
actually fixed a positive root system
$\Delta^{+}(\mathfrak{g},\mathfrak{h}_{f})$. Restricting it to
$\mathfrak{t}_{f}^{*}$, we get
$(\Delta^{+})^{(0)}(\mathfrak{g},\mathfrak{t}_{f})$ which contains a
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$. We fix this
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ once for all. Let
$\alpha_{1},\dots,\alpha_{l}$ be the simple roots of
$(\Delta^{+})^{(0)}(\mathfrak{g},\mathfrak{t}_{f})$, and let
$\zeta_{1},\dots,\zeta_{l}$ be the corresponding fundamental weights. There
are $s$ ways of choosing positive root systems for
$\Delta(\mathfrak{g},\mathfrak{t}_{f})$ containing
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$, where
$s=|W(\mathfrak{g},\mathfrak{t}_{f})|/|W(\mathfrak{k},\mathfrak{t}_{f})|$. We
enumerate them as

$(\Delta^{+})^{(j)}(\mathfrak{g},\mathfrak{t}_{f})=\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})\cup(\Delta^{+})^{(j)}(\mathfrak{p},\mathfrak{t}_{f}),\quad
0\leq j\leq s-1.$

Let us denote the half sum of roots of
$(\Delta^{+})^{(j)}(\mathfrak{g},\mathfrak{t}_{f})$ by $\rho^{(j)}$, and put
$\rho_{n}^{(j)}:=\rho^{(j)}-\rho_{c}$. Then $\rho_{n}^{(j)}$ is the half sum
of roots in $(\Delta^{+})^{(j)}(\mathfrak{p},\mathfrak{t}_{f})$. Let $w^{(j)}$
be the unique element in $W(\mathfrak{g},\mathfrak{h}_{f})$ such that
$w^{(j)}\rho^{(0)}=\rho^{(j)}$, and collect them as the set
$W(\mathfrak{g},\mathfrak{t}_{f})^{1}$. Then $w^{(0)}=e$, and by a result of
Kostant [17], the multiplication map induces a bijection from
$W(\mathfrak{g},\mathfrak{t}_{f})^{1}\times W(\mathfrak{k},\mathfrak{t}_{f})$
to $W(\mathfrak{g},\mathfrak{t}_{f})$. For any $0\leq j\leq s-1$,
$w^{(j)}\alpha_{1},\dots,w^{(j)}\alpha_{l}$ are the simple roots of
$(\Delta^{+})^{(j)}(\mathfrak{g},\mathfrak{t}_{f})$, and
$w^{(j)}\zeta_{1},\dots,w^{(j)}\zeta_{l}$ are the corresponding fundamental
weights.

Choose an irreducible module ${\rm Spin}_{G}$ of the Clifford algebra
$C(\mathfrak{p})$. Then as $\mathfrak{k}$-modules, we have the following
decomposition

(4) ${\rm
Spin}_{G}=2^{[\frac{l_{0}}{2}]}\bigoplus_{j=0}^{s-1}E_{\rho_{n}^{(j)}},$

where $l_{0}=\mathop{\hbox{dim}}\nolimits_{\mathbb{C}}\mathfrak{a}_{f}$. See
Lemma 9.3.2 of [27].

Let us denote the dominant Weyl chamber for
$(\Delta^{+})^{(j)}(\mathfrak{g},\mathfrak{t}_{f})$ (resp.,
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$) by $\mathcal{C}^{(j)}$ (resp.,
$\mathcal{C}$). Then

(5) $\mathcal{C}=\bigcup_{j=0}^{s-1}\mathcal{C}^{(j)}.$

Note that $w^{(j)}\mathcal{C}^{(0)}=\mathcal{C}^{(j)}$ for $0\leq j\leq s-1$.
The convex hull formed by the $W(\mathfrak{k},\mathfrak{t}_{f})$ orbits of the
points $\rho_{n}^{(j)}$, $0\leq j\leq s-1$, is called the _unitarily small
convex hull_ , and a $K$-type is said to be _unitariy small_ (_u-small_ for
short henceforth) if its highest weight lies in this polyhedron. See
Salamanca-Riba and Vogan [21]. Otherwise, we will call this $K$-type _u-
large_.

###  2.1. Lambda norm and spin norm

The following geometric way of describing the lambda norm in [23] is due to
Carmona [5]. Consider an arbitrary $K$-type $E_{\mu}$. Choose $0\leq j\leq
s-1$ such that $\mu+2\rho_{c}$ is dominant for
$(\Delta^{+})^{(j)}(\mathfrak{g},\mathfrak{t}_{f})$. Put

(6) $\lambda_{a}(\mu):=P(\mu+2\rho_{c}-\rho^{(j)}).$

Here for any vector $\eta\in i\mathfrak{t}_{f,0}^{*}$, $P(\eta)$ stands for
the projection of $\eta$ to the cone $\mathcal{C}^{(j)}$. Namely, $P(\eta)$ is
the unique point in $\mathcal{C}^{(j)}$ which is closest to $\eta$. By
Proposition 1.1(c) of [21], the two vectors $\eta-P(\eta)$ and $P(\eta)$ are
orthogonal to each other. Therefore,

(7) $\|\eta\|^{2}-\|P(\eta)\|^{2}=\|\eta-P(\eta)\|^{2}.$

The lambda norm of $\mu$ is defined as

(8) $\|\mu\|_{\rm lambda}:=\|\lambda_{a}(\mu)\|.$

It turns out that $\|\mu\|_{\rm lambda}$ is independent of the choice of an
allowable $j$ and is thus well-defined.

On the other hand, the spin norm introduced in [8] is

(9) $\|\mu\|_{\rm spin}:=\min_{0\leq j\leq
s-1}\|\\{\mu-\rho_{n}^{(j)}\\}+\rho_{c}\|.$

Here $\\{\mu-\rho_{n}^{(j)}\\}$ is the unique dominant weight in the
$W(\mathfrak{k},\mathfrak{t}_{f})$ orbit of $\mu-\rho_{n}^{(j)}$. We emphasize
that $\\{\mu-\rho_{n}^{(j)}\\}$, $0\leq j\leq s-1$, are precisely all the PRV
components [20] of the tensor product $E_{\mu}\otimes{\rm Spin}_{G}$ as
$\mathfrak{k}$-modules. It has been shown in [8] that

(10) $\|\mu\|_{\rm spin}\geq\|\mu\|_{\rm lambda}.$

###  2.2. Pathasarathy’s Dirac operator inequality and a reformulation

Let $Z_{1},\dots,Z_{m}$ be an orthonormal basis of $\mathfrak{p}_{0}$ under
the Killing form. Then

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

is the Dirac operator introduced by Parthasarathy [18]. For any irreducible
$(\mathfrak{g},K)$-module $X$ with infinitesimal character (1), the Dirac
operator $D$ acts on $X\otimes{\rm Spin}_{G}$. When $X$ is unitary, then
$D^{2}\geq 0$ on $X\otimes{\rm Spin}_{G}$. Writing out $D^{2}$ carefully, one
will get Parthasarathy’s Dirac operator inequality [19]

(11) $\|\gamma+\rho_{c}\|\geq\|\Lambda\|,$

where $\gamma$ is the highest weight of any $\widetilde{K}$-type occurring in
$X\otimes{\rm Spin}_{G}$. Here $\widetilde{K}$ is the spin covering group of
$K$. This inequality is very effective in non-unitarity test: once a
$\widetilde{K}$-type $\gamma$ in $X\otimes{\rm Spin}_{G}$ has been detected
such that (11) fails, then $\pi$ must be non-unitary. Finding such a $\gamma$
maybe skillful, yet the paper [20] guarantees that it suffices to consider the
PRV components of $X\otimes{\rm Spin}_{G}$. Keeping this in mind, one sees
that the inequality (11) is equivalent to

(12) $\|\mu\|_{\rm spin}\geq\|\Lambda\|,$

where $\mu$ is the highest weight of any $K$-type of $\pi$.

###  2.3. Non-decreasable $K$-types should be u-small

For any $0\leq j\leq s-1$, collect all the $K$-types $\mu$ such that
$\mu+2\rho_{c}\in\mathcal{C}^{(j)}$ as $\Omega(j)$. We call a $K$-type
$\mu\in\Omega(j)$ _decreasable_ if $\mu-w^{(j)}\zeta_{p}\in\mathcal{C}^{(j)}$
for certain $1\leq p\leq l$. Otherwise, $\mu$ is called _non-decreasable_.

######  Conjecture 2.1.

Assume that the $K$-type $\mu\in\Omega(j)$ is non-decreasable. Then $\mu$
should be u-small.

We will give explicit examples of non-decreasable $K$-types in Sections 5 and
6. The above conjecture has been verified for all exceptional groups that we
shall address.

##  3\. Sharpening the Helgason-Johnson bound

Learning from Vogan’s lectures [25, 26], this section aims to give a way to
improve the Helgason-Johnson bound for infinite-dimensional unitary
representations. Indeed, let $\pi$ be such a representation with a lowest
$K$-type $E_{\mu}$, and let its infinitesimal character be given as in (1).
Now by (12),

$\|\Lambda\|^{2}=\|\lambda_{a}(\mu)\|^{2}+\|\nu\|^{2}\leq\|\mu+m\beta\|^{2}_{\rm
spin},\quad\forall m\in\mathbb{Z}_{\geq 0}.$

Therefore,

$\|\nu\|^{2}\leq\|\mu+m\beta\|^{2}_{\rm
spin}-\|\lambda_{a}(\mu)\|^{2},\quad\forall m\in\mathbb{Z}_{\geq 0}.$

In other words,

(13) $\|\nu\|^{2}\leq{\rm MP}(\mu)-\|\mu\|_{\rm lambda}^{2}.$

To get a uniform bound of $\|\nu\|^{2}$ for all the infinite-dimensional
irreducible unitary representations, it suffices to obtain the maximum of the
RHS of (13) over all the $K$-types $\mu$.

Now let us explain the idea of handling the RHS of (13). Theorem 1.1 of [9]
and Theorem C of [10] suggest that when $\mu$ is u-large, we should have ${\rm
MP(\mu)}=\|\mu\|_{\rm spin}^{2}$. Thus we should firstly understand the
behavior of $\|\mu\|_{\rm spin}^{2}-\|\mu\|_{\rm lambda}^{2}$.

Take any $\mu\in\Omega(j)$. We have that

$\displaystyle\|\mu\|_{\rm spin}^{2}-\|\mu\|_{\rm lambda}^{2}$
$\displaystyle=\|\mu\|_{\rm spin}^{2}-\|P(\mu+2\rho_{c}-\rho^{(j)})\|$
$\displaystyle\leq\|\\{\mu-\rho_{n}^{(j)}\\}+\rho_{c}\|^{2}-\|P(\mu+\rho_{c}-\rho_{n}^{(j)})\|$
$\displaystyle\leq{\rm I}(\mu)+{\rm II}(\mu),$

where

(14) ${\rm
I}(\mu)=\|\\{\mu-\rho_{n}^{(j)}\\}+\rho_{c}\|^{2}-\|\mu-\rho_{n}^{(j)}+\rho_{c}\|^{2}=2\langle\rho_{c},\\{\mu-\rho_{n}^{(j)}\\}-(\mu-\rho_{n}^{(j)})\rangle$

and

(15) ${\rm
II}(\mu)=\|\mu+\rho_{c}-\rho_{n}^{(j)}\|^{2}-\|P(\mu+\rho_{c}-\rho_{n}^{(j)})\|^{2}.$

Here recall that $P(\mu+\rho_{c}-\rho_{n}^{(j)})$ is the projection of
$\mu+\rho_{c}-\rho_{n}^{(j)}$ to the cone $\mathcal{C}^{(j)}$. By (7),

(16) ${\rm
II}(\mu)=\|\mu+\rho_{c}-\rho_{n}^{(j)}-P(\mu+\rho_{c}-\rho_{n}^{(j)})\|^{2}.$

It follows that both ${\rm I}(\mu)$ and ${\rm II}(\mu)$ are non-negative.

Take any vector $\delta\in\mathcal{C}^{(j)}$, Lemma 3.3 will guarantee that
${\rm I}(\mu+\delta)\leq{\rm I}(\mu)$ since $\delta$ must be dominant for
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$. On the other hand,

$\displaystyle{\rm II}(\mu+\delta)$
$\displaystyle=\|\mu+\delta+\rho_{c}-\rho_{n}^{(j)}-P(\mu+\delta+\rho_{c}-\rho_{n}^{(j)})\|^{2}$
$\displaystyle\leq\|\mu+\delta+\rho_{c}-\rho_{n}^{(j)}-(\delta+P(\mu+\rho_{c}-\rho_{n}^{(j)}))\|^{2}$
$\displaystyle=\|\mu+\rho_{c}-\rho_{n}^{(j)}-P(\mu+\rho_{c}-\rho_{n}^{(j)})\|^{2}$
$\displaystyle={\rm II}(\mu),$

where the second step holds since
$\delta+P(\mu+\rho_{c}-\rho_{n}^{(j)})\in\mathcal{C}^{(j)}$.

Now we can state the following way of sharpening the Helgason-Johnson bound:

(a) For any $0\leq j\leq s-1$, select finitely many u-small $K$-types
$\mu_{1},\dots,\mu_{m(j)}$ in $\Omega(j)$ so that for any u-large
$\mu\in\Omega(j)$, there exists $1\leq i\leq m(j)$ such that
$\mu-\mu_{i}\in\mathcal{C}^{(j)}$. Then as deduced above,

${\rm I}(\mu)+{\rm II}(\mu)\leq{\rm I}(\mu_{i})+{\rm II}(\mu_{i}).$

Let

(17) $A_{j}:=\max\\{{\rm I}(\mu_{i})+{\rm II}(\mu_{i})\mid 1\leq i\leq
m(j)\\}.$

Therefore, for any u-large $\mu\in\Omega(j)$, we have that $\|\mu\|_{\rm
spin}^{2}-\|\mu\|_{\rm lambda}^{2}\leq A_{j}$.

(b) We directly compute

(18) $B=\max_{\mu}\left\\{\min_{m}\|\mu+m\beta\|^{2}_{\rm
spin}-\|\mu\|^{2}_{\rm lambda}\mid m\in\mathbb{Z}_{\geq 0}\mbox{ such that
}\mu+m\beta\mbox{ is u-small}\right\\},$

where $\mu$ runs over all the u-small $K$-types in $\Omega(j)$ other than
those $\mu_{1},\dots,\mu_{m(j)}$ in step (a).

To sum up, $\|\nu\|^{2}$ is upper bounded by
$\max\\{A_{0},A_{1},\dots,A_{s-1},B\\}$.

###  3.1. Technical lemmas for ${\rm I}(\mu)$

This subsection aims to prepare some technical lemmas for analyzing the term
${\rm I}(\mu)$ defined in (14). Let $\gamma_{1},\dots,\gamma_{l}$ be the
simple roots of $\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$, and let
$\varpi_{1},\dots,\varpi_{l}$ be the corresponding fundamental weights.

Let $\eta=\sum_{i=1}^{l}\eta_{i}\varpi_{i}$ be an arbitrary weight. We utilize
the following _negative index algorithm_ to conjugate $\eta$ to $\mathcal{C}$:
if the weight is not yet dominant, select an index $i$ such that $\eta_{i}<0$,
then apply the simple reflection $s_{\gamma_{i}}$. We record the corresponding
sequence of simple reflections, and collect all of them as $S(\lambda)$.

######  Lemma 3.1.

_(Theorem 4.3.1(iv) of Björner and Brenti[2], Proposition 4.1 of Eriksson
[13])_ Let $\eta$ be any weight. Let $s_{\beta_{1}},\dots,s_{\beta_{n}},\dots$
be any sequence in $S(\eta)$, then the word $s_{\beta_{n}}\cdots
s_{\beta_{1}}$ is reduced. Here each $\beta_{i}$ is a simple root of
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$. In particular, each sequence in
$S(\eta)$ is of finite length.

######  Lemma 3.2.

_(Theorem 4.5 of Eriksson[13])_ Assume that $\eta_{1}$ and $\eta_{2}$ are two
weights such that $\eta_{1}-\eta_{2}$ is dominant for
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$. Then every sequence
$s_{\beta_{1}},\dots,s_{\beta_{n}}$ in $S(\eta_{1})$ can be extended to a
sequence $s_{\beta_{1}},\dots,s_{\beta_{n}},\dots,s_{\beta_{N}}$ in
$S(\eta_{2})$. Here $N\geq n$.

###### Proof.

Note that for any $1\leq k\leq n$, we have

$\langle\eta_{1}-\eta_{2},s_{\beta_{1}}s_{\beta_{2}}\cdots
s_{\beta_{k-1}}(\beta_{k})\rangle\geq 0$

since $\eta_{1}-\eta_{2}$ is dominant for
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ and
$s_{\beta_{1}}s_{\beta_{2}}\cdots
s_{\beta_{k-1}}(\beta_{k})\in\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$.
Therefore,

$\langle s_{\beta_{k-1}}\cdots
s_{\beta_{2}}s_{\beta_{1}}(\eta_{1}-\eta_{2}),\beta_{k}\rangle\geq 0.$

Thus for any $1\leq k\leq n$,

$\langle s_{\beta_{k-1}}\cdots
s_{\beta_{2}}s_{\beta_{1}}(\eta_{2}),\beta_{k}^{\vee}\rangle\leq\langle
s_{\beta_{k-1}}\cdots
s_{\beta_{2}}s_{\beta_{1}}(\eta_{1}),\beta_{k}^{\vee}\rangle<0.$

Now the negative index algorithm finishes the proof. ∎

######  Lemma 3.3.

Assume that $\eta_{1}$ and $\eta_{2}$ are two weights such that
$\eta_{1}-\eta_{2}$ is dominant for
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$. Then
$\langle\rho_{c},\\{\eta_{1}\\}-\eta_{1}\rangle\leq\langle\rho_{c},\\{\eta_{2}\\}-\eta_{2}\rangle$.

###### Proof.

Take a sequence $s_{\beta_{1}},\dots,s_{\beta_{n}}$ in $S(\eta_{1})$. By Lemma
3.2, it extends to a sequence
$s_{\beta_{1}},\dots,s_{\beta_{n}},\dots,s_{\beta_{N}}$ in $S(\eta_{2})$. Note
that $s_{\beta_{n}}\cdots s_{\beta_{1}}$ and $s_{\beta_{N}}\cdots
s_{\beta_{n}}\cdots s_{\beta_{1}}$ are reduced expressions due to Lemma 3.1.

By Lemma 5.5 of [11],

$\displaystyle\\{\eta_{1}\\}-\eta_{1}$ $\displaystyle=s_{\beta_{n}}\cdots
s_{\beta_{1}}(\eta_{1})-\eta_{1}$
$\displaystyle=\sum_{k=1}^{n}\left(s_{\beta_{n}}\cdots
s_{\beta_{k}}(\eta_{1})-s_{\beta_{n}}\cdots s_{\beta_{k+1}}(\eta_{1})\right)$
$\displaystyle=\sum_{k=1}^{n}s_{\beta_{n}}\cdots
s_{\beta_{k+1}}(\eta_{1}-s_{\beta_{k}}(\eta_{1}))$
$\displaystyle=\sum_{k=1}^{n}\langle-\eta_{1},\beta_{k}^{\vee}\rangle
s_{\beta_{n}}\cdots s_{\beta_{k+1}}(\beta_{k}).$

Therefore,

(19)
$\langle\rho_{c},\\{\eta_{1}\\}-\eta_{1}\rangle=\sum_{k=1}^{n}\langle-\eta_{1},\beta_{k}^{\vee}\rangle\langle\rho_{c},s_{\beta_{n}}\cdots
s_{\beta_{k+1}}(\beta_{k})\rangle.$

On the other hand,

$\displaystyle\\{\eta_{2}\\}-\eta_{2}$
$\displaystyle=\sum_{k=1}^{n}\left(s_{\beta_{n}}\cdots
s_{\beta_{k}}(\eta_{2})-s_{\beta_{n}}\cdots s_{\beta_{k+1}}(\eta_{2})\right)$
$\displaystyle+\sum_{k=n+1}^{N}\left(s_{\beta_{k}}\cdots
s_{\beta_{1}}(\eta_{2})-s_{\beta_{k-1}}\cdots s_{\beta_{1}}(\eta_{2})\right)$
$\displaystyle=\sum_{k=1}^{n}\langle-\eta_{2},\beta_{k}^{\vee}\rangle
s_{\beta_{n}}\cdots s_{\beta_{k+1}}(\beta_{k})-\sum_{k=n+1}^{N}\langle
s_{\beta_{k-1}}\cdots
s_{\beta_{1}}(\eta_{2}),\beta_{k}^{\vee}\rangle\beta_{k}$

Therefore,

$\displaystyle\langle\rho_{c},(\\{\eta_{2}\\}-\eta_{2})-(\\{\eta_{1}\\}-\eta_{1})\rangle$
$\displaystyle=\sum_{k=1}^{n}\langle\eta_{1}-\eta_{2},\beta_{k}^{\vee}\rangle\langle\rho_{c},s_{\beta_{n}}\cdots
s_{\beta_{k+1}}(\beta_{k})\rangle$ $\displaystyle-\sum_{k=n+1}^{N}\langle
s_{\beta_{k-1}}\cdots
s_{\beta_{1}}(\eta_{2}),\beta_{k}^{\vee}\rangle\langle\rho_{c},\beta_{k}\rangle.$

Since the sequence $s_{\beta_{1}},\dots,s_{\beta_{n}},\dots,s_{\beta_{N}}$
belongs to $S(\eta_{2})$, we have that

$\langle s_{\beta_{k-1}}\cdots
s_{\beta_{1}}(\eta_{2}),\beta_{k}^{\vee}\rangle<0,\quad\forall n+1\leq k\leq
N$

Now the desired conclusion follows. ∎

##  4\. Complex exceptional Lie groups

This section aims to prove Theorem 1.1. Assume that $G$ is complex, let
$\alpha_{1},\dots,\alpha_{l}$ be the simple roots for the fixed
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$. Let $\varpi_{1},\dots,\varpi_{l}$
be the corresponding fundamental weights. The highest weight $\mu$ of a
$\mathfrak{k}$-type is a non-negative integer combination of
$\varpi_{1},\dots,\varpi_{l}$. For convenience, we use
$[a_{1},a_{2},\dots,a_{l}]$ to stand for the vector
$\sum_{i=1}^{l}a_{i}\varpi_{i}$.

Note that $s=1$, $\rho^{(0)}=2\rho_{c}$ and $\rho_{n}^{(0)}=\rho_{c}$. Collect
all the u-small $\mathfrak{k}$-types as $\Omega_{\rm us}(0)$. Let $B$ be the
maximum of the expression ${\rm MP}(\mu)-\|\mu\|^{2}_{\rm lambda}$ over the
set $\Omega_{\rm us}(0)$. Define

(20) $\partial\Omega_{\rm us}(0)=\\{\mu\in\Omega_{\rm us}(0)\mid\exists\,1\leq
i\leq l\mbox{ such that }\mu+\varpi_{i}\mbox{ is not u-small}\\}.$

Let $A_{0}$ be the maximum of the expression $\|\mu\|^{2}_{\rm
spin}-\|\mu\|^{2}_{\rm lambda}$ over the set $\partial\Omega_{\rm us}(0)$.
Note that for any u-large $\mathfrak{k}$-type $\mu$, there exists
$\mu_{1}\in\partial\Omega_{\rm us}(0)$ such that $\mu-\mu_{1}\in\mathcal{C}$.
Indeed, we can subtract some $\varpi_{i}$ from $\mu$ step by step, but stay
within the cone $\mathcal{C}$. Then by Conjecture 2.1 (which is easily seen to
be true for complex groups), eventually we will come down to the u-small
convex hull. The first $\mathfrak{k}$-type that we meet in the u-small convex
hull can be chosen as the desired $\mu_{1}$. Therefore, due to the discussion
around (18), $\max\\{A_{0},B\\}$ is an upper bound of $\|\nu\|^{2}$ in Theorem
1.1.

We will handle the complex exceptional groups one by one. It turns out that
$B$ is attained at the trivial $\mathfrak{k}$-type in each case.

###  4.1. Complex $G_{2}$

Let $\alpha_{1}=(1,-1,0)$ and $\alpha_{2}=(-2,1,1)$ be the two simple roots.
The Dynkin diagram is presented in Fig. 1.

Figure 1. The Dynkin diagram for $G_{2}$

In this case, $\Omega_{\rm us}(0)$ has fourteen $\mathfrak{k}$-types and

$\partial\Omega_{\rm us}(0)=\\{[0,3],[1,2],[2,2],[3,1],[4,0],[5,0]\\}.$

In Fig. 2, we use circles to denote the $\mathfrak{k}$-types in
$\partial\Omega_{\rm us}(0)$, while the other u-small $\mathfrak{k}$-types are
denoted by dots.

Figure 2. The complex $G_{2}$ case

We compute that $A_{0}=6$ and $B=26$, which is attained at the trivial
$\mathfrak{k}$-type. Thus the complex $G_{2}$ case in Theorem 1.1 follows.

###  4.2. Complex $F_{4}$

Let $\alpha_{1}=(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2})$,
$\alpha_{2}=(0,0,0,1)$, $\alpha_{3}=(0,0,1,-1)$ and $\alpha_{4}=(0,1,-1,0)$ be
the simple roots. The Dynkin diagram is presented in Fig. 3.

Figure 3. The Dynkin diagram for $F_{4}$

We compute that $|\Omega_{\rm us}(0)|=451$ and $|\partial\Omega_{\rm
us}(0)|=214$. Moreover, $A_{0}=35$ and $B=78$, which is attained at the
trivial $\mathfrak{k}$-type. The complex $F_{4}$ case in Theorem 1.1 follows.

###  4.3. Complex $E_{6}$

We label the Dynkin diagram in Fig. 4.

Figure 4. The Dynkin diagram for $E_{6}$

We compute that $|\Omega_{\rm us}(0)|=13007$ and $|\partial\Omega_{\rm
us}(0)|=6249$. Moreover, $A_{0}=120$ and $B=170$, which is attained at the
trivial $\mathfrak{k}$-type. The complex $E_{6}$ case in Theorem 1.1 follows.

###  4.4. Complex $E_{7}$

We label the Dynkin diagram in Fig. 5.

Figure 5. The Dynkin diagram for $E_{7}$

We compute that $|\Omega_{\rm us}(0)|=105951$ and $|\partial\Omega_{\rm
us}(0)|=52630$. Moreover, $A_{0}=312$ and $B=464$, which is attained at the
trivial $\mathfrak{k}$-type. The complex $E_{7}$ case in Theorem 1.1 follows.

###  4.5. Complex $E_{8}$

We label the Dynkin diagram in Fig. 6.

Figure 6. The Dynkin diagram for $E_{8}$

We compute that $|\Omega_{\rm us}(0)|=950077$ and $|\partial\Omega_{\rm
us}(0)|=486023$. Moreover, $A_{0}=800$ and $B=1522$, which is attained at the
trivial $\mathfrak{k}$-type. The complex $E_{8}$ case in Theorem 1.1 follows.

##  5\. Real exceptional Lie groups

This section aims to prove Theorem 1.2. Recall that
$\alpha_{1},\dots,\alpha_{l}$ are the simple roots of
$(\Delta^{+})^{(0)}(\mathfrak{g},\mathfrak{t}_{f})$, with
$\zeta_{1},\dots,\zeta_{l}$ being the corresponding fundamental weights. For
convenience, we will use $s_{i}$ to denote the simple reflection
$s_{\alpha_{i}}$, and use $[[a_{1},\dots,a_{l}]]$ to denote the vector
$a_{1}\zeta_{1}+\cdots+a_{l}\zeta_{l}$.

Since $\mathfrak{k}$ is assumed to have no center,
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ has $l$ simple roots as well. We
denote them by $\gamma_{1},\dots,\gamma_{l}$, and let the corresponding
fundamental weights be $\varpi_{1},\dots,\varpi_{l}$. The highest weight of
any $\mathfrak{k}$-type is a non-negative integer combination of
$\varpi_{1},\dots,\varpi_{l}$. For convenience, we will write
$[b_{1},\dots,b_{l}]$ to stand for the vector
$b_{1}\varpi_{1}+\cdots+b_{l}\varpi_{l}$.

Fix any $0\leq j\leq s-1$. Collect all the $\mathfrak{k}$-types $\mu$ such
that $\mu+2\rho_{c}\in\mathcal{C}^{(j)}$ as $\Omega(j)$. Let $\Omega_{\rm
us}(j)$ consist of the u-small members of $\Omega(j)$. Define

(21) $\partial\Omega_{\rm us}(j)=\\{\mu\in\Omega_{\rm us}(j)\mid\exists\,1\leq
i\leq l\mbox{ such that }\mu+w^{(j)}\zeta_{i}\mbox{ is not u-small}\\}.$

Let $A_{j}$ be the maximum of
$\|\\{\mu-\rho_{n}^{(j)}\\}+\rho_{c}\|^{2}-\|\mu\|^{2}_{\rm lambda}$ over the
set $\partial\Omega_{\rm us}(j)$.

We claim that for any u-large $\mathfrak{k}$-type $\mu\in\Omega(j)$, there
exists $\mu_{1}\in\partial\Omega_{\rm us}(j)$ such that
$\mu-\mu_{1}\in\mathcal{C}^{(j)}$. Indeed, it follows from Conjecture 2.1
(once verified) that $\mu\in\Omega(j)$ is decreasable. Namely, we can subtract
some $w^{(j)}\zeta_{i}$ from $\mu$ so that $\mu-w^{(j)}\zeta_{i}\in\Omega(j)$.
Doing this step by step, eventually we will come down to a non-decreasable
$\mathfrak{k}$-type in $\Omega(j)$ which must be u-small. The first
$\mathfrak{k}$-type that we meet in the u-small convex hull during this going
down process can be chosen as the desired $\mu_{1}$. Thus the claim holds.

Let $B$ be the maximum of the expression

$\min_{m}\\{\|\mu+m\beta\|^{2}_{\rm spin}-\|\mu\|^{2}_{\rm lambda}\mid
m\in\mathbb{Z}_{\geq 0}\mbox{ such that }\mu+m\beta\mbox{ is u-small}\\}$

with $\mu$ running over $\Omega_{\rm us}(j)\setminus\partial\Omega_{\rm
us}(j)$. Due to the discussion around (18), $\max\\{A_{0},\dots,A_{s-1},B\\}$
is an upper bound of $\|\nu\|^{2}$ in Theorem 1.2.

In the following, we will handle the real exceptional Lie groups one by one.
As mentioned in the introduction, the value $B$ is attained at the trivial
$K$-type if and only if $G/K$ is _not_ quaternionic in the sense of [16].

###  5.1. $G_{2(2)}$

The Vogan diagram of this simple Lie algebra is obtained from Fig. 1 by
painting the simple root $\alpha_{2}$. Then $\gamma_{1}=\alpha_{1}$ and
$\gamma_{2}=3\alpha_{1}+2\alpha_{2}$ are the simple roots for
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$, which has type $A_{1}\times
A_{1}$. Let $\mu=[a,b]$ be a $\mathfrak{k}$-type. Let us enumerate the set
$W(\mathfrak{g},\mathfrak{t}_{f})^{1}$ in order as $w^{(0)}=e$,
$w^{(1)}=s_{2}$ and $w^{(2)}=s_{2}s_{1}$. Then

$\rho_{n}^{(0)}=[0,2],\quad\rho_{n}^{(1)}=\beta=[3,1],\quad\rho_{n}^{(2)}=[4,0].$

Moreover, $\mu+2\rho_{c}$ has coordinates

$(a+2,\frac{b}{2}-\frac{a}{2}),\quad(-\frac{a}{2}+\frac{3b}{2}+2,\frac{a}{2}-\frac{b}{2}),\quad(\frac{a}{2}-\frac{3b}{2}-2,b+2)$

with respect to the basis $\\{w^{(j)}\zeta_{1},w^{(j)}\zeta_{2}\\}$ for
$j=0,1,2$, respectively.

We move on according to the following three cases:

  * $\bullet$

$\mu\in\Omega(0)$ if and only if $b\geq a$. Note that $\zeta_{1}=[1,1]$ and
$\zeta_{2}=[0,2]$. We compute that

$\partial\Omega_{\rm us}(0)=\\{[0,3],[0,4],[1,2],[1,3],[2,2],[2,3],[3,3]\\}$

and that $A_{0}=\frac{3}{2}$.

  * $\bullet$

$\mu\in\Omega(1)$ if and only if $b\leq a\leq 3b+4$.

Note that $w^{(1)}\zeta_{1}=[1,1]$ and $w^{(1)}\zeta_{2}=[3,1]$. We compute
that

$\partial\Omega_{\rm
us}(1)=\\{[4,1],[5,1],[6,1],[7,1],[2,2],[3,2],[4,2],[5,2],[6,2],[3,3]\\}$

and that $A_{1}=\frac{7}{2}$.

  * $\bullet$

$\mu\in\Omega(2)$ if and only if $a\geq 3b+4$.

Note that $w^{(2)}\zeta_{1}=[2,0]$ and $w^{(2)}\zeta_{2}=[3,1]$. We compute
that

$\partial\Omega(2)_{\rm us}=\\{[5,0],[6,0],[7,0],[8,0],[7,1]\\}$

and that $A_{2}=\frac{1}{2}$.

Moreover, we compute that $B=\frac{51}{8}$, which happens at the
$\mathfrak{k}$-type $[0,1]$. The $G_{2(2)}$ case in Theorem 1.2 follows.

In Fig. 7, the $\mathfrak{k}$-types in $\partial\Omega_{\rm us}(i)$ for $0\leq
i\leq 2$ are denoted by circles, while the other u-small $\mathfrak{k}$-types
are denoted by dots.

Figure 7. The $G_{2(2)}$ case

######  Example 5.1.

The non-decreasable $\mathfrak{k}$-types in $\Omega(j)$ are

$j=0:[0,0],[0,1];\quad j=1:[0,0],[1,0],[2,0],[3,0],[4,0];\quad
j=2:[4,0],[5,0].$

They are all u-small. ∎

###  5.2. ${\rm FI}=F_{4(4)}$

The Vogan diagram of this simple Lie algebra is obtained from Fig. 3 by
painting the simple root $\alpha_{4}$. Then $\gamma_{i}=\alpha_{i}$ for $1\leq
i\leq 3$ and $\gamma_{4}:=2\alpha_{1}+4\alpha_{2}+3\alpha_{3}+2\alpha_{4}$ are
the simple roots for $\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$, which has
type $C_{3}\times A_{1}$. Note that $\beta=[0,0,1,1]$.

Table 1. The $F_{4(4)}$ case $j$ | $w^{(j)}$ | $\rho_{n}^{(j)}$ | $\\#\Omega_{\rm us}(j)$ | $\\#\partial\Omega_{\rm us}(j)$ | $A_{j}$
---|---|---|---|---|---
$0$ | $e$ | $[0,0,0,7]$ | $12$ | $11$ | $0.5$
$1$ | $s_{4}$ | $[0,0,1,6]$ | $68$ | $53$ | $8.5$
$2$ | $s_{4}s_{3}$ | $[0,2,0,5]$ | $116$ | $84$ | $9.5$
$3$ | $s_{4}s_{3}s_{2}$ | $[1,2,0,4]$ | $193$ | $132$ | $9.5$
$4$ | $s_{4}s_{3}s_{2}s_{1}$ | $[0,3,0,3]$ | $105$ | $74$ | $7$
$5$ | $s_{4}s_{3}s_{2}s_{3}$ | $[3,0,1,3]$ | $166$ | $114$ | $9$
$6$ | $s_{4}s_{3}s_{2}s_{1}s_{3}$ | $[2,1,1,2]$ | $339$ | $215$ | $9$
$7$ | $s_{4}s_{3}s_{2}s_{3}s_{4}$ | $[5,0,0,2]$ | $41$ | $34$ | $0.75$
$8$ | $s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}$ | $[2,0,2,1]$ | $215$ | $142$ | $9.75$
$9$ | $s_{4}s_{3}s_{2}s_{1}s_{3}s_{4}$ | $[4,1,0,1]$ | $130$ | $94$ | $6.25$
$10$ | $s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{3}$ | $[0,0,3,0]$ | $43$ | $34$ | $2.5$
$11$ | $s_{4}s_{3}s_{2}s_{1}s_{3}s_{2}s_{4}$ | $[4,0,1,0]$ | $87$ | $63$ | $6.25$

We summarize the information in Table 1. Moreover, $B=\frac{39}{2}$ and it is
attained at the $\mathfrak{k}$-type $\mu=[0,0,0,1]$. Indeed, $\mu$ belongs to
$\Omega(6)$, and $\mu+2\rho_{c}-\rho^{(6)}$ has coordinates
$(0,0,-\frac{1}{2},\frac{1}{2})$ w.r.t. the basis
$\\{w^{(6)}\zeta_{1},\dots,w^{(6)}\zeta_{4}\\}$. Therefore, projecting
$\mu+2\rho_{c}-\rho^{(6)}$ to $\mathcal{C}^{(6)}$ is equivalent to projecting
$[[0,0,-\frac{1}{2},\frac{1}{2}]]$ to $\mathcal{C}^{(0)}$, which produces the
zero vector. Therefore, $\|\mu\|_{\rm lambda}=0$. On the other hand, ${\rm
MP}(\mu)=\|\mu+\beta\|_{\rm spin}=\frac{39}{2}$. The $F_{4(4)}$ case in
Theorem 1.2 follows.

######  Example 5.2.

Take $j=6$ and consider the $\mathfrak{k}$-type $\mu=[a,b,c,d]$. Then
$\mu+2\rho_{c}$ has coordinates

$(b+c-d+2,d-c,a/2+c/2-d/2+1,-a/2+c/2+d/2+1)$

w.r.t. the basis $\\{w^{(j)}\zeta_{1},\dots,w^{(j)}\zeta_{4}\\}$. Therefore,
$\mu\in\Omega(j)$ if and only if

$d\leq b+c+2,\quad c\leq d,\quad d\leq a+c+2,\quad a\leq c+d+2.$

On the other hand,

$w^{(j)}\zeta_{1}=[0,1,0,0],\quad w^{(j)}\zeta_{2}=[1,1,0,1],\quad
w^{(j)}\zeta_{3}=[2,0,1,1],\quad w^{(j)}\zeta_{4}=[0,0,1,1].$

We compute that the non-decreasable $\mathfrak{k}$-types in $\Omega(j)$ are

$\displaystyle[0,0,0,0],[0,0,0,1],[0,0,0,2],[1,0,0,0],[1,0,0,1],[1,0,0,2],$
$\displaystyle[2,0,0,0],[2,0,0,1],[2,0,0,2],[3,0,0,1],[3,0,0,2],[4,0,0,2].$

They are all u-small. ∎

###  5.3. ${\rm FII}=F_{4(-20)}$

The Vogan diagram of this simple Lie algebra is obtained from Fig. 3 by
painting the simple root $\alpha_{1}$. Then $\gamma_{1}=(1,-1,0,0)$,
$\gamma_{2}=\alpha_{4}$, $\gamma_{3}=\alpha_{3}$ and $\gamma_{4}=\alpha_{2}$
are the simple roots for $\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$, which is
of type $B_{4}$. Let $\mu=[a,b,c,d]$ be a $\mathfrak{k}$-type. Note that
$\beta=[0,0,0,1]$.

Table 2. The $F_{4(-20)}$ case $j$ | $w^{(j)}$ | $\rho_{n}^{(j)}$ | $\\#\Omega_{\rm us}(j)$ | $\\#\partial\Omega_{\rm us}(j)$ | $A_{j}$
---|---|---|---|---|---
$0$ | $e$ | $[2,0,0,0]$ | $5$ | $5$ | $0.25$
$1$ | $s_{1}$ | $[1,0,0,1]$ | $21$ | $18$ | $11.25$
$2$ | $s_{1}s_{2}$ | $[0,0,1,0]$ | $14$ | $12$ | $10.25$

We summarize the information in Table 2. Moreover, $B=\frac{81}{4}$, which is
attained at the trivial $\mathfrak{k}$-type. The $F_{4(-20)}$ case in Theorem
1.2 follows.

###  5.4. ${\rm EI}=E_{6(6)}$

The Vogan diagram of this simple Lie algebra is presented in Fig. 8. The
positive root system $(\Delta^{+})^{(0)}(\mathfrak{g},\mathfrak{t}_{f})$ is of
type $F_{4}$, and has simple roots
$\alpha_{1}=\frac{1}{2}(\beta_{1}+\beta_{6})$,
$\alpha_{2}=\frac{1}{2}(\beta_{3}+\beta_{5})$, $\alpha_{3}=\beta_{4}$,
$\alpha_{4}=\beta_{2}$. Here $\alpha_{1}$ and $\alpha_{2}$ are short, while
$\alpha_{3}$ and $\alpha_{4}$ are long. On the other hand, the positive root
system $\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ is of type $C_{4}$, and has
simple roots $\gamma_{1}=\alpha_{2}+\alpha_{3}+\alpha_{4}$,
$\gamma_{2}=\alpha_{1}$, $\gamma_{3}=\alpha_{2}$, $\gamma_{4}=\alpha_{3}$.
Here $\gamma_{4}$ is long. Note that $\beta=[0,0,0,1]$.

Figure 8. The Vogan diagram for EI

We summarize the information in Table 3. Moreover, $B=42$, which is attained
at the trivial $\mathfrak{k}$-type. The $E_{6(6)}$ case in Theorem 1.2
follows.

Table 3. The $E_{6(6)}$ case $j$ | $w^{(j)}$ | $\rho_{n}^{(j)}$ | $\\#\Omega_{\rm us}(j)$ | $\\#\partial\Omega_{\rm us}(j)$ | $A_{j}$
---|---|---|---|---|---
$0$ | $e$ | $[5,1,1,0]$ | $192$ | $124$ | $16$
$1$ | $s_{4}$ | $[3,1,1,1]$ | $539$ | $295$ | $18.5$
$2$ | $s_{4}s_{3}$ | $[1,1,3,0]$ | $354$ | $202$ | $16.5$

###  5.5. ${\rm EII}=E_{6(2)}$

The Vogan diagram of this simple Lie algebra is obtained by painting the
simple root $\alpha_{2}$ in Fig. 4. Then
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ is of type $A_{5}\times A_{1}$,
and has the following simple roots

$\gamma_{i}=\alpha_{7-i},\quad 1\leq i\leq
4;\quad\gamma_{5}=\alpha_{1};\quad\gamma_{6}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6}.$

We have that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=36$ and
$\beta=[0,0,1,0,0,1]$.

Table 4. The $E_{6(2)}$ case $\rho_{n}^{(j)}$ | $\\#\Omega_{\rm us}(j)$ | $\\#\partial\Omega_{\rm us}(j)$ | $A_{j}$ | $\rho_{n}^{(j)}$ | $\\#\Omega_{\rm us}(j)$ | $\\#\partial\Omega_{\rm us}(j)$ | $A_{j}$
---|---|---|---|---|---|---|---
$[0,0,0,0,0,10]$ | $19$ | $18$ | $0.5$ | $[0,0,1,0,0,9]$ | $166$ | $144$ | $12.5$
$[0,1,0,1,0,8]$ | $693$ | $548$ | $21$ | $[0,2,0,0,1,7]$ | $553$ | $437$ | $25$
$[1,0,0,2,0,7]$ | $484$ | $379$ | $25$ | $[0,3,0,0,0,6]$ | $151$ | $130$ | $9$
$[1,1,0,1,1,6]$ | $2178$ | $1574$ | $29$ | $[0,0,0,3,0,6]$ | $118$ | $100$ | $9$
$[1,2,0,1,0,5]$ | $1175$ | $869$ | $29$ | $[0,1,0,2,1,5]$ | $1035$ | $756$ | $29$
$[2,0,1,0,2,5]$ | $1146$ | $841$ | $29$ | $[0,2,0,2,0,4]$ | $674$ | $487$ | $17.5$
$[2,1,1,0,1,4]$ | $2134$ | $1523$ | $29$ | $[3,0,0,0,3,4]$ | $235$ | $193$ | $9$
$[1,0,1,1,2,4]$ | $1899$ | $1333$ | $29$ | $[3,1,0,0,2,3]$ | $800$ | $605$ | $25$
$[1,1,1,1,1,3]$ | $3909$ | $2590$ | $30$ | $[2,0,0,1,3,3]$ | $727$ | $541$ | $25$
$[3,0,2,0,0,3]$ | $497$ | $375$ | $25$ | $[0,0,2,0,3,3]$ | $454$ | $339$ | $25$
$[2,1,0,1,2,2]$ | $2108$ | $1436$ | $32$ | $[0,1,2,0,2,2]$ | $1333$ | $918$ | $30$
$[4,0,1,0,1,2]$ | $1116$ | $832$ | $21$ | $[1,0,1,0,4,2]$ | $1008$ | $742$ | $21$
$[2,0,2,1,0,2]$ | $1333$ | $918$ | $30$ | $[1,1,1,0,3,1]$ | $2268$ | $1554$ | $32$
$[3,0,1,1,1,1]$ | $2268$ | $1554$ | $32$ | $[1,0,3,0,1,1]$ | $1308$ | $920$ | $30$
$[5,0,0,1,0,1]$ | $413$ | $332$ | $13$ | $[0,1,0,0,5,1]$ | $375$ | $299$ | $13$
$[0,2,0,0,4,0]$ | $242$ | $192$ | $17$ | $[2,0,2,0,2,0]$ | $861$ | $598$ | $32$
$[0,0,0,0,6,0]$ | $30$ | $27$ | $0.5$ | $[4,0,0,2,0,0]$ | $242$ | $192$ | $17$
$[0,0,4,0,0,0]$ | $75$ | $66$ | $0.5$ | $[6,0,0,0,0,0]$ | $37$ | $34$ | $0.5$

We summarize the information in Table 4. Moreover, $B=42$, which is attained
at the $\mathfrak{k}$-type $[0,0,0,0,0,2]$. The $E_{6(2)}$ case in Theorem 1.2
follows.

###  5.6. ${\rm EIV}=E_{6(-26)}$

The Vogan diagram of this simple Lie algebra is presented in Fig. 9. The
positive root systems $(\Delta^{+})^{(0)}(\mathfrak{g},\mathfrak{t}_{f})$ and
$(\Delta^{+})^{(0)}(\mathfrak{k},\mathfrak{t}_{f})$ are both of type $F_{4}$.
The simple roots are
$\alpha_{1}=\gamma_{1}:=\frac{1}{2}(\beta_{1}+\beta_{6})$,
$\alpha_{2}=\gamma_{2}:=\frac{1}{2}(\beta_{3}+\beta_{5})$,
$\alpha_{3}=\gamma_{3}:=\beta_{4}$, $\alpha_{4}=\gamma_{4}:=\beta_{2}$. The
set $W(\mathfrak{g},\mathfrak{t}_{f})^{1}=\\{e\\}$. We have that
$\rho_{n}^{(0)}=[1,1,0,0]$ and that $\beta=[1,0,0,0]$.

Figure 9. The Vogan diagram for EIV

We compute that $\\#\Omega_{\rm us}(0)=37$, $\\#\partial\Omega_{\rm
us}(0)=28$, $A_{0}=21$ and $B=37$, which is attained at the trivial
$\mathfrak{k}$-type. The $E_{6(-26)}$ case in Theorem 1.2 follows.

###  5.7. ${\rm EV}=E_{7(7)}$

The Vogan diagram of this simple Lie algebra is obtained by painting the
simple root $\alpha_{2}$ in Fig. 5. The positive roots system
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ is of type $A_{7}$. Indeed, it has
simple roots

$\gamma_{1}:=\alpha_{1};\quad\gamma_{i}:=\alpha_{i+1},\,2\leq i\leq
6;\quad\gamma_{7}:=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6}.$

Note that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=72$ and
$\beta=[0,0,0,1,0,0,0]$.

We calculate that $\max\\{A_{j}\mid 0\leq j\leq 71\\}=78$. Moreover,
$B=\frac{231}{2}$, which is attained at the trivial $\mathfrak{k}$-type. The
$E_{7(7)}$ case in Theorem 1.2 follows.

###  5.8. ${\rm EVI}=E_{7(-5)}$

The Vogan diagram of this simple Lie algebra is obtained by painting the
simple root $\alpha_{1}$ in Fig. 5. The positive roots system
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ is of type $D_{6}\times A_{1}$.
Indeed, it has simple roots

$\gamma_{i}:=\alpha_{8-i},\,1\leq i\leq
4;\,\gamma_{5}=\alpha_{2};\,\gamma_{6}=\alpha_{3};\,\gamma_{7}:=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}.$

Note that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=63$ and
$\beta=[0,0,0,0,0,1,1]$.

We compute that $\max\\{A_{j}\mid 0\leq j\leq 62\\}=82$. Moreover,
$B=\frac{227}{2}$, which is attained at the $\mathfrak{k}$-type
$\mu=[0,0,0,0,0,0,4]$. Indeed, ${\rm MP}(\mu)=\|\mu+\beta\|_{\rm
spin}=\frac{231}{2}$, while $\|\mu\|_{\rm lambda}=2$. The $E_{7(-5)}$ case in
Theorem 1.2 follows.

###  5.9. ${\rm EVIII}=E_{8(8)}$

The Vogan diagram of this simple Lie algebra is obtained by painting the
simple root $\alpha_{1}$ in Fig. 6. The positive roots system
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ is of type $D_{8}$. Indeed, it has
simple roots

$\gamma_{1}:=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7};\,\gamma_{i}:=\alpha_{10-i},\,2\leq
i\leq 6;\,\gamma_{7}=\alpha_{2};\,\gamma_{8}=\alpha_{3}.$

Note that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=135$ and
$\beta=[0,0,0,0,0,0,1,0]$.

We compute that $\max\\{A_{j}\mid 0\leq j\leq 134\\}=\frac{411}{2}$. Moreover,
$B=380$, which is attained at the trivial $\mathfrak{k}$-type. The $E_{8(8)}$
case in Theorem 1.2 follows.

###  5.10. ${\rm EIX}=E_{8(-24)}$

The Vogan diagram of this simple Lie algebra is obtained by painting the
simple root $\alpha_{8}$ in Fig. 6. The positive roots system
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f})$ is of type $E_{7}\times A_{1}$.
Indeed, it has simple roots

$\gamma_{i}:=\alpha_{i},\,1\leq i\leq
7;\quad\gamma_{8}:=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}+2\alpha_{8}.$

Note that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=120$ and that
$\beta=[0,0,0,0,0,0,1,1]$.

We compute that $\max\\{A_{j}\mid 0\leq j\leq 119\\}=212$. Moreover,
$B=\frac{723}{2}$, which is attained at the $\mathfrak{k}$-type
$[0,0,0,0,0,0,0,8]$. The $E_{8(-24)}$ case in Theorem 1.2 follows.

##  6\. Hermitian symmetric case

In this section, we consider the case that $(G,K)$ is a Hermitian symmetric
pair. Then the $K$-types of an infinite-dimensional $(\mathfrak{g},K)$ module
$\pi$ may _not_ be a union of Vogan pencils. Therefore, if $\pi$ is a
$(\mathfrak{g},K)$ module with a lowest $K$-type $E_{\mu}$ and with
infinitesimal character (1), we can only conclude that

(22) $\|\nu\|^{2}\leq\|\mu\|_{\rm spin}^{2}-\|\mu\|_{\rm lambda}^{2}.$

Thus our job is to get the maximum of the RHS of (22) over all the $K$-types.
Our main result of this section is summarized as follows.

######  Theorem 6.1.

Let $G$ be the Lie group _E6_h_ or _E7_h_ in _atlas_ [1, 29]. Let $\pi$ be any
irreducible unitary $(\mathfrak{g},K)$ module whose infinitesimal character is
given by (1). Then we have the following bounds for $\|\nu\|$:

  * $\bullet$

_E6_h :_ $\|\nu\|\leq\sqrt{73}$, while $\|\rho(G)\|=\sqrt{78}$.

  * $\bullet$

_E7_h :_ $\|\nu\|\leq\sqrt{\frac{371}{2}}$, while
$\|\rho(G)\|=\sqrt{\frac{399}{2}}$.

In each case, the bound is attained at the trivial representation.

Let $\mathfrak{t}_{f}$ be the fundamental Cartan subalgebra of $\mathfrak{g}$.
We may and we will identify $\mathfrak{t}_{f}$ with $\mathfrak{t}_{f}^{*}$ via
the Killing form $B(\cdot,\cdot)$. Let $\mathbb{R}\zeta$ be the one-
dimensional center of $\mathfrak{k}_{f}$, and let $\mathfrak{t}_{f}^{-}$ be
the orthogonal complement of $\mathbb{R}\zeta$ in $\mathfrak{t}_{f}$ under
$B(\cdot,\cdot)$.

Let $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=s$, and enumerate its elements as
$w^{(j)}$ for $0\leq j\leq s-1$. We adopt the notation $\Omega(j)$,
$\Omega_{\rm us}(j)$, $\partial\Omega_{\rm us}(j)$, $A_{j}$ etc as in Section
3, with the discrepancy that instead of $\mathfrak{k}$-types, now we use
$K$-types. Put

(23) $B=\max_{\mu}\left\\{\|\mu\|^{2}_{\rm spin}-\|\mu\|^{2}_{\rm
lambda}\right\\},$

where $\mu$ runs over all the $K$-types in $\Omega_{\rm
us}(j)\setminus\partial\Omega_{\rm us}(j)$. Then $\|\nu\|^{2}$ is upper
bounded by $\max\\{A_{0},A_{1},\dots,A_{s-1},B\\}$.

###  6.1. E6_h

This non-compact simple linear Lie group in atlas has center
$\mathbb{Z}/3\mathbb{Z}$ and Lie algebra ${\rm EIII}=E_{6(-14)}$. Its Vogan
diagram is obtained from Fig. 4 by painting the simple root $\alpha_{6}$. Let
$\zeta_{1},\dots,\zeta_{6}:=\zeta$ be the fundamental weights corresponding to
the simple roots $\alpha_{1},\dots,\alpha_{6}$ of
$\Delta^{+}(\mathfrak{g},\mathfrak{t}_{f})$. Put $\gamma_{i}=\alpha_{i}$ for
$1\leq i\leq 5$. Then $\gamma_{1},\dots,\gamma_{5}$ are the simple roots of
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f}^{-})$, which is of type $A_{5}$. Let
$\varpi_{1},\dots,\varpi_{5}\in(\mathfrak{t}_{f}^{-})^{*}$ be the
corresponding fundamental weights. For $a,b,c,d,e\in\mathbb{Z}_{\geq 0}$ and
$f\in\mathbb{Z}$, let $[a,b,c,d,e,f]$ stand for the vector
$a\varpi_{1}+b\varpi_{2}+c\varpi_{3}+d\varpi_{4}+e\varpi_{5}+\frac{f}{4}\zeta$.
Then ${[a,b,c,d,e,f]}$ is the highest weight of a $K$-type if and only if

(24)
$-\frac{3a}{4}-\frac{5b}{4}-\frac{3c}{2}-d-\frac{e}{2}+\frac{f}{4}\in\mathbb{Z}.$

We have that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=27$. We compute that
$\max\\{A_{j}\mid 0\leq j\leq 26\\}=33$. Moreover, $B=73$, which is attained
at the trivial $K$-type. The E6_h case in Theorem 6.1 follows.

######  Example 6.2.

Take the index $j$ so that $\rho_{n}^{(j)}=[1,0,1,0,1,3]$ and consider the
$K$-type $\mu=[a,b,c,d,e,f]$. Then $\mu\in\Omega(j)$ if and only if

$\displaystyle b+2e\leq a+2c+f,a+f\leq b+2c+2e+8,b+2c\leq a+2e+f,2e+f\leq
3a+b+2c+8.$

On the other hand,

$\displaystyle w^{(j)}\zeta_{1}=[0,1,0,0,0,1],\quad
w^{(j)}\zeta_{2}=[0,0,0,1,0,0],\quad w^{(j)}\zeta_{3}=[0,0,1,0,0,2],$
$\displaystyle w^{(j)}\zeta_{4}=[0,0,1,0,1,0],\quad
w^{(j)}\zeta_{5}=[1,0,0,0,1,1],\quad w^{(j)}\zeta_{6}=[1,0,0,0,0,-1].$

We compute that the non-decreasable $K$-types in $\Omega(j)$ are

$\displaystyle[0,0,0,0,0,0],[0,0,0,0,0,4],[0,0,0,0,0,8],[0,0,0,0,1,2],[0,0,0,0,1,6],[0,0,0,0,2,4].$

They are all u-small. ∎

###  6.2. E7_h

This non-compact simple linear Lie group in atlas has center
$\mathbb{Z}/2\mathbb{Z}$ and Lie algebra ${\rm EVII}=E_{7(-25)}$. Its Vogan
diagram is obtained from Fig. 5 by painting the simple root $\alpha_{7}$. Let
$\zeta_{1},\dots,\zeta_{7}:=\zeta$ be the fundamental weights corresponding to
the simple roots $\alpha_{1},\dots,\alpha_{7}$ of
$\Delta^{+}(\mathfrak{g},\mathfrak{t}_{f})$. Put $\gamma_{i}=\alpha_{i}$ for
$1\leq i\leq 6$. Then $\gamma_{1},\dots,\gamma_{6}$ are the simple roots of
$\Delta^{+}(\mathfrak{k},\mathfrak{t}_{f}^{-})$, which is of type $E_{6}$. Let
$\varpi_{1},\dots,\varpi_{6}\in(\mathfrak{t}_{f}^{-})^{*}$ be the
corresponding fundamental weights. For $a,b,c,d,e,f\in\mathbb{Z}_{\geq 0}$ and
$g\in\mathbb{Z}$, let $[a,b,c,d,e,f,g]$ stand for the vector
$a\varpi_{1}+b\varpi_{2}+c\varpi_{3}+d\varpi_{4}+e\varpi_{5}+f\varpi_{6}+\frac{g}{3}\zeta$.
Then $[a,b,c,d,e,f,g]$ is the highest weight of a $K$-type if and only if

(25)
$-\frac{2a}{3}-b-\frac{4c}{3}-2d-\frac{5e}{3}-\frac{4f}{3}+\frac{g}{3}\in\mathbb{Z}.$

We have that $|W(\mathfrak{g},\mathfrak{t}_{f})^{1}|=56$. We compute that
$\max\\{A_{j}\mid 0\leq j\leq 55\\}=79$. Moreover, $B=\frac{371}{2}$, which is
attained at the trivial $K$-type. The E7_h case in Theorem 6.1 follows.

Funding

Dong is supported by the National Natural Science Foundation of China (grant
12171344, 2022-2025).

Acknowledgements

I thank Daniel Wong for helpful discussions. I thank an anonymous referee for
giving us expertise suggestions.

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# OPTIMIZED LEARNED ENTROPY CODING PARAMETERS FOR PRACTICAL NEURAL-BASED IMAGE
AND VIDEO COMPRESSION

###### Abstract

Neural-based image and video codecs are significantly more power-efficient
when weights and activations are quantized to low-precision integers. While
there are general-purpose techniques for reducing quantization effects, large
losses can occur when specific entropy coding properties are not considered.
This work analyzes how entropy coding is affected by parameter quantizations,
and provides a method to minimize losses. It is shown that, by using a certain
type of coding parameters to be learned, uniform quantization becomes
practically optimal, also simplifying the minimization of code memory
requirements. The mathematical properties of the new representation are
presented, and its effectiveness is demonstrated by coding experiments,
showing that good results can be obtained with precision as low as 4 bits per
network output, and practically no loss with 8 bits.

Index Terms—  Learned image and video compression, neural network
quantization, entropy coding.

##  1 Introduction

After years of research showing the advantages of neural-based techniques for
video coding [1, 2], and recent demonstrations of real-time decoding in mobile
devices [3, 4], there is growing interest in their standardization and
commercial deployment [5], and a main practical implementation challenge is
the minimization of decoding power requirements.

Implementing neural network (NN) computations with low-precision integers
significantly reduces power needs, but requires minimizing the effects of
quantizing networks weights and activations [6, 7, 8]. For neural image and
video codecs, efficacy is strongly defined by the entropy coding stage and
related networks, and can be severely degraded.

Analyzing quantization effects in NNs can be difficult because they implement
non-linear processes, and learned variables have unknown properties.
Techniques based on re-training under quantization constraints help to reduce
the losses, but have limited efficacy.

In this work it is shown that NN outputs used for entropy coding have well-
defined meaning and properties, which can be used for predicting quantization
effects. Furthermore, those properties can be used for modifying the loss
function used while learning, and it is demonstrated that networks can be
trained to be minimally sensitive to quantization.

Further analysis shows that this optimized representation is remarkably
independent of the quantization, and thus networks do not need to be retrained
even after significant changes to entropy coding design choices. Thus, in
those networks the uniform quantization can also be extended beyond the
hardware requirements, and used to minimize the memory needed for storing
entropy coding tables.

Experimental results show that the proposed solution can guarantee good
performance ($\approx$ 2% redundancy) with precision as low as 4 bits, and
with relative redundancy reduction by a factor of 4 for each additional bit,
reaching less than 0.01% for 8 bits.

##  2 Entropy coding for NN-based codecs

Entropy coding in conventional codecs is based on estimating the probabilities
of variables to be coded. For example, in video standards like HEVC and VVC
[9, 10, 11], adaptive binary arithmetic coding contexts are used for
estimating probabilities related to data elements.

Neural-based codecs use a different principle: a parameterized probability
distribution (pdf) is chosen for training, and a learned scheme (table or NN)
is used to determine the parameter value to be used for coding a quantized
random variable. Fig. 1 shows an example of the commonly used normal
distribution, and unit-step quantization.

Fig. 1: Quantized values from normal distribution with std. dev. $\sigma$,
commonly used in neural-based codecs. Fig. 2: Example of a neural-based codec
architecture, including scale hyperprior networks for entropy coding.

The sequence of probabilities of random variable $y\sim{\cal
N}(0,\sigma^{2})$, quantized to value $n\in\mathbb{Z}$, is

$p_{n}(\sigma)=\frac{1}{2}\left[{\rm
erfc}\\!\left(\frac{2n-1}{\sqrt{8}{\sigma}}\right)-{\rm
erfc}\\!\left(\frac{2n+1}{\sqrt{8}{\sigma}}\right)\right],$ (1)

with entropy

$H(\sigma)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}-\sum_{n=-\infty}^{\infty}p_{n}(\sigma)\log_{2}\\!\left(p_{n}(\sigma)\right),$
(2)

where the complementary error function is

${\rm erfc}\\!\left({x}\right)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}\>\mathrm{d}{t}.$
(3)

In practical implementations special symbols are used to code values in
underflow and overflow infinite sets, which have very small but nonzero
probabilities, as shown in Fig. 1. Within those sets the values are sub-
optimally coded with, for example, Elias universal codes [12].

Also shown in Fig. 1 are the elements of code vector ${\bf c}(\sigma)$ (CV),
which contain the probabilities to be used for arithmetic coding (AC), within
a given range $R$

$c_{i}(\sigma)=\begin{cases}\sum_{n=-\infty}^{-R}p_{n}(\sigma),&i=0,\\\
p_{i-R}(\sigma),&i=1,2,\ldots,2R-1,\\\
\sum_{n=R}^{\infty}p_{n}(\sigma),&i=2R.\end{cases}$ (4)

To simplify the notation we ignore implementation details like conversions to
integer-valued cumulative sums for AC [13, 14]. It is assumed that $R$ and AC
precision are chosen to obtain compression very close to entropy.

This type of entropy coding is used within a NN-based codec as shown in Fig.
2, corresponding to the scale hyperprior architecture [15]. There are other
codec configurations, but our main interest is in the per-element coding
stages shown in Fig. 3, which is used with all factorized priors.

The latent $y$ to be coded is quantized with $Q_{y\rightarrow q}$, which is
taken into account during floating-point training. The problems addressed here
are caused by the quantization stage $Q_{\sigma\rightarrow\hat{\sigma}}$,
representing the use of low-precision network outputs during coding, but not
during training.

Since code vectors must be bitwise identical at the encoder and decoder [16],
given a range ${\cal U}=[\sigma_{\min},\sigma_{\max}]$, we assume they share a
monotonic sequence $\\{t_{k}\\}_{k=0}^{N}$ partitioning ${\cal U}$, and that
latent $y$ is coded using ${\bf c}(\rho^{*}_{k})$ whenever
$\hat{\sigma}\in[t_{k},t_{k+1})$.

To determine the optimal values of $\rho^{*}_{k}$ for code vector computation
we measure the average redundancy, defined by the Kullback-Leibler divergence

$R(\sigma,\rho)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\sum_{n=-\infty}^{\infty}p_{n}(\sigma)\log_{2}\\!\left(\frac{p_{n}(\sigma)}{p_{n}(\rho)}\right).$
(5)

Fig. 3: Per-element entropy coding, including quantization of pdf parameter
$\sigma$. Fig. 4: Example of nonuniform quantization of parameter $\sigma$,
designed for constant maximum relative redundancy.

Since practical image and video codecs are meant to be efficient in a wide
range of bit rates, it is interesting to have entropy coding designed to
minimize relative redundancy

$L(\sigma,\rho)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\frac{R(\sigma,\rho)}{H(\sigma)}.$ (6)

and to enable good performance for all use cases use

$\rho^{*}_{k}=\mathop{\rm
arg\,min}_{\rho}\int_{t_{k}}^{t_{k+1}}L(\sigma,\rho)\>\mathrm{d}{\sigma}.$ (7)

##  3 New pdf parameterization

From these definitions it is possible to optimize non-uniform quantization
schemes as shown in the example of Fig. 4, where similarly to other
quantization solutions, all intervals have the same maximum relative
redundancy and averages

$\bar{L}_{k}\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\frac{1}{t_{k+1}-t_{k}}\int_{t_{k}}^{t_{k+1}}L(\sigma,\rho^{*}_{k})\>\mathrm{d}{\sigma}.$
(8)

With sufficiently fine quantization of $\sigma$, $L(\sigma,\rho^{*}_{k})$ is
approximately quadratic, and we can use

$\displaystyle\rho^{*}_{k}$ $\displaystyle\approx$
$\displaystyle\left\\{\rho:L(t_{k},\rho)=L(t_{k+1},\rho)\right\\},$ (9)
$\displaystyle\bar{L}_{k}$ $\displaystyle\approx$
$\displaystyle\frac{L(t_{k},\rho^{*}_{k})}{3}=\frac{L(t_{k+1},\rho^{*}_{k})}{3}.$
(10)

Fig. 5: Proposed modification of codec in Fig. 3, replacing pdf parameter
$\sigma$ with parameter $u$ optimized for quantization.

Neural-based codecs using precise floating-point computations can be
implemented with arrays of pre-computed code vectors, and the approach shown
in the example of Fig. 4. However, for practical codecs, which must handle a
wide range of bit rates, it is necessary to support, for example,
$\sigma\in[0.1,1000]$ with the precision of hyperprior network output
$\hat{\sigma}$ constrained to only 8 bits. This means that
$\\{t_{k}\\}_{k=0}^{N}$ must also use only 8 bits, severely constraining the
design and resulting in significant losses.

Fig. 6: Mapping from non-uniform to uniform quantization.

To solve this problem we can exploit the fact that $\sigma$ is only used for
rate estimation during training, and while computing code vectors. The
proposed solution is shown in Fig. 5: hyperprior networks are trained to
compute a new pdf parameter $u\in[0,1]$ that is converted as $\sigma=T(u)$.
For analysis we define $u\in\mathbb{R}$, but note that during actual coding
the quantized networks only need to generate integer $k$. This approach is
used in [16], where the conversion is chosen to be in the form
$\sigma=e^{\alpha u+\beta}$.

Fig. 6 shows how optimal $T(u,N)$ can be defined, for a given $N$, to obtain
constant maximum relative redundancy, and the following algorithm can be used
for its computation.

  1. 1.

Given $N$ and $[\sigma_{\min},\sigma_{\max}]$, choose initial maximum
redundancy $\epsilon$; initialize $t_{0}=\sigma_{\min}$.

  2. 2.

For $n=1,2,\ldots,N$:

    1. (a)

Set $\rho^{*}_{n-1}=\\{\rho>t_{n-1}:L(t_{n-1},\rho)=\epsilon\\}$;

    2. (b)

Set $t_{n}=\\{t>\rho^{*}_{n-1}:L(t,\rho^{*}_{n-1})=\epsilon\\}$.

  3. 3.

If $t_{N}\approx\sigma_{max}$ then stop.

  4. 4.

If $t_{N}<\sigma_{max}$ then increase $\epsilon$, otherwise decrease
$\epsilon$, using a method for unidimensional search.

  5. 5.

Go to step 2.

The surprising conclusion from computing $T(u,N)$ for several values of $N$,
and interpolating between the sampled values, is that all results are
practically identical. This means that if we compute

$T(u)=\lim_{N\rightarrow\infty}T(u,N),$ (11)

we obtain a single reference function that can be used for all network
trainings, and the value of $N$ can be chosen and modified later, without the
need for re-training.

The rationale for determining $T(u)$ can be seen from the lower part of Fig.
6. For increasing values of $N$ the redundancy curves in all intervals should
approximate quadratic equations on $u$, with same second derivatives. This can
be achieved if we can find $T(u)$ and constant $\alpha$ such that

$\left.{\frac{\partial^{2}}{{\partial{u}}^{2}}\,{L(T(u),\rho)}}\right|_{\rho=T(u)}=\alpha,\quad
u\in[0,1].$ (12)

Defining

$\psi(\sigma)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\left.{\frac{\partial^{2}}{{\partial{\sigma}}^{2}}\,{L(\sigma,\rho)}}\right|_{\rho=\sigma}$
(13)

it can be shown that, thanks to properties of $L(\sigma,\rho)$, eq. (12) is
equivalent to

$\frac{\mathrm{d}}{\mathrm{d}{u}}\,{T(u)}=\sqrt{\frac{\alpha}{\psi(T(u))}},$
(14)

and thus function $T(u)$ can be determined by solving this ordinary
differential equation (ODE) using boundary conditions
$T(0)=\sigma_{\min},\;T(1)=\sigma_{\max}$.

For any sequence of differentiable parameterized probabilities it can be shown
that

$\psi(\sigma)=\frac{1}{\ln(2)H(\sigma)}\sum_{n=-\infty}^{\infty}\frac{1}{p_{n}(\sigma)}\;\left[\frac{\mathrm{d}{p_{n}(\sigma)}}{\mathrm{d}{\sigma}}\right]^{2},$
(15)

and for normal pdf we have

$\displaystyle\psi(\sigma)$ $\displaystyle=$
$\displaystyle\frac{1}{\ln(2)H(\sigma)}\left\\{\frac{\left[\phi_{1}(\sigma)\right]^{2}}{1-\omega_{1}(\sigma)}+\right.$
$\displaystyle\left.\sum_{n=1}^{\infty}\frac{\left[\phi_{2n-1}(\sigma)-\phi_{2n+1}(\sigma)\right]^{2}}{\omega_{2n-1}(\sigma)-\omega_{2n+1}(\sigma)}\right\\},$

where

$\omega_{k}(\sigma)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}{\rm
erfc}\\!\left(\frac{k}{\sqrt{8}{\sigma}}\right),\quad\phi_{k}(\sigma)\stackrel{{\scriptstyle\text{\tiny
def}}}{{=}}\frac{k\,e^{-k^{2}/(8\sigma^{2})}}{\sqrt{2\pi}\sigma^{2}}.$ (17)

##  4 Experimental results

Fig. 7: The optimal pdf parameter conversion function $T(u)$ and two
approximations. Fig. 8: Derivatives of pdf parameter conversion function
$T(u)$ and its approximations.

The ODE of eq. (14) was solved using the 4th-order Runge-Kutta method [17,
§17.1], and boundary conditions $\sigma\in[0.1,1000]$. The solution $T(u)$ is
shown in Fig. 7. A quick observation may lead to the incorrect conclusion that
it can be approximated by a function like $10^{4u^{2}-1}$ (red line).

Since $T(u)$ is defined by a differential equation, the quality of the
approximation must be measured by relative errors of derivatives. Those are
shown in Fig. 8, where we can also see a much better approximation
$T_{\pi}(u)=10^{\pi(u)}$, where polynomial

$\pi(u)=2.49284\,u^{3}+0.93703\,u^{2}+0.57013\,u-1,$ (18)

minimizes the maximum relative derivative error.

Fig. 9 shows how the average relative redundancy $\bar{L}(\sigma)$ varies with
$\sigma$ when uniform quantization is applied to pdf parameter $u$, and $N$
code vectors are used for entropy coding latent variables $y$, as shown in
Fig. 5. Following the condition defined by eq. (13), $\bar{L}(\sigma)$ does
not change when $T(u)$ is used for pdf parameter conversion. On the other
hand, there are relatively small variations when approximation $T_{\pi}(u)$ is
used instead, and those deviations correspond to errors in approximating the
derivatives (cf. Fig. 8).

We also observe that the curves are nearly perfectly parallel, confirming the
predominance of the quadratic term in the Taylor series of the relative
redundancy. This is also verified by the fact that $\bar{L}(\sigma)$
quadruples when the $N$ is halved.

Non-uniform quantizations defined by transformation $T(u)$ were used for
entropy coding values from a floating-point implementation of the scale
hyperprior codec [15], in the Kodak image dataset [18]. Redundancy results are
shown in Table 1, for average bit rates between 0.25 and 2.5 bits/pixel.

For values of $N$ larger than about 128 (7 bit representation) the redundancy
from quantization is 0.01% or below, which is within the experimental
variation. For smaller values of $N$ we can observe that

  * •

The redundancies are practically constant for all bit rates, meaning that the
proposed coding method achieves the design objective of having constant
relative redundancies.

  * •

The average redundancy levels match those of Fig. 9, showing that features
predicted by theory match the practical implementation.

Fig. 9: Average relative redundancies from $T(u)$ (dashed) and approximation $10^{\pi(u)}$ (solid), for different number of quantization intervals $N$. Table 1: Coding results obtained using scale hyperprior codec applied to 24 images of Kodak dataset. $N$ | CV mem. | Rel. redundancy (%) @ bit rate
---|---|---
| (Kbytes) | 0.25 | 0.50 | 1.00 | 1.50 | 2.50
256 | 51.6 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01
192 | 38.7 | 0.00 | 0.00 | 0.00 | 0.01 | 0.01
128 | 25.8 | 0.04 | 0.04 | 0.04 | 0.03 | 0.03
96 | 19.4 | 0.03 | 0.04 | 0.04 | 0.04 | 0.05
64 | 12.9 | 0.13 | 0.12 | 0.12 | 0.12 | 0.11
48 | 9.7 | 0.24 | 0.23 | 0.22 | 0.22 | 0.20
32 | 6.5 | 0.38 | 0.38 | 0.40 | 0.41 | 0.42
24 | 4.9 | 0.85 | 0.81 | 0.81 | 0.82 | 0.79
16 | 3.4 | 1.79 | 1.76 | 1.76 | 1.78 | 1.75

The second column of Table 1 contains the total amount of memory needed to
store the code vectors, when using 16 bits/element. This shows that the
proposed method can be directly used to minimize memory. For example, even if
the network is able to support 8 bit outputs, and redundancy below 0.5% is
acceptable, the pdf parameter can be further quantized to 5 bits, by simply
discarding the 3 least significant bits, to reduce CV memory from 51.6 to 6.5
Kbytes.

##  5 Conclusions

It is shown that the effects of quantization on the entropy coding can be
analyzed by evaluating coding redundancy, and this can be used for changing
the pdf parameter to be learned so that average relative redundancy becomes
practically constant. In consequence, networks do not need to be re-trained
when the pdf parameter quantization is changed, which we show can also be done
intentionally to minimize memory for code tables.

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