# On the sharpness of Strichartz estimates and spectrum of compact Lie groups

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS>,  Brian Grajales  Brian Grajales: Department
of Mathematics Universidad de Pamplona, Colombia E-mail address
<EMAIL_ADDRESS>and  Michael Ruzhansky  Michael Ruzhansky:
Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent
University, Belgium and School of Mathematical Sciences Queen Mary University
of London United Kingdom E-mail address<EMAIL_ADDRESS>
###### Abstract.

We prove Strichartz estimates on any compact connected simple Lie group. In
the diagonal case of Bourgain's exponents $p=q,$ we substantially improve the
regularity orders showing the existence of indices $s<s_{0}(d)$ below the
Sobolev exponent $s_{0}(d)=\frac{d}{2}-\frac{d+2}{p}.$ Motivated by the recent
progress in the field, in the spirit of the analytical number theory methods
developed by Bourgain in the analysis of periodic Schrödinger equations, we
link the problem of finding Strichartz estimates on compact Lie groups with
the problem of counting the number of representations $r_{s,2}(R)$ of an
integer number $R$ as a sum of $s$ squares, and then, we implicitly use the
very well known bounds for $r_{s,2}(R)$ from the Hardy-Littlewood-Ramanujan
circle method. In our analysis, we explicitly compute the parametrisation of
the spectrum of the Laplacian (modulo a factor depending on the geometry of
the group) in terms of sums of squares. As a byproduct, our approach also
yields explicit expressions for the spectrum of the Laplacian on all compact
connected simple Lie groups, providing also a number of sharp results for
Strichartz estimates in the borderline case $p=2.$

###### Key words and phrases:

Strichartz estimate, compact Lie group, Waring problem, Vinogradov's
conjecture, analytic number theory

###### 2020 Mathematics Subject Classification:

35S30, 42B20; Secondary 42B37, 42B35

The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis
and Partial Differential Equations, by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021). B. Grajales
has been partially supported by Universidad de Pamplona. M. Ruzhansky is also
supported by EPSRC grant EP/R003025/2.

###### Contents

  1. 1 Introduction
    1. 1.1 Outline
    2. 1.2 Main result
  2. 2 Tools from Lie theory and from analytic number theory
    1. 2.1 Spectrum of the Laplacian on compact simple Lie groups
    2. 2.2 Tools from analytic number theory
  3. 3 Strichartz estimates
    1. 3.1 Data statement

##  1\. Introduction

###  1.1. Outline

Through J. Bourgain's work about dispersive equations on rational and
irrational torus, the role of the analytic number theory and in particular,
the analysis of the Waring problem via the recent complete solution in [6, 7]
of Vinogradov's mean value conjecture, has shown to be relevant for several
problems in harmonic analysis and dispersive PDE. In this work we follow the
approach of Bourgain in order to investigate the extension of Strichartz
estimates on an arbitrary simple compact Lie group $G$ (endowed with a bi-
invariant Riemannian metric $g$) connecting this problem to the one of
computing the number of representations $r_{s,2}(R)$ of an integer $R,$ as
sums of $s\geq 2$ squares.

The validity of Strichartz estimates for the Schr̈odinger equation for the
positive Laplacian $\Delta_{g}$ on a compact Riemannian manifold $(M,g)$ is
very well known from the work of Burq, Gérard and Tzvetkov [8]. Their estimate
is as follows.

Let $d\geq 1.$ The solution to the linear Schrodinger equation

$iu_{t}+\Delta_{g}u=0,\,\,u(0,x)=u_{0}(x),\,\,(t,x)\in[0,1]\times M,$ (1.1)

satisfies the estimate

$\|u(t,x)\|_{L^{p}([0,1],L^{q}(M))}\leq C\|u_{0}\|_{H^{\frac{1}{p}}(M)},\,\,$
(1.2)

where $(p,q)$ are Lebesgue indices satisfying the admissibility condition
$2/p+d/q=d/2.$ The optimality of (1.2) in the end-point case $p=2$ was proved
in [8] for spheres $\mathbb{S}^{d}.$ Moreover, they have proved the following
general statement when adapting Bourgain's argument to this generality level.

Let $d\geq 2.$ Assume that $(M,g)$ is a compact Riemannian manifold whose
geodesics are closed and with a common period. Then the estimate (1.2) can be
improved as

$\|u(t,x)\|_{L^{4}([0,1]\times M)}\leq
C\|u_{0}\|_{H^{s}(M)},\,\,s>s_{0}(d),\,\,$ (1.3)

where

$s_{0}(2)=1/8,\,s_{0}(d)=\frac{d}{4}-\frac{1}{2},\,\forall d\geq 3.$ (1.4)

Moreover, if $M=\mathbb{S}^{d}$ is endowed with the canonical metric, the
Sobolev regularity order $s>s_{0}(d)$ in (1.3) is sharp.

We note that the the regularity order in (1.4) agrees with the one obtained by
Bourgain in the case of the torus only when $d\geq 3.$ Indeed, on the flat
torus $\mathbb{T}^{d},$ and by computing explicitly

$\|u\|_{L^{4}(\mathbb{T}\times\mathbb{T}^{d})}=\|u^{2}\|_{L^{2}(\mathbb{T}\times\mathbb{T}^{d})}.$

Bourgain showed that:

if $d=2,$ for every $\varepsilon>0,$
$\|u\|_{L^{4}(\mathbb{T}\times\mathbb{T}^{d})}\leq
C_{\varepsilon}\|u_{0}\|_{H^{\varepsilon}(\mathbb{T}^{d})},$ and when $d\geq
3,$ the following estimate

$\|u(t,x)\|_{L^{4}([0,1]\times M)}\leq
C\|u_{0}\|_{H^{s}(M)},\,\,s>s_{0}(d),\,\,$ (1.5)

holds for $s_{0}(d)={d}/{4}-{1}/{2}.$

Due to the general perspective of the results in (1.2) or in (1.3) the
sharpness of these results is an interesting problem. The best-studied
situation is when $M=\mathbb{R}^{d}/\Gamma$ is a torus for some rank $d$
lattice. Improvements of (1.5) for general indices of $p$ is a significant
problem due to its connections with weak turbulence for non-linear Schrödinger
equations, see e.g. [10, Page 703].

Recently, in a series of works, Zhang in [33, 34, 35, 36] has obtained a
number of improvements in the setting of simple compact Lie groups for
Strichartz estimates of the Schrödinger equation by obtaining analogues of
(1.3) and extensions of it for all $1<p<\infty.$ Zhang's result can be stated
as follows, see [36, Theorem 1] in the compact Lie group setting.

Let $(G,g)$ be a compact Lie group of rank $\ell\geq 2$ equipped with the
canonical Killing metric $g$ and let $I\subset\mathbb{R}^{+}_{0}$ be a closed
interval. Then the solution to (1.1) satisfies the estimate

$\|u(t,x)\|_{L^{p}(I\times G)}\lesssim\|u\|_{H^{s}(G)},$ (1.6)

provided that $p\geq 2+\frac{8}{\ell},$ for all $s$ such that

$s\geq\frac{d}{2}-\frac{d+2}{p}.$ (1.7)

The aim of this work is to analyse when on a compact Lie group the validity of
Strichartz estimates holds for the Sobolev indices $s$ in the interval
$\\{s:s_{G}(d)<s<s_{0}(d)\\},$ for some threshold value of regularity
$s_{G}(d)>0,$ where $d$ is the dimension of the group and
$s_{0}(d)=\frac{d}{2}-\frac{d+2}{p}$. Moreover, we consider only the case
where $G$ is a simple and simply connected Lie group, because we make use of
explicit representations of the spectrum of the Laplace operator, which modulo
a factor, can be written in terms of a sums of squares. Indeed, it is a very
well-known fact that for a fixed set of simple positive roots $\Sigma$ with
corresponding fundamental weights
$\overline{\omega}_{1},...,\overline{\omega}_{n},$ and each highest weight

$\small{{\tilde{\nu}}=\sum\limits_{j=1}^{l}{\tilde{\nu}}_{j}\overline{\omega}_{j}},\,{\tilde{\nu}}_{j}\in\mathbb{N}_{0},$

on a compact Lie group $G,$ one can associate an eigenvalue
$\lambda_{{\tilde{\nu}}}$ for the positive Laplacian $\mathcal{L}_{G}$ given
by

$\lambda_{{\tilde{\nu}}}=(\eta,\eta)-({\tilde{\nu}}+\eta,{\tilde{\nu}}+\eta),$
(1.8)

where $\eta:=\frac{1}{2}\sum\limits_{\alpha\in\Pi^{+}}\alpha,$ and the sum
runs over the system of positive roots $\Pi^{+}.$ Conversely, every eigenvalue
of $\mathcal{L}_{G}$ can be written in the form of (2.4). Due to the
importance that it has for us the explicit construction of the spectrum of the
Laplace operator in terms of the root systems, we compute it on every simple
and simply connected $G.$ Since these groups are classified by the root system
associated with their Cartan subalgebras, in terms of their Dynkin diagrams,
we split our analysis into the following cases:

  * •

The Lie algebra $\mathfrak{g}$ of $G$ is $A_{\ell},$ in which case
$G\cong\textnormal{SU}(\ell+1).$

  * •

The Lie algebra $\mathfrak{g}$ of $G$ is $B_{\ell},$ in which case
$G\cong\textnormal{Spin}(2\ell+1).$

  * •

The Lie algebra $\mathfrak{g}$ of $G$ is $C_{\ell},$ in which case
$G\cong\textnormal{Sp}(\ell).$

  * •

The Lie algebra $\mathfrak{g}$ of $G$ is $D_{\ell},$ in which case
$G\cong\textnormal{Spin}(2\ell).$

  * •

The Lie algebra $\mathfrak{g}$ of $G$ is one of the exceptional Lie algebras
$\mathfrak{g}_{2},\mathfrak{f}_{4}$ or $\mathfrak{e}_{8},$ in which case
$G\cong G_{2},$ $G\cong F_{4},$ or $G\cong E_{8},$ respectively.

For our further analysis, we exclude the case $\ell=1$ since
$\textnormal{Spin(3)}\cong\textnormal{SU}(2)\cong\mathbb{S}^{3}$ and the
Strichartz estimates in the case of the spheres have been analysed in the work
of Burq, Gerard, and Tzvetkov [8]. Having explained this classification, we
are ready to present our result.

###  1.2. Main result

In the case where $p=q,$ our results of this paper can be summarised in
Theorem 1.1 below (see Theorem 3.4 for different values of $p$ and $q,$ where
we allow $p\neq q$). In view of Remark 2.4, we only consider the validity of
the Strichartz estimates in the case where $G$ is a connected simple and
simply connected compact Lie group. However, our analysis also covers the case
where $G$ is not simply connected. Indeed, if $G$ is a connected (not
necessarily simply-connected) compact simple Lie group, we have that

$G\cong\tilde{G}/\Gamma,$

where $\tilde{G}$ is a connected simply-connected compact simple Lie group
(diffeomorphic to the universal cover of $G$) and $\Gamma$ is a discrete
subgroup of $\tilde{G}.$ This property allows us to conclude the inclusion

$\textnormal{Spec}(G)\subseteq\textnormal{Spec}(\tilde{G}).$ (1.9)

As in our approach we apply an explicit construction of the spectrum of the
Laplacian we have proved the following theorem.

######  Theorem 1.1.

Let $2\leq p<\infty.$ Let $(G,g)$ be a compact connected simple Lie group of
rank $\ell\geq 2$ endowed with a Riemmanian bi-invariant metric $g$ and let
$\tilde{G}$ be its universal cover. Let

$m=\left\\{\begin{array}[]{lll}\ell+1,&&\textnormal{if}\
\tilde{G}\cong\textnormal{SU($\ell+1$)}\ \textnormal{or}\ \tilde{G}\cong
G_{2},\\\ \ell,&&\textnormal{otherwise}.\end{array}\right.$ (1.10)

Let $T=2\pi b\gamma,$ where $b$ is a non-negative integer depending on $G$ and
$\gamma$ is a geometric constant defined by the metric $g.$ Then

$\left|\left|u\right|\right|_{L^{p}([0,2\pi b\gamma]\times
G)}\lesssim_{s}\|u_{0}\|_{H^{s}(G)},$ (1.11)

where $s\geq\frac{m}{4}-\frac{1}{p}+s(p)$ if $m\geq 5,$
$s>\frac{m}{4}-\frac{1}{p}+s(p)$ if $m=2,3,4$ and

$s(p):=\left\\{\begin{array}[]{lll}\frac{d-1}{4}\left(\frac{1}{2}-\frac{1}{p}\right),&&\textnormal{if}\
2\leq p\leq\frac{2(d+1)}{d-1},\\\ \\\
\frac{d-1}{4}-\frac{d}{2p},&&\textnormal{if}\
\frac{2(d+1)}{d-1}<p\leq\frac{4d}{d-\ell},\\\ \\\
\frac{d-\ell}{4}-\frac{d}{2p},&&\textnormal{if}\
\frac{4d}{d-\ell}<p\leq\infty.\end{array}\right.$ (1.12)

Now, we briefly discuss this result and its relation to the existent results
on compact manifolds.

######  Remark 1.2.

Note that when $\tilde{G}\cong\textnormal{SU}(\ell+1)$ or if
$\tilde{G}\cong\textnormal{G}_{2},$ the root system of the Lie algebra is
constructed in an Euclidean space of dimension $\ell+1.$ Also, if $\tilde{G}$
is not diffeomorphic to $\textnormal{SU}(\ell+1)$ or to $\textnormal{G}_{2},$
its root system is constructed in an Euclidean $\ell$-dimensional space. These
facts justify the two values $m=\ell,$ or $m=\ell+1,$ of $m$ defined in
(1.10).

######  Remark 1.3.

On $\mathbb{R}^{n},$ the sharp Strichartz estimate

$\|u(t,x)\|_{L^{p}_{t}L^{q}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\leq
C\|u_{0}\|_{L^{2}(\mathbb{R}^{d})},$ (1.13)

is valid for all the admissible pairs $(p,q),$ i.e. $p$ and $q$ satisfying
$2/p+d/q=d/2,$ see Ginibre and Velo [11] and Keel and Tao [22]. Note that by
the Sobolev embedding theorem one has that the estimate

$\|u(t,x)\|_{L^{p}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\leq
C\|u_{0}\|_{H^{s}(\mathbb{R}^{d})},$ (1.14)

is valid with

$s=\frac{d}{2}-\frac{2}{p}-\frac{d}{r}\geq 0,$ (1.15)

where $p,q\geq 2,$ and $(p,r,d)\neq(2,\infty,2).$ The triples $(p,q,r)$
satisfying (1.14) and (1.15) are usually called scale-invariant. However, on
compact manifolds the Strichartz estimates as in (1.13) are expected to fail,
while in the setting of compact manifolds the validity of an estimate of the
type (1.14) is expected to hold only for positive values of $s>0.$ Moreover,
is also expected to have Strichartz estimates for $(p,q,r)$ being a non-
admissible triple and with $s>\frac{d}{2}-\frac{2}{p}-\frac{d}{r},$ see e.g.
the discussion in Zhang [34].

######  Remark 1.4.

Recently Zhang in [37] has improved the validity of Strichartz estimates (1.6)
where $s\geq\frac{d}{2}-\frac{d+2}{p},$ and $p\geq
2+\frac{8(s-1)}{s\ell}=2+\frac{4(d+\ell)}{d\ell}.$ So, the contributions of
our result for $p$ in the interval

$[2,2+\frac{4(d+\ell)}{d\ell}]$

are new. Now, let us follow the notation in Theorem 1.1. In view of the
improved Sogge index $s(p)$ (which is always a non-negative function of
$p\in[2,\infty]$, see Remark 3.2) for the norm of the spectral projections for
the Laplacian on a compact Lie group obtained by Zhang in [37, Theorem 1.1],
(see also [36, Theorem 1]), namely, that $s(p)=\frac{d-\ell}{4}-\frac{d}{2p}$
in (1.12) holds with $\frac{4d}{d-\ell}<p\leq\infty,$ we have that (1.11)
holds with $s>s_{p}$ if $m=2,3,4,$ or with $s\geq s_{p},$ for $m\geq 5,$ where

$s_{p}=\frac{m}{4}-\frac{1}{p}+s(p).$ (1.16)

Note that if $p>\frac{4d}{d-\ell},$ we have that

$s_{p}=\left\\{\begin{array}[]{lll}\frac{\ell+1}{4}-\frac{1}{p}+s(p)=\frac{1}{4}-\frac{1}{p}+\frac{d}{2}\left(\frac{1}{2}-\frac{1}{p}\right),&&\textnormal{if}\
\tilde{G}\cong\textnormal{SU($\ell+1$)}\ \textnormal{or}\ \tilde{G}\cong
G_{2},\\\
\frac{\ell}{4}-\frac{1}{p}+s(p)=-\frac{1}{p}+\frac{d}{2}\left(\frac{1}{2}-\frac{1}{p}\right),&&\textnormal{otherwise},\end{array}\right.$

which improves the Sobolev order in (1.7). We also note that for $p$ in the
intervals $[2+\frac{4(\ell+d)}{\ell d},\frac{2(d+1)}{d-1}],$
$[\frac{2(d+1)}{d-1},\frac{4d}{d-\ell}],$ our result in Theorem 1.1 and the
aforementioned results by Zhang are not comparable. In some regions, their
index (1.7) is better than the ours, and in other regions, our index $s_{p}$
is sharper than Zhang's one.

##  2\. Tools from Lie theory and from analytic number theory

###  2.1. Spectrum of the Laplacian on compact simple Lie groups

We will find the spectrum of the Laplacian operator on a Lie group via
representations of its Lie algebra and provide some explicit examples of the
eigenvalues in compact and simply-connected Lie groups. For this, we shall
follow [2, 3, 30, 38, 39].

Let $G$ be a $n$-dimensional connected and simply connected compact simple Lie
group whose corresponding Lie algebra $\mathfrak{g}$ is the real compact form
of a complex simple Lie algebra $\mathfrak{k}.$ Let us endow $\mathfrak{g}$
with the Ad-invariant inner product $g$ given by a multiple $\gamma>0$ of the
negative of the Killing form of $\mathfrak{g}$, that is, $g=-\gamma\kappa,\
\textnormal{where}\
\kappa(X,Y):=\textnormal{Tr(ad(}X\textnormal{)ad(}Y\textnormal{))}$. It is
known that $g$ induces a bi-invariant Riemannian metric on $G$ which will be
also denoted by $g$. The Laplacian on $G$ is defined by

$\mathcal{L}_{G}:=-\sum\limits_{j=1}^{n}X_{j}^{2},$

where $\mathbb{X}=\\{X_{1},...,X_{n}\\}$ is a $g$-orthonormal basis of left-
invariant vector fields. Let $\mathfrak{t}$ be a Cartan subalgebra of
$\mathfrak{k}$ (i.e. $\mathfrak{t}$ is nilpotent and self-normalising) and
denote by $\mathfrak{t}(\mathbb{R})$ its corresponding real form so that we
can consider the root system

$\Pi:=\\{\alpha\in\mathfrak{t}(\mathbb{R})^{*}\setminus\\{0\\}|\exists
X\in\mathfrak{g}\setminus\\{0\\}\ \textnormal{such that}\ [H,X]=\alpha(H)X,\
\forall H\in\mathfrak{t}(\mathbb{R})\\}\subseteq\mathfrak{t}(\mathbb{R})^{*}.$

Fix subsets $\Pi^{+}\subseteq\Pi$ and
$\Sigma=\\{\alpha_{1},...,\alpha_{\ell}\\}\subseteq\Pi^{+}$ of positive roots
and positive simple roots respectively. The restriction
$(\cdot,\cdot):=\gamma\kappa\left|{}_{\mathfrak{t}(\mathbb{R})\times\mathfrak{t}(\mathbb{R})}\right.$
on $\mathfrak{t}(\mathbb{R})$ is non-degenerate and it induces a quadratic
form on $\mathfrak{t}(\mathbb{R})^{*}$ (also denoted by $(\cdot,\cdot)$) as
follows: since $(\cdot,\cdot)$ is non-degenerate, then the map
$H\mapsto\alpha_{H}:=(H,\cdot)$ is an isomorphism between
$\mathfrak{t}(\mathbb{R})$ and $\mathfrak{t}(\mathbb{R})^{*}$; this means that
for each $\alpha\in\mathfrak{t}(\mathbb{R})$ there exists a unique
$H_{\alpha}\in\mathfrak{t}(\mathbb{R})$ satisfying $\alpha(H)=(H_{\alpha},H),\
\forall H\in\mathfrak{t}(\mathbb{R})$, so we define
$(\alpha,\beta):=(H_{\alpha},H_{\beta}).$

The fundamental weights
$\overline{\omega}_{1},...,\overline{\omega}_{\ell}\in\mathfrak{t}(\mathbb{R})^{*}$
of $\mathfrak{g}$ with respect to $\Sigma$ are defined by the relations

$(\overline{\omega}_{i},\alpha_{j})=\delta_{ij}\frac{(\alpha_{j},\alpha_{j})}{2},\
i,j=1,...,\ell,$

where $\delta_{ij}$ is the Kronecker symbol. For a given a complex finite
dimensional representation $\rho:\mathfrak{k}\rightarrow\mathfrak{gl}(V)$,
there exists a decomposition

$V=\bigoplus\limits_{\alpha\in\Phi_{\rho}}V_{\alpha},$

where $V_{\alpha}:=\\{v\in V:\rho(H)v=\alpha^{\mathbb{C}}(H)v,\forall
H\in\mathfrak{t}\\}$ and
$\Phi_{\rho}:=\\{\alpha\in\mathfrak{t}(\mathbb{R})^{*}:V_{\alpha}\neq\\{0\\}\\}.$
The elements of $\Phi_{\rho}$ and the subspaces $V_{\alpha},\
\alpha\in\Phi_{\rho},$ are called the weights and the weight subspaces of the
representation $\rho$ respectively. If $\alpha,\beta\in\Phi_{\rho}$, we say
that $\alpha$ is higher than $\beta$ whenever $\alpha-\beta$ is a linear
combination of positive roots with non-negative coefficients. A highest weight
of $\rho$ is a weight $\tilde{\nu}$ which is higher than every other weight.
The following is classical result by Cartan [9].

######  Theorem 2.1 (Theorem of the highest weight).

With the notations above we have that the following assertions hold:

  * (1)

Every finite-dimensional irreducible complex representation
$\rho:\mathfrak{k}\rightarrow\mathfrak{gl}(V)$ has a unique highest weight
$\tilde{\nu}$ which can be written as a linear combination of the fundamental
weights $\overline{\omega}_{1},...,\overline{\omega}_{\ell}$ with integral
non-negative coefficients, that is,

$\tilde{\nu}=\sum\limits_{j=1}^{\ell}\tilde{\nu}_{j}\overline{\omega}_{j},\
\tilde{\nu}_{j}\in\mathbb{N}_{0}.$ (2.1)

Moreover, two irreducible finite-dimensional representations of $\mathfrak{k}$
with the same highest weight are equivalent.

  * (2)

Every functional of the form (2.1) is the highest weight of an irreducible
complex representation of $\mathfrak{k}.$

It is a known fact that if $\theta:G\rightarrow GL(V)$ is an irreducible
complex finite-dimensional representation of the group $G$, then the function

$\varphi=\frac{1}{d}\chi_{\theta},$ (2.2)

where $d=\textnormal{dim}(V)$ and $\chi_{\theta}$ is the character of the
representation $\theta,$ is an eigenfunction of $\mathcal{L}_{G}$ and
conversely, for each
$\lambda\in\textnormal{Spec}(\mathcal{L}_{G})\setminus\\{0\\}$ there exists a
eigenfunction of the form (2.2) associated to the eigenvalue $\lambda.$
Furthermore, for each
$\lambda\in\textnormal{Spec}(\mathcal{L}_{G})\setminus\\{0\\}$, there exists a
real irreducible representation $\theta_{0}:G\rightarrow GL(V)$ such that
$\varphi=\frac{1}{d}\chi_{\theta_{0}}$ is an eigenfunction of
$\mathcal{L}_{G}$ corresponding to $\lambda$ (see [2, Corollary 3.2]). The
next proposition provides a formula to compute
$\textnormal{Spec}(\mathcal{L}_{G})$ in terms of real irreducible
representations of $G.$ We refer to [2, Theorem 3.2] for a proof.

######  Proposition 2.2.

Let $\theta_{0}:G\rightarrow GL(V)$ be an irreducible real $d$-dimensional
representation of $G$ and
$\rho_{0}:=(d\theta_{0})_{e}:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ its
corresponding infinitesimal representation. Then

$\lambda_{\theta_{0}}:=-\frac{1}{d}\sum\limits_{j=1}^{n}\textnormal{Tr}[\rho_{0}(X_{j})^{2}]\in\textnormal{Spec}(\mathcal{L}_{G}).$
(2.3)

Since we are assuming that $G$ is simply connected, then the restriction
$\rho\left|{}_{\mathfrak{g}}\right.$ of an irreducible complex representation
$\rho$ of $\mathfrak{k}$ induces an irreducible real representation $\rho_{0}$
of $\mathfrak{g}$ whose character is uniquely determined by the highest weight
$\tilde{\nu}$ of $\rho.$ On the other hand, irreducible real finite-
dimensional representations of the Lie algebra $\mathfrak{g}$ are in bijective
correspondence with irreducible real finite-dimensional representations of the
group $G.$ So we can parametrise the nonzero eigenvalues of $\mathcal{L}_{G}$
with the set of all highest weights $\tilde{\nu}$ of irreducible complex
finite-dimensional representations of $\mathfrak{k}$ which, due to Theorem
2.1, is nothing but the set of all linear combinations of the fundamental
weights with non-negative integral coefficients. Moreover, we have the
following proposition:

######  Proposition 2.3.

([2, Theorem 5.20]) For each highest weight
$\tilde{\nu}=\sum\limits_{j=1}^{l}\tilde{\nu}_{j}\overline{\omega}_{j},\
\tilde{\nu}_{j}\in\mathbb{N}_{0},$ the associated eigenvalue of
$\mathcal{L}_{G}$ is given by

$\lambda_{\tilde{\nu}}=(\eta,\eta)-(\tilde{\nu}+\eta,\tilde{\nu}+\eta),$ (2.4)

where

$\eta:=\frac{1}{2}\sum\limits_{\alpha\in\Pi^{+}}\alpha.$

######  Remark 2.4.

It is well-known that if $p:M\rightarrow N$ is a Riemannian submersion with
totally geodesic fibers and $\varphi$ is an eigenfunction of the Laplacian
operator in $N$, then $\varphi\circ\pi$ is an eigenfunction of the Laplacian
operator in $M$ corresponding to the same eigenvalue [4]. As a consequence of
this fact we have that $\textnormal{Spec}(N)\subseteq\textnormal{Spec}(M)$. In
particular, if $G$ is a connected (not necessarily simply-connected) compact
simple Lie group, then $G\cong\tilde{G}/\Gamma$, where $\tilde{G}$ is a
connected simply-connected compact simple Lie group (diffeomorphic to the
universal cover of $G$) and $\Gamma$ is a discrete subgroup of $\tilde{G}$,
every bi-invariant metric $g$ on $G$ is induced by a bi-invariant metric
$\tilde{g}$ on $\tilde{G}$ in such a way that the projection
$\pi:\tilde{G}\rightarrow\tilde{G}/\Gamma\cong G$ is a Riemannian submersion
with totally geodesic fibers. This allows us to conclude that, in this
situation we have

$\textnormal{Spec}(G)\subseteq\textnormal{Spec}(\tilde{G}),$ (2.5)

With the notations in [20, Chapter 12], we can construct all the irreducible
root systems starting from an Euclidean space endowed with an inner product
$\langle\cdot,\cdot\rangle$ and a $\langle\cdot,\cdot\rangle$-orthonormal
basis $\\{\varepsilon_{j}\\}$. From this construction we have that
$(\cdot,\cdot)=-\frac{1}{b\gamma}\langle\cdot,\cdot\rangle$ with
$b=\langle\tilde{\alpha},\tilde{\alpha}\rangle+2\langle\tilde{\alpha},\eta\rangle$
and $\tilde{\alpha}$ the highest long root so that

$\lambda_{\tilde{\nu}}=(\eta,\eta)-(\tilde{\nu}+\eta,\tilde{\nu}+\eta)=\frac{1}{b\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right].$
(2.6)

We shall use the formula in (2.6) to compute explicitly the eigenvalues of
$\mathcal{L}_{G}$ for the connected and simply-connected compact simple Lie
groups classified by their root system.

$\bullet$ Diagram $A_{\ell}.$ In this case, one has that
$G\cong\textnormal{SU}(\ell+1).$ The spectrum of the Laplacian is computed in
the following theorem.

######  Theorem 2.5.

The spectrum of the Laplacian operator on SU($\ell+1$) ($\ell\geq 1$) is given
by

$\textnormal{Spec(SU(}\ell+1\textnormal{))}=\\{\lambda_{\nu}:\nu=(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell}\\},$

where

$\displaystyle\lambda_{\nu}=$
$\displaystyle\frac{1}{2(\ell+1)^{3}\gamma}\left[\left(\sum\limits_{j=1}^{\ell}(\ell+1-j)\nu_{j}\right)^{2}\right.+\sum\limits_{k=2}^{\ell}\left(-\sum\limits_{j=1}^{k-1}j\nu_{j}+\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j}\right)^{2}$
$\displaystyle\hskip
56.9055pt+\left.\left(\sum\limits_{j=1}^{\ell}j\nu_{j}\right)^{2}-\frac{1}{4}\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}\right].$
(2.7)

###### Proof.

The root system associated to SU($\ell+1$) is $A_{\ell}$. The standard
realisation of $A_{\ell}$ is given by a $(\ell+1)-$dimensional Euclidean space
with an inner product $\langle\cdot,\cdot\rangle$ and an orthonormal basis
$\\{\varepsilon_{1},...,\varepsilon_{\ell+1}\\}$. The sets of positive roots
and simple positive roots are

$\Pi^{+}=\\{\varepsilon_{i}-\varepsilon_{j}:1\leq i<j\leq\ell+1\\}\
\textnormal{and}\ \Sigma=\\{\alpha_{j}=\varepsilon_{j}-\varepsilon_{j+1}:1\leq
j\leq\ell\\}$

respectively. The fundamental weights with respect to $\Sigma$ are

$\overline{\omega}_{j}=\frac{1}{\ell+1}\left[(\ell-j+1)(\varepsilon_{1}+\cdots+\varepsilon_{j})-j(\varepsilon_{j+1}+\cdots+\varepsilon_{\ell+1})\right],\
j=1,...,\ell,$

the highest root is

$\tilde{\alpha}=\alpha_{1}+\cdots+\alpha_{\ell}=\varepsilon_{1}-\varepsilon_{\ell+1}$

and

$\eta=\frac{1}{2}\sum\limits_{1\leq
i<j\leq\ell+1}(\varepsilon_{i}-\varepsilon_{j})=\frac{1}{2}\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))\varepsilon_{k}.$

Then, we can compute

$b=\langle\varepsilon_{1}-\varepsilon_{\ell+1},\varepsilon_{1}-\varepsilon_{\ell+1}\rangle+2\left\langle\varepsilon_{1}-\varepsilon_{\ell+1},\frac{1}{2}\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))\varepsilon_{k}\right\rangle=2\ell+2.$

Consider a highest weight
$\tilde{\nu}=\tilde{\nu}_{1}\overline{\omega}_{1}+\cdots+\tilde{\nu}_{1}\overline{\omega}_{\ell}$,
then we have that

$\displaystyle\tilde{\nu}=$
$\displaystyle\frac{\left(\sum\limits_{j=1}^{\ell}(\ell+1-j)\tilde{\nu}_{j}\right)\varepsilon_{1}+\sum\limits_{k=2}^{\ell}\left(-\sum\limits_{j=1}^{k-1}j\tilde{\nu}_{j}+\sum\limits_{j=k}^{\ell}(\ell+1-j)\tilde{\nu}_{j}\right)\varepsilon_{k}-\left(\sum\limits_{j=1}^{\ell}j\tilde{\nu}_{j}\right)\varepsilon_{\ell+1}}{\ell+1}$
$\displaystyle=$
$\displaystyle\frac{\sum\limits_{k=1}^{\ell+1}c_{k}\varepsilon_{k}}{\ell+1},$

where $c_{1}:=\sum\limits_{j=1}^{\ell}(\ell+1-j)\tilde{\nu}_{j},\
c_{k}:=-\sum\limits_{j=1}^{k-1}j\tilde{\nu}_{j}+\sum\limits_{j=k}^{\ell}(\ell+1-j)\tilde{\nu}_{j},\
k=2,...,\ell,$ and $c_{\ell+1}:=-\sum\limits_{j=1}^{\ell}j\tilde{\nu}_{j}.$
Therefore

$\displaystyle\tilde{\nu}+\eta=\sum\limits_{k=1}^{\ell+1}\left(\frac{2c_{k}}{2(\ell+1)}+\frac{(\ell-2(k-1))(\ell+1)}{2(\ell+1)}\right)\varepsilon_{k},$
$\displaystyle\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle=\frac{1}{4(\ell+1)^{2}}\sum\limits_{k=1}^{\ell+1}\left(2c_{k}+(\ell-2(k-1))(\ell+1)\right)^{2},$
$\displaystyle\langle\eta,\eta\rangle=\frac{1}{4(\ell+1)^{2}}\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2},$

and the eigenvalue $\lambda_{\tilde{\nu}}$ associated to $\tilde{\nu}$ is
given by

$\displaystyle\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{b\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{8(\ell+1)^{3}\gamma}\left[\sum\limits_{k=1}^{\ell+1}\left(2c_{k}+(\ell-2(k-1))(\ell+1)\right)^{2}-\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{8(\ell+1)^{3}\gamma}\left[\left(2\sum\limits_{j=1}^{\ell}(\ell+1-j)\tilde{\nu}_{j}+\ell(\ell+1)\right)^{2}\right.$
$\displaystyle\hskip
56.9055pt+\sum\limits_{k=2}^{\ell}\left(-2\sum\limits_{j=1}^{k-1}j\tilde{\nu}_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\tilde{\nu}_{j}+(\ell-2(k-1))(\ell+1)\right)^{2}$
$\displaystyle\hskip
56.9055pt+\left.\left(2\sum\limits_{j=1}^{\ell}j\tilde{\nu}_{j}+\ell(\ell+1)\right)^{2}-\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}\right].$

Let $\nu_{j}=\tilde{\nu}_{j}+1,\ j=1,...,\ell.$ The highest weight
$\tilde{\nu}$ is uniquely determined by
$\nu=(\nu_{1},...,\nu_{l})\in\mathbb{N}^{\ell}$. Let us write
$\lambda_{\tilde{\nu}}$ in terms of $\nu:$

$\displaystyle
2\sum\limits_{j=1}^{\ell}(\ell+1-j)\tilde{\nu}_{j}+\ell(\ell+1)=$
$\displaystyle 2\sum\limits_{j=1}^{\ell}(\ell+1-j)(\nu_{j}-1)+\ell(\ell+1)$
$\displaystyle=$ $\displaystyle
2\sum\limits_{j=1}^{\ell}(\ell+1-j)\nu_{j}-2\sum\limits_{j=1}^{\ell}(\ell+1-j)+\ell(\ell+1)$
$\displaystyle=$ $\displaystyle
2\sum\limits_{j=1}^{\ell}(\ell+1-j)\nu_{j}-\ell(\ell+1)+\ell(\ell+1)$
$\displaystyle=$ $\displaystyle 2\sum\limits_{j=1}^{\ell}(\ell+1-j)\nu_{j},$
$\displaystyle-2\sum\limits_{j=1}^{k-1}j\tilde{\nu}_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\tilde{\nu}_{j}+(\ell-2(k-1))(\ell+1)$
$\displaystyle=$
$\displaystyle-2\sum\limits_{j=1}^{k-1}j(\nu_{j}-1)+2\sum\limits_{j=k}^{\ell}(\ell+1-j)(\nu_{j}-1)+(\ell-2(k-1))(\ell+1)$
$\displaystyle=$
$\displaystyle-2\sum\limits_{j=1}^{k-1}j\nu_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j}+2\sum\limits_{j=1}^{k-1}j-2\sum\limits_{j=k}^{\ell}(\ell+1-j)$
$\displaystyle+(\ell-2(k-1))(\ell+1)$ $\displaystyle=$
$\displaystyle-2\sum\limits_{j=1}^{k-1}j\nu_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j}+k(k-1)-(\ell+1-k)(\ell+2-k)$
$\displaystyle+(\ell-2(k-1))(\ell+1)$ $\displaystyle=$
$\displaystyle-2\sum\limits_{j=1}^{k-1}j\nu_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j}-(\ell-2(k-1))(\ell+1)+(\ell-2(k-1))(\ell+1)$
$\displaystyle=$
$\displaystyle-2\sum\limits_{j=1}^{k-1}j\nu_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j},$
$\displaystyle 2\sum\limits_{j=1}^{\ell}j\tilde{\nu}_{j}+\ell(\ell+1)=$
$\displaystyle 2\sum\limits_{j=1}^{\ell}j(\nu_{j}-1)+\ell(\ell+1)$
$\displaystyle=$ $\displaystyle
2\sum\limits_{j=1}^{\ell}j\nu_{j}-2\sum\limits_{j=1}^{\ell}j+\ell(\ell+1)$
$\displaystyle=$ $\displaystyle
2\sum\limits_{j=1}^{\ell}j\nu_{j}-\ell(\ell+1)+\ell(\ell+1)$ $\displaystyle=$
$\displaystyle 2\sum\limits_{j=1}^{\ell}j\nu_{j},\hskip 230.46748pt$

then

$\displaystyle\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{8(\ell+1)^{3}\gamma}\left[\left(2\sum\limits_{j=1}^{\ell}(\ell+1-j)\tilde{\nu}_{j}+\ell(\ell+1)\right)^{2}\right.$
$\displaystyle\hskip
56.9055pt+\sum\limits_{k=2}^{\ell}\left(-2\sum\limits_{j=1}^{k-1}j\tilde{\nu}_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\tilde{\nu}_{j}+(\ell-2(k-1))(\ell+1)\right)^{2}$
$\displaystyle\hskip
56.9055pt+\left.\left(2\sum\limits_{j=1}^{\ell}j\tilde{\nu}_{j}+\ell(\ell+1)\right)^{2}-\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{8(\ell+1)^{3}\gamma}\left[\left(2\sum\limits_{j=1}^{\ell}(\ell+1-j)\nu_{j}\right)^{2}\right.+\sum\limits_{k=2}^{\ell}\left(-2\sum\limits_{j=1}^{k-1}j\nu_{j}+2\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j}\right)^{2}$
$\displaystyle\hskip
56.9055pt+\left.\left(2\sum\limits_{j=1}^{\ell}j\nu_{j}\right)^{2}-\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2(\ell+1)^{3}\gamma}\left[\left(\sum\limits_{j=1}^{\ell}(\ell+1-j)\nu_{j}\right)^{2}\right.+\sum\limits_{k=2}^{\ell}\left(-\sum\limits_{j=1}^{k-1}j\nu_{j}+\sum\limits_{j=k}^{\ell}(\ell+1-j)\nu_{j}\right)^{2}$
$\displaystyle\hskip
56.9055pt+\left.\left(\sum\limits_{j=1}^{\ell}j\nu_{j}\right)^{2}-\frac{1}{4}\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}\right].$

Since $\tilde{\nu}_{j}\geq 0$, then $\nu_{j}\geq 1.$ By setting
$\nu=(\nu_{1},...,\nu_{\ell})$ and $\lambda_{\nu}=\lambda_{\tilde{\nu}}$ we
obtain the result. ∎

$\bullet$ Diagram $B_{\ell}$ In this case, one has that
$G\cong\textnormal{Spin}(2\ell+1).$ The spectrum of the Laplacian is computed
in the following theorem. We exclude the case $\ell=1$ since
$\textnormal{Spin(3)}\cong\textnormal{SU}(2)\cong\mathbb{S}^{3}$ and the
Strichartz estimates in the case of the spheres have been analysed in the work
of Burq, Gerard, and Tzvetkov [8].

######  Theorem 2.6.

The spectrum of the Laplacian operator on Spin($2\ell+1$) ($\ell\geq 2$) is
given by

$\textnormal{Spec(Spin(}2\ell+1\textnormal{))}=\\{\lambda_{\nu}:\nu=(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell}\\},$

where

$\lambda_{\nu}=\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\nu_{j}+\nu_{\ell}\right)^{2}+\nu_{\ell}^{2}-\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right).$
(2.8)

###### Proof.

The root system associated to Spin($2\ell+1$) is $B_{\ell}$ which can be
constructed by considering a $\langle\cdot,\cdot\rangle$-orthonormal basis
$\\{\varepsilon_{1},...,\varepsilon_{\ell}\\}$ on a $\ell$-dimensional
Euclidean space. The positive roots are $\varepsilon_{j},\ 1\leq j\leq\ell$
(the short roots) and $\varepsilon_{i}\pm\varepsilon_{j},\ 1\leq i<j\leq\ell$
(the long roots). The set
$\Sigma=\\{\alpha_{1}=\varepsilon_{1}-\varepsilon_{2},,...,\alpha_{\ell-1}=\varepsilon_{\ell-1}-\varepsilon_{\ell},\alpha_{\ell}=\varepsilon_{\ell}\\}$
is a system of simple positive roots and the corresponding fundamental weights
are $\overline{\omega}_{j}=\varepsilon_{1}+\cdots+\varepsilon_{j},$ $1\leq
j<\ell$ and
$\overline{\omega}_{\ell}=\frac{\varepsilon_{1}+\cdots+\varepsilon_{\ell}}{2}$.
The highest long root is
$\tilde{\alpha}=\alpha_{1}+2(\alpha_{2}+\cdots+\alpha_{\ell})=\varepsilon_{1}+\varepsilon_{2}$
and

$\eta=\frac{1}{2}\left(\sum\limits_{j=1}^{\ell}\varepsilon_{j}+\sum\limits_{1\leq
i<j\leq\ell}(\varepsilon_{i}+\varepsilon_{j})+\sum\limits_{1\leq
i<j\leq\ell}(\varepsilon_{i}-\varepsilon_{j})\right)=\frac{1}{2}\left(\sum\limits_{k=1}^{\ell}(2(\ell-k)+1)\varepsilon_{k}\right),$

thus

$b=\langle\varepsilon_{1}+\varepsilon_{2},\varepsilon_{1}+\varepsilon_{2}\rangle+2\left\langle\varepsilon_{1}+\varepsilon_{2},\frac{1}{2}\left(\sum\limits_{k=1}^{\ell}(2(\ell-k)+1)\varepsilon_{k}\right)\right\rangle=4\ell-2.$

Given a highest weight

$\tilde{\nu}=\tilde{\nu}_{1}\overline{\omega}_{1}+\cdots+\tilde{\nu}_{\ell}\overline{\omega}_{\ell}=\sum\limits_{k=1}^{\ell-1}\left[\sum\limits_{j=k}^{\ell-1}\tilde{\nu}_{j}+\frac{\tilde{\nu}_{\ell}}{2}\right]\varepsilon_{k}+\frac{\tilde{\nu}_{\ell}}{2}\varepsilon_{\ell},\
\tilde{\nu}_{j}\in\mathbb{N}_{0},\ j=1,...,l;$

we have (by formula (2.6)) that the eigenvalue $\lambda_{\tilde{\nu}}$
associated to $\tilde{\nu}$ is

$\lambda_{\tilde{\nu}}=\frac{1}{(4\ell-2)\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$

but

$\tilde{\nu}+\eta=\sum\limits_{k=1}^{\ell-1}\left\\{\left[\sum\limits_{j=k}^{\ell-1}\tilde{\nu}_{j}+\frac{\tilde{\nu}_{\ell}}{2}\right]+\frac{(2(\ell-k)+1)}{2}\right\\}\varepsilon_{k}+\frac{\tilde{\nu}_{\ell}+1}{2}\varepsilon_{\ell}$

so that

$\displaystyle\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle=$
$\displaystyle\frac{1}{4}\left\\{\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\tilde{\nu}_{j}+\tilde{\nu}_{\ell}+(2(\ell-k)+1)\right)^{2}+(\tilde{\nu}_{\ell}+1)^{2}\right\\}.$

On the other hand

$\langle\eta,\eta\rangle=\frac{1}{4}\sum\limits_{k=1}^{l}\left(2(l-k)+1\right)^{2},$

hence

$\displaystyle\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\tilde{\nu}_{j}+\tilde{\nu}_{\ell}+(2(\ell-k)+1)\right)^{2}+(\tilde{\nu}_{\ell}+1)^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right).$

If $\nu_{j}:=\tilde{\nu}_{j}+1,\ j=1,...,\ell$,
$\nu:=(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell},$ and
$\lambda_{\nu}:=\lambda_{\tilde{\nu}}$ we have

$\displaystyle\lambda_{\nu}=\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\tilde{\nu}_{j}+\tilde{\nu}_{\ell}+(2(\ell-k)+1)\right)^{2}+(\tilde{\nu}_{\ell}+1)^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}(\nu_{j}-1)+(\nu_{\ell}-1)+(2(\ell-k)+1)\right)^{2}+\nu_{\ell}^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\nu_{j}-2\sum\limits_{j=k}^{\ell-1}1+\nu_{\ell}-1+(2(\ell-k)+1)\right)^{2}+\nu_{\ell}^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\nu_{j}+\nu_{\ell}-2(\ell-k)-1+(2(\ell-k)+1)\right)^{2}+\nu_{\ell}^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-8)\gamma}\left(\sum\limits_{k=1}^{\ell-1}\left(2\sum\limits_{j=k}^{\ell-1}\nu_{j}+\nu_{\ell}\right)^{2}+\nu_{\ell}^{2}-\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}\right),$

completing the proof. ∎

$\bullet$ Diagram $C_{\ell}$. In this case, one has that
$G\cong\textnormal{Sp}(\ell).$ Observe that
$\textnormal{Sp(1)}\cong\textnormal{SU(2)}$ was treated in Theorem 2.5 and
$\textnormal{Sp(2)}\cong\textnormal{SO(5)}$ has a universal cover
diffeomorphic to Spin(5) whose spectrum was described in Theorem 2.6. For
$\ell\geq 3$, the spectrum of the Laplacian is computed in the following
theorem.

######  Theorem 2.7.

The spectrum of the Laplacian operator on Sp($\ell$), ($\ell\geq 3$) is given
by

$\textnormal{Spec(Sp(}\ell\textnormal{))}=\\{\lambda_{\nu}:\nu=(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell}\\},$

where

$\lambda_{\nu}=\frac{1}{4(\ell+1)\gamma}\left[\sum\limits_{k=1}^{\ell}\left(\sum\limits_{j=k}^{\ell}\nu_{j}\right)^{2}-\frac{\ell(\ell+1)(2\ell+1)}{6}\right].$
(2.9)

###### Proof.

The root system associated to the group Sp($\ell$) is $C_{\ell}$ which is
obtained by considering a $\ell$-dimensional Euclidean space with an inner
product $\langle\cdot,\cdot\rangle$ and a orthonormal basis
$\\{\varepsilon_{1},...,\varepsilon_{\ell}\\}$. The simple positive roots are
$\alpha_{j}=\varepsilon_{j}-\varepsilon_{j+1},\ j=1,...,\ell-1$ and
$\alpha_{\ell}=2\varepsilon_{\ell}.$ The positive roots are
$2\varepsilon_{i},\ i=1,...,\ell,\ \varepsilon_{i}\pm\varepsilon_{j},\ 1\leq
i<j\leq\ell$ and the fundamental weights are
$\overline{w}_{j}=\varepsilon_{1}+\cdots+\varepsilon_{j},\ j=1,...,\ell.$ In
this case

$\displaystyle\tilde{\alpha}=$ $\displaystyle
2(\alpha_{1}+\cdots+\alpha_{\ell-1})+\alpha_{\ell}=2\varepsilon_{1}\
\textnormal{and}$ $\displaystyle\eta=$
$\displaystyle\frac{1}{2}\left(\sum\limits_{1\leq
i<j\leq\ell}(\varepsilon_{i}+\varepsilon_{j})+\sum\limits_{1\leq
i<j\leq\ell}(\varepsilon_{i}-\varepsilon_{j})+\sum\limits_{i=1}^{\ell}2\varepsilon_{i}\right)$
$\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{\ell}(\ell-k+1)\varepsilon_{k},$

thus

$\displaystyle b=$
$\displaystyle\langle\tilde{\alpha},\tilde{\alpha}\rangle+2\left\langle\tilde{\alpha},\eta\right\rangle$
$\displaystyle=$ $\displaystyle
4\langle\varepsilon_{1},\varepsilon_{1}\rangle+4\left\langle\varepsilon_{1},\sum\limits_{k=1}^{\ell}(\ell-k+1)\varepsilon_{k}\right\rangle$
$\displaystyle=$ $\displaystyle 4(\ell+1).$

A highest weight has the form

$\tilde{\nu}=\sum\limits_{k=1}^{\ell}\tilde{\nu}_{k}\overline{\omega}_{k}=\sum\limits_{k=1}^{\ell}\tilde{\nu}_{k}\sum\limits_{j=1}^{k}\varepsilon_{j}=\sum\limits_{k=1}^{\ell}\sum\limits_{j=1}^{k}\tilde{\nu}_{k}\varepsilon_{j}=\sum\limits_{k=1}^{\ell}\left[\sum\limits_{j=k}^{\ell}\tilde{\nu}_{j}\right]\varepsilon_{k},$

so

$\tilde{\nu}+\eta=\sum\limits_{k=1}^{\ell}\left[\sum\limits_{j=k}^{\ell}\tilde{\nu}_{j}\right]\varepsilon_{k}+\sum\limits_{k=1}^{\ell}(\ell-k+1)\varepsilon_{k}=\sum\limits_{k=1}^{\ell}\left[\sum\limits_{j=k}^{\ell}\tilde{\nu}_{j}+(\ell-k+1)\right]\varepsilon_{k}$

which implies

$\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle=\sum\limits_{k=1}^{\ell}\left(\sum\limits_{j=k}^{\ell}\tilde{\nu}_{j}+(\ell-k+1)\right)^{2}.$

On the other hand

$\langle\eta,\eta\rangle=\sum\limits_{k=1}^{\ell}(\ell-k+1)^{2}=\frac{\ell(\ell+1)(2\ell+1)}{6}.$

Hence, the eigenvalue $\lambda_{\tilde{\nu}}$ associated to $\tilde{\nu}$ is

$\displaystyle\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{b\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4(\ell+1)\gamma}\left[\sum\limits_{k=1}^{\ell}\left(\sum\limits_{j=k}^{\ell}\tilde{\nu}_{j}+(\ell-k+1)\right)^{2}-\frac{\ell(\ell+1)(2\ell+1)}{6}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4(\ell+1)\gamma}\left[\sum\limits_{k=1}^{\ell}\left(\sum\limits_{j=k}^{\ell}\tilde{\nu}_{j}+\sum\limits_{j=k}^{\ell}1\right)^{2}-\frac{\ell(\ell+1)(2\ell+1)}{6}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4(\ell+1)\gamma}\left[\sum\limits_{k=1}^{\ell}\left(\sum\limits_{j=k}^{\ell}(\tilde{\nu}_{j}+1)\right)^{2}-\frac{\ell(\ell+1)(2\ell+1)}{6}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4(\ell+1)\gamma}\left[\sum\limits_{k=1}^{\ell}\left(\sum\limits_{j=k}^{\ell}\nu_{j}\right)^{2}-\frac{\ell(\ell+1)(2\ell+1)}{6}\right],$

where $\nu_{j}=\tilde{\nu}_{j}+1,\ j=1,...,\ell$. By setting
$\nu:=(\nu_{1},...,\nu_{\ell})$ and $\lambda_{\nu}:=\lambda_{\tilde{\nu}}$ we
complete the proof. ∎

$\bullet$ Diagram $D_{\ell}.$ In this case, one has that
$G\cong\textnormal{Spin}(2\ell).$ Note that
$\textnormal{Spin}(2)\cong\mathbb{S}^{1}$ was treated in [8],
$\textnormal{Spin}(4)\cong\textnormal{SU}(2)\times\textnormal{SU}(2)$ is not
simple and $\textnormal{Spin}(6)\cong\textnormal{SU}(4)$ was considered in
Theorem 2.5. The spectrum of the Laplacian is computed in the following
theorem for $\ell\geq 4.$

######  Theorem 2.8.

The spectrum of the Laplacian operator on Spin($2\ell$), ($\ell\geq 4$) is
given by

$\textnormal{Spec(Spin(}2\ell\textnormal{))}=\\{\lambda_{\nu}:\nu=(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell}\\},$

where

$\displaystyle\lambda_{\nu}=$
$\displaystyle\frac{1}{(16\ell-16)\gamma}\left(\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}\nu_{j}+\nu_{\ell-1}+\nu_{\ell}\right)^{2}+(\nu_{\ell}+\nu_{\ell-1})^{2}+(\nu_{\ell}-\nu_{\ell-1})^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}\right).$
(2.10)

###### Proof.

The root system associated to Spin($2\ell$) is $D_{\ell}$ which can be
constructed by considering a $\langle\cdot,\cdot\rangle$-orthonormal basis
$\\{\varepsilon_{1},...,\varepsilon_{\ell}\\}$ on a $\ell$-dimensional
Euclidean space. The positive roots are $\varepsilon_{i}\pm\varepsilon_{j},\
1\leq i<j\leq\ell$. The set
$\Sigma=\\{\alpha_{1}=\varepsilon_{1}-\varepsilon_{2},,...,\alpha_{\ell-1}=\varepsilon_{\ell-1}-\varepsilon_{\ell},\alpha_{\ell}=\varepsilon_{\ell-1}+\varepsilon_{\ell}\\}$
is a system of simple positive roots and the corresponding fundamental weights
are $\overline{\omega}_{j}=\varepsilon_{1}+\cdots+\varepsilon_{j},$ $1\leq
j<\ell-1,$
$\overline{\omega}_{\ell-1}=\frac{\varepsilon_{1}+\cdots+\varepsilon_{\ell-1}-\varepsilon_{\ell}}{2},$
and
$\overline{\omega}_{\ell}=\frac{\varepsilon_{1}+\cdots+\varepsilon_{\ell-1}+\varepsilon_{\ell}}{2}$.
The highest root is

$\tilde{\alpha}=\alpha_{1}+2(\alpha_{2}+\cdots+\alpha_{\ell-2})+\alpha_{\ell-1}+\alpha_{\ell}=\varepsilon_{1}+\varepsilon_{2}$

and

$\eta=\frac{1}{2}\left(\sum\limits_{1\leq
i<j\leq\ell}(\varepsilon_{i}+\varepsilon_{j})+\sum\limits_{1\leq
i<j\leq\ell}(\varepsilon_{i}-\varepsilon_{j})\right)=\sum\limits_{k=1}^{\ell}(\ell-k)\varepsilon_{k},$

thus

$b=\langle\varepsilon_{1}+\varepsilon_{2},\varepsilon_{1}+\varepsilon_{2}\rangle+2\left\langle\varepsilon_{1}+\varepsilon_{2},\sum\limits_{k=1}^{\ell}(\ell-k)\varepsilon_{k}\right\rangle=4\ell-4.$

A highest weight has the form

$\tilde{\nu}=\tilde{\nu}_{1}\overline{\omega}_{1}+\cdots+\tilde{\nu}_{\ell}\overline{\omega}_{\ell}=\sum\limits_{k=1}^{\ell-2}\left[\sum\limits_{j=k}^{\ell-2}\tilde{\nu}_{j}+\frac{\tilde{\nu}_{\ell-1}+\tilde{\nu}_{\ell}}{2}\right]\varepsilon_{k}+\frac{\tilde{\nu}_{\ell-1}+\tilde{\nu}_{\ell}}{2}\varepsilon_{\ell-1}+\frac{\tilde{\nu}_{\ell}-\tilde{\nu}_{\ell-1}}{2}\varepsilon_{\ell},$

with $\tilde{\nu}_{j}\in\mathbb{N}_{0},\ j=1,...,\ell.$ Let us compute
$\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle$ and
$\langle\eta,\eta\rangle:$

$\tilde{\nu}+\eta=\sum\limits_{k=1}^{\ell-2}\left\\{\left[\sum\limits_{j=k}^{\ell-2}\tilde{\nu}_{j}+\frac{\tilde{\nu}_{\ell-1}+\tilde{\nu}_{\ell}}{2}\right]+(\ell-k)\right\\}\varepsilon_{k}+\frac{\tilde{\nu}_{\ell-1}+\tilde{\nu}_{\ell}+2}{2}+\frac{\tilde{\nu}_{\ell}-\tilde{\nu}_{\ell-1}}{2}\varepsilon_{\ell}$

so that

$\displaystyle\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle$
$\displaystyle=\frac{1}{4}\left\\{\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}\tilde{\nu}_{j}+\tilde{\nu}_{\ell-1}+\tilde{\nu}_{\ell}+2(\ell-k)\right)^{2}+(\tilde{\nu}_{\ell}+\tilde{\nu}_{\ell-1}+2)^{2}+(\tilde{\nu}_{\ell}-\tilde{\nu}_{\ell-1})^{2}\right\\}$

and

$\langle\eta,\eta\rangle=\sum\limits_{k=1}^{\ell}(\ell-k)^{2}=\frac{1}{4}\sum\limits_{k=1}^{\ell}(2(\ell-k))^{2}.$

By formula (2.6) we have

$\displaystyle\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{(4\ell-4)\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-16)\gamma}\left(\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}\tilde{\nu}_{j}+\tilde{\nu}_{\ell-1}+\tilde{\nu}_{\ell}+2(\ell-k)\right)^{2}+(\tilde{\nu}_{\ell}+\tilde{\nu}_{\ell-1}+2)^{2}\right.$
$\displaystyle\hskip
56.9055pt+(\tilde{\nu}_{\ell}-\tilde{\nu}_{\ell-1})^{2}-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}\right).$

Set $\nu_{j}:=\tilde{\nu}_{j}+1,\ j=1,...,\ell$,
$\nu:=(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell}$, and
$\lambda_{\nu}:=\lambda_{\tilde{\nu}}$, so that

$\displaystyle\lambda_{\nu}=\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{(16\ell-16)\gamma}\left(\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}(\nu_{j}-1)+(\nu_{\ell-1}-1)+(\nu_{\ell}-1)+2(\ell-k)\right)^{2}\right.$
$\displaystyle\hskip
56.9055pt+(\nu_{\ell}-1+\nu_{\ell-1}-1+2)^{2}+(\nu_{\ell}-\nu_{\ell-1})^{2}-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-16)\gamma}\left(\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}\nu_{j}-2\sum\limits_{j=k}^{\ell-2}1+\nu_{\ell-1}+\nu_{\ell}-2+2(\ell-k)\right)^{2}\right.$
$\displaystyle\hskip
56.9055pt+(\nu_{\ell}+\nu_{\ell-1})^{2}+(\nu_{\ell}-\nu_{\ell-1})^{2}-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-16)\gamma}\left(\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}\nu_{j}+\nu_{\ell-1}+\nu_{\ell}-2(\ell-k-1)-2+2(\ell-k)\right)^{2}\right.$
$\displaystyle\hskip
56.9055pt+(\nu_{\ell}+\nu_{\ell-1})^{2}+(\nu_{\ell}-\nu_{\ell-1})^{2}-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(16\ell-16)\gamma}\left(\sum\limits_{k=1}^{\ell-2}\left(2\sum\limits_{j=k}^{\ell-2}\nu_{j}+\nu_{\ell-1}+\nu_{\ell}\right)^{2}+(\nu_{\ell}+\nu_{\ell-1})^{2}+(\nu_{\ell}-\nu_{\ell-1})^{2}\right.$
$\displaystyle\hskip
56.9055pt-\left.\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}\right),$

completing the proof. ∎

$\bullet$ Diagram $\textnormal{E}_{8}.$ Now, we consider the exceptional case
of the compact Lie group $\textnormal{E}_{8}.$

######  Theorem 2.9.

The spectrum of the Laplacian operator on $\textnormal{E}_{8}$ is given by

$\textnormal{Spec(E}_{8}\textnormal{)}=\\{\lambda_{\nu}:\nu=(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5},\nu_{6},\nu_{7},\nu_{8})\in\mathbb{N}^{8}\\}$

where

$\displaystyle\lambda_{\nu}=$
$\displaystyle\frac{1}{240\gamma}\left[\left(\nu_{2}-\nu_{3}\right)^{2}+\left(\nu_{2}+\nu_{3}\right)^{2}+\left(\nu_{2}+\nu_{3}+2\nu_{4}\right)^{2}+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}\right)^{2}+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}+2\nu_{8}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(4\nu_{1}+5\nu_{2}+7\nu_{3}+10\nu_{4}+8\nu_{5}+6\nu_{6}+4\nu_{7}+2\nu_{8}\right)^{2}-2480\right].$
(2.11)

###### Proof.

We can construct $\textnormal{E}_{8}$ on a 8-dimensional linear space with an
inner product $\langle\cdot,\cdot\rangle$ and an orthonormal basis
$\\{\varepsilon_{1},\cdots,\varepsilon_{8}\\}.$ The positive roots are given
by

$\displaystyle\varepsilon_{j}\pm\varepsilon_{i},\ 1\leq i<j\leq 8,\
\textnormal{and}$
$\displaystyle\frac{1}{2}\left((-1)^{i_{1}}\varepsilon_{1}+\cdots+(-1)^{i_{7}}\varepsilon_{7}+\varepsilon_{8}\right),\
i_{k}\in\\{0,1\\},\ i_{1}+\cdots+i_{7}\equiv 0\ \textnormal{mod}\ 2$

and the simple roots are

$\displaystyle\alpha_{1}=\frac{1}{2}\left(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}-\varepsilon_{5}-\varepsilon_{6}-\varepsilon_{7}+\varepsilon_{8}\right),$
$\displaystyle\alpha_{2}=\varepsilon_{1}+\varepsilon_{2}$
$\displaystyle\alpha_{j}=\varepsilon_{j-1}-\varepsilon_{j-2},\ j=3,4,5,6,7,8.$

The corresponding fundamental weights are given by

$\displaystyle\overline{\omega}_{1}=$ $\displaystyle
4\alpha_{1}+5\alpha_{2}+7\alpha_{3}+10\alpha_{4}+8\alpha_{5}+6\alpha_{6}+4\alpha_{7}+2\alpha_{8}=2\varepsilon_{8}$
$\displaystyle\overline{\omega}_{2}=$ $\displaystyle
5\alpha_{1}+8\alpha_{2}+10\alpha_{3}+15\alpha_{4}+12\alpha_{5}+9\alpha_{6}+6\alpha_{7}+3\alpha_{8}$
$\displaystyle=$
$\displaystyle\frac{\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}+5\varepsilon_{8}}{2}$
$\displaystyle\overline{\omega}_{3}=$ $\displaystyle
7\alpha_{1}+10\alpha_{2}+14\alpha_{3}+20\alpha_{4}+16\alpha_{5}+12\alpha_{6}+8\alpha_{7}+4\alpha_{8}$
$\displaystyle=$
$\displaystyle\frac{-\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}+7\varepsilon_{8}}{2}$
$\displaystyle\overline{\omega}_{4}=$ $\displaystyle
10\alpha_{1}+15\alpha_{2}+20\alpha_{3}+30\alpha_{4}+24\alpha_{5}+18\alpha_{6}+12\alpha_{7}+6\alpha_{8}$
$\displaystyle=$
$\displaystyle\varepsilon_{3}+\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}+5\varepsilon_{8}$
$\displaystyle\overline{\omega}_{5}=$ $\displaystyle
8\alpha_{1}+12\alpha_{2}+16\alpha_{3}+24\alpha_{4}+20\alpha_{5}+15\alpha_{6}+10\alpha_{7}+5\alpha_{8}$
$\displaystyle=$
$\displaystyle\varepsilon_{4}+\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}+4\varepsilon_{8}$
$\displaystyle\overline{\omega}_{6}=$ $\displaystyle
6\alpha_{1}+9\alpha_{2}+12\alpha_{3}+18\alpha_{4}+15\alpha_{5}+12\alpha_{6}+8\alpha_{7}+4\alpha_{8}$
$\displaystyle=$
$\displaystyle\varepsilon_{5}+\varepsilon_{6}+\varepsilon_{7}+3\varepsilon_{8}$
$\displaystyle\overline{\omega}_{7}=$ $\displaystyle
4\alpha_{1}+6\alpha_{2}+8\alpha_{3}+12\alpha_{4}+10\alpha_{5}+8\alpha_{6}+6\alpha_{7}+3\alpha_{8}$
$\displaystyle=$
$\displaystyle\varepsilon_{6}+\varepsilon_{7}+2\varepsilon_{8}$
$\displaystyle\overline{\omega}_{8}=$ $\displaystyle
2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}+2\alpha_{8}$
$\displaystyle=$ $\displaystyle\varepsilon_{7}+\varepsilon_{8},$

the highest long root is $\tilde{\alpha}=\varepsilon_{7}+\varepsilon_{8}$ and

$\displaystyle\eta=$
$\displaystyle\frac{1}{2}\sum\limits_{\alpha\in\Pi^{+}}\alpha$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(\sum\limits_{1\leq i<j\leq
8}(\varepsilon_{j}+\varepsilon_{i})+\sum\limits_{1\leq i<j\leq
8}(\varepsilon_{j}-\varepsilon_{i})+\frac{1}{2}\left[\sum\limits_{\begin{subarray}{c}i_{1},...,i_{7}=0\\\
i_{1}+\cdots+i_{7}\equiv 0\ \textnormal{mod}\
2\end{subarray}}^{1}\left(\sum\limits_{k=1}^{7}(-1)^{i_{k}}\varepsilon_{k}+\varepsilon_{8}\right)\right]\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(\sum\limits_{1\leq i<j\leq
8}2\varepsilon_{j}+32\varepsilon_{8}\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=2}^{7}(k-1)\varepsilon_{k}+23\varepsilon_{8},$

so

$\displaystyle
b=\langle\tilde{\alpha},\tilde{\alpha}\rangle+2\langle\tilde{\alpha},\eta\rangle=2+2(6+23)=60.$

Let $\tilde{\nu}=\sum\limits_{k=1}^{8}\tilde{\nu}_{j}\overline{\omega}_{j}$ be
a highest weight, then

$\displaystyle\tilde{\nu}+\eta=$
$\displaystyle\left(\frac{\tilde{\nu}_{2}-\tilde{\nu}_{3}}{2}\right)\varepsilon_{1}+\left(\frac{\tilde{\nu}_{2}+\tilde{\nu}_{3}+2}{2}\right)\varepsilon_{2}+\left(\frac{\tilde{\nu}_{2}+\tilde{\nu}_{3}+2\tilde{\nu}_{4}+4}{2}\right)\varepsilon_{3}$
$\displaystyle+\left(\frac{\tilde{\nu}_{2}+\tilde{\nu}_{3}+2\tilde{\nu}_{4}+2\tilde{\nu}_{5}+6}{2}\right)\varepsilon_{4}+\left(\frac{\tilde{\nu}_{2}+\tilde{\nu}_{3}+2\tilde{\nu}_{4}+2\tilde{\nu}_{5}+2\tilde{\nu}_{6}+8}{2}\right)\varepsilon_{5}$
$\displaystyle+\left(\frac{\tilde{\nu}_{2}+\tilde{\nu}_{3}+2\tilde{\nu}_{4}+2\tilde{\nu}_{5}+2\tilde{\nu}_{6}+2\tilde{\nu}_{7}+10}{2}\right)\varepsilon_{6}$
$\displaystyle+\left(\frac{\tilde{\nu}_{2}+\tilde{\nu}_{3}+2\tilde{\nu}_{4}+2\tilde{\nu}_{5}+2\tilde{\nu}_{6}+2\tilde{\nu}_{7}+2\tilde{\nu}_{8}+12}{2}\right)\varepsilon_{7}$
$\displaystyle+\left(\frac{4\tilde{\nu}_{1}+5\tilde{\nu}_{2}+7\tilde{\nu}_{3}+10\tilde{\nu}_{4}+8\tilde{\nu}_{5}+6\tilde{\nu}_{6}+4\tilde{\nu}_{7}+2\tilde{\nu}_{8}+46}{2}\right)\varepsilon_{8}$
$\displaystyle=$
$\displaystyle\left(\frac{\nu_{2}-\nu_{3}}{2}\right)\varepsilon_{1}+\left(\frac{\nu_{2}+\nu_{3}}{2}\right)\varepsilon_{2}+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}}{2}\right)\varepsilon_{3}$
$\displaystyle+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}}{2}\right)\varepsilon_{4}+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}}{2}\right)\varepsilon_{5}$
$\displaystyle+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}}{2}\right)\varepsilon_{6}$
$\displaystyle+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}+2\nu_{8}}{2}\right)\varepsilon_{7}$
$\displaystyle+\left(\frac{4\nu_{1}+5\nu_{2}+7\nu_{3}+10\nu_{4}+8\nu_{5}+6\nu_{6}+4\nu_{7}+2\nu_{8}}{2}\right)\varepsilon_{8},$

with $\nu_{j}=\tilde{\nu}_{j}+1,\ j=1,2,3,4,5,6,7,8.$ By setting
$\nu:=(\nu_{1},...,\nu_{8})$ and $\lambda_{\nu}:=\lambda_{\tilde{\nu}}$ we
have that

$\displaystyle\lambda_{\nu}=$
$\displaystyle\frac{1}{b\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{60\gamma}\left[\left(\frac{\nu_{2}-\nu_{3}}{2}\right)^{2}+\left(\frac{\nu_{2}+\nu_{3}}{2}\right)^{2}+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}}{2}\right)^{2}+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}}{2}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}}{2}\right)^{2}+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}}{2}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\frac{\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}+2\nu_{8}}{2}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\frac{4\nu_{1}+5\nu_{2}+7\nu_{3}+10\nu_{4}+8\nu_{5}+6\nu_{6}+4\nu_{7}+2\nu_{8}}{2}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt-\left(\sum\limits_{k=2}^{7}(k-1)^{2}+23^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{240\gamma}\left[\left(\nu_{2}-\nu_{3}\right)^{2}+\left(\nu_{2}+\nu_{3}\right)^{2}+\left(\nu_{2}+\nu_{3}+2\nu_{4}\right)^{2}+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}\right)^{2}+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(\nu_{2}+\nu_{3}+2\nu_{4}+2\nu_{5}+2\nu_{6}+2\nu_{7}+2\nu_{8}\right)^{2}\right.$
$\displaystyle\left.\hskip
28.45274pt+\left(4\nu_{1}+5\nu_{2}+7\nu_{3}+10\nu_{4}+8\nu_{5}+6\nu_{6}+4\nu_{7}+2\nu_{8}\right)^{2}-2480\right],$

as we wanted to prove. ∎

$\bullet$ Diagram $\textnormal{F}_{4}.$ Now, we consider the exceptional case
of the compact Lie group $\textnormal{F}_{4}.$

######  Theorem 2.10.

The spectrum of the Laplacian operator on $\textnormal{F}_{4}$ is given by

$\textnormal{Spec(F}_{4}\textnormal{)}=\\{\lambda_{\nu}:\nu=(\nu_{1},\nu_{2},\nu_{3},\nu_{4})\in\mathbb{N}^{4}\\},$

where

$\lambda_{\nu}=\frac{1}{72\gamma}\left[(2\nu_{1}+4\nu_{2}+3\nu_{3}+2\nu_{4})^{2}+(2\nu_{1}+2\nu_{2}+\nu_{3})^{2}+(2\nu_{2}+\nu_{3})^{2}+\nu_{4}^{2}-156\right].$
(2.12)

###### Proof.

Let $\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\\}$ an
orthonormal basis of a 4-dimensional vector space endowed with an inner
product $\langle\cdot,\cdot\rangle.$ The simple roots are
$\alpha_{1}=\varepsilon_{2}-\varepsilon_{3},\
\alpha_{2}=\varepsilon_{3}-\varepsilon_{4},\ \alpha_{3}=\varepsilon_{4},$ and
$\alpha_{4}=\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4})$
and the positive roots are $\varepsilon_{i},\ 1\leq i\leq 4,\
\varepsilon_{i}\pm\varepsilon_{j},\ 1\leq i<j\leq 4,$ and
$\frac{1}{2}(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4}).$
The fundamental weights with respect to
$\Sigma=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$ are

$\overline{\omega}_{1}=\varepsilon_{1}+\varepsilon_{2},\
\overline{\omega}_{2}=2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3},\
\overline{\omega}_{3}=\frac{3\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}}{2},\
\overline{\omega}_{4}=\varepsilon_{1},$

the highest long root is

$\tilde{\alpha}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}=\varepsilon_{1}+\varepsilon_{2},$

and

$\displaystyle\eta=$
$\displaystyle\frac{1}{2}\sum\limits_{\alpha\in\Pi^{+}}\alpha$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sum\limits_{i=1}^{4}\varepsilon_{i}+\sum\limits_{1\leq
i<j\leq 4}(\varepsilon_{i}+\varepsilon_{j})+\sum\limits_{1\leq i<j\leq
4}(\varepsilon_{i}-\varepsilon_{j})\right.$ $\displaystyle\left.\hskip
14.22636pt+\frac{1}{2}\sum\limits_{i,j,k=0}^{1}(\varepsilon_{1}+(-1)^{i}\varepsilon_{2}+(-1)^{j}\varepsilon_{3}+(-1)^{k}\varepsilon_{4}\right)$
$\displaystyle=$
$\displaystyle\frac{11}{2}\varepsilon_{1}+\frac{5}{2}\varepsilon_{2}+\frac{3}{2}\varepsilon_{3}+\frac{1}{2}\varepsilon_{4},$
$\displaystyle\Longrightarrow b=$
$\displaystyle\langle\tilde{\alpha},\tilde{\alpha}\rangle+2\langle\tilde{\alpha},\eta\rangle=2+2\left(\frac{11}{2}+\frac{5}{2}\right)=18.$

For a highest weight

$\displaystyle\tilde{\nu}=$ $\displaystyle\
\tilde{\nu}_{1}\overline{\omega}_{1}+\tilde{\nu}_{2}\overline{\omega}_{2}+\tilde{\nu}_{3}\overline{\omega}_{3}+\tilde{\nu}_{4}\overline{\omega}_{4}$
$\displaystyle=$
$\displaystyle\left(\tilde{\nu}_{1}+2\tilde{\nu}_{2}+\frac{3\tilde{\nu}_{3}}{2}+\tilde{\nu}_{4}\right)\varepsilon_{1}+\left(\tilde{\nu}_{1}+\tilde{\nu}_{2}+\frac{\tilde{\nu}_{3}}{2}\right)\varepsilon_{2}+\left(\tilde{\nu}_{2}+\frac{\tilde{\nu}_{3}}{2}\right)\varepsilon_{3}+\frac{\tilde{\nu}_{4}}{2}\varepsilon_{4}$
$\displaystyle=$
$\displaystyle\left(\frac{2\tilde{\nu}_{1}+4\tilde{\nu}_{2}+3\tilde{\nu}_{3}+2\tilde{\nu}_{4}}{2}\right)\varepsilon_{1}+\left(\frac{2\tilde{\nu}_{1}+2\tilde{\nu}_{2}+\tilde{\nu}_{3}}{2}\right)\varepsilon_{2}+\left(\frac{2\tilde{\nu}_{2}+\tilde{\nu}_{3}}{2}\right)\varepsilon_{3}+\frac{\tilde{\nu}_{4}}{2}\varepsilon_{4}$

we have that

$\displaystyle\tilde{\nu}+\eta=$
$\displaystyle\left(\frac{2\tilde{\nu}_{1}+4\tilde{\nu}_{2}+3\tilde{\nu}_{3}+2\tilde{\nu}_{4}+11}{2}\right)\varepsilon_{1}+\left(\frac{2\tilde{\nu}_{1}+2\tilde{\nu}_{2}+\tilde{\nu}_{3}+5}{2}\right)\varepsilon_{2}$
$\displaystyle+\left(\frac{2\tilde{\nu}_{2}+\tilde{\nu}_{3}+3}{2}\right)\varepsilon_{3}+\left(\frac{\tilde{\nu}_{4}+1}{2}\right)\varepsilon_{4}$
$\displaystyle=$
$\displaystyle\left(\frac{2(\tilde{\nu}_{1}+1)+4(\tilde{\nu}_{2}+1)+3(\tilde{\nu}_{3}+1)+2(\tilde{\nu}_{4}+1)}{2}\right)\varepsilon_{1}$
$\displaystyle+\left(\frac{2(\tilde{\nu}_{1}+1)+2(\tilde{\nu}_{2}+1)+(\tilde{\nu}_{3}+1)}{2}\right)\varepsilon_{2}+\left(\frac{2(\tilde{\nu}_{2}+1)+(\tilde{\nu}_{3}+1)}{2}\right)\varepsilon_{3}$
$\displaystyle+\left(\frac{\tilde{\nu}_{4}+1}{2}\right)\varepsilon_{4}$
$\displaystyle=$
$\displaystyle\left(\frac{2\nu_{1}+4\nu_{2}+3\nu_{3}+2\nu_{4}}{2}\right)\varepsilon_{1}+\left(\frac{2\nu_{1}+2\nu_{2}+\nu_{3}}{2}\right)\varepsilon_{2}+\left(\frac{2\nu_{2}+\nu_{3}}{2}\right)\varepsilon_{3}+\frac{\nu_{4}}{2}\varepsilon_{4}$

where $\nu_{j}=\tilde{\nu}_{j}+1,\ j=1,2,3,4.$ Hence,
$\nu:=(\nu_{1},\nu_{2},\nu_{3},\nu_{4})\in\mathbb{N}^{4}$ and the
corresponding eigenvalue $\lambda_{\nu}:=\lambda_{\tilde{\nu}}$ is

$\displaystyle\lambda_{\nu}=$
$\displaystyle\frac{1}{b\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{18\gamma}\left[\left(\frac{2\nu_{1}+4\nu_{2}+3\nu_{3}+2\nu_{4}}{2}\right)^{2}+\left(\frac{2\nu_{1}+2\nu_{2}+\nu_{3}}{2}\right)^{2}+\left(\frac{2\nu_{2}+\nu_{3}}{2}\right)^{2}+\left(\frac{\nu_{4}}{2}\right)^{2}\right.$
$\displaystyle\left.\hskip
22.76228pt-\left(\left(\frac{11}{2}\right)^{2}+\left(\frac{5}{2}\right)^{2}+\left(\frac{3}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{72\gamma}\left[(2\nu_{1}+4\nu_{2}+3\nu_{3}+2\nu_{4})^{2}+(2\nu_{1}+2\nu_{2}+\nu_{3})^{2}+(2\nu_{2}+\nu_{3})^{2}+\nu_{4}^{2}-156\right],$

which completes the proof. ∎

$\bullet$ Diagram $\textnormal{G}_{2}.$ Now, we consider the exceptional case
of the compact Lie group $\textnormal{G}_{2}.$

######  Theorem 2.11.

The spectrum of the Laplacian operator on $\textnormal{G}_{2}$ is given by

$\textnormal{Spec(G}_{2}\textnormal{)}=\\{\lambda_{\nu}:\nu=(\nu_{1},\nu_{2})\in\mathbb{N}\times\mathbb{N}\\},$

where

$\lambda_{\nu}=\frac{1}{24\gamma}\left[(\nu_{1}+\nu_{2})^{2}+(\nu_{1}+2\nu_{2})^{2}+\nu_{2}^{2}-14\right].$
(2.13)

###### Proof.

The root system $\textnormal{G}_{2}$ is constructed in a 3-dimensional space
by considering the subspace
$\\{\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\\}^{\perp}$ with respect
to an inner product $\langle\cdot,\cdot\rangle$ and a
$\langle\cdot,\cdot\rangle$-orthonormal basis
$\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\\}.$ The sets of positive
roots and simple positive roots are

$\Pi^{+}=\\{\varepsilon_{1}-\varepsilon_{2},\varepsilon_{3}-\varepsilon_{2},\varepsilon_{3}-\varepsilon_{1},\varepsilon_{2}+\varepsilon_{3}-2\varepsilon_{1},\varepsilon_{1}+\varepsilon_{3}-2\varepsilon_{2},2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2}\\}$

and

$\Sigma=\\{\alpha_{1}=\varepsilon_{1}-\varepsilon_{2},\alpha_{2}=\varepsilon_{2}+\varepsilon_{3}-2\varepsilon_{1}\\}$

respectively. The fundamental weights with respect to $\Sigma$ are

$\overline{\omega}_{1}=2\alpha_{1}+\alpha_{2}=\varepsilon_{3}-\varepsilon_{2},\
\overline{\omega}_{2}=3\alpha_{1}+2\alpha_{2}=2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2},$

the highest long root is

$\tilde{\alpha}=3\alpha_{1}+2\alpha_{2}=2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2}$

and

$\displaystyle\eta=$
$\displaystyle\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}+\varepsilon_{3}-\varepsilon_{2}+\varepsilon_{3}-\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}-2\varepsilon_{1}+\varepsilon_{1}+\varepsilon_{3}-2\varepsilon_{2}+2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2})$
$\displaystyle=$
$\displaystyle-\varepsilon_{1}-2\varepsilon_{2}+3\varepsilon_{3}$

so that

$b=\langle\tilde{\alpha},\tilde{\alpha}\rangle+2\left\langle\tilde{\alpha},\eta\right\rangle=6+2(9)=24.$

A highest weight
$\tilde{\nu}=\tilde{\nu}_{1}\overline{\omega}_{1}+\tilde{\nu}_{2}\overline{\omega}_{2}$
is written in the basis
$\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\\}$ as

$\tilde{\nu}=\tilde{\nu}_{1}(\varepsilon_{3}-\varepsilon_{2})+\tilde{\nu}_{2}(2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2})=-\tilde{\nu}_{2}\varepsilon_{1}-(\tilde{\nu}_{1}+\tilde{\nu}_{2})\varepsilon_{2}+(\tilde{\nu}_{1}+2\tilde{\nu}_{2})\varepsilon_{3},$

hence

$\tilde{\nu}+\eta=-(\tilde{\nu}_{2}+1)\varepsilon_{1}-(\tilde{\nu}_{1}+\tilde{\nu}_{2}+2)\varepsilon_{2}+(\tilde{\nu}_{1}+2\tilde{\nu}_{2}+3)\varepsilon_{3}.$

We can now find the eigenvalue $\lambda_{\tilde{\nu}}$ associated to
$\tilde{\nu}:$

$\displaystyle\lambda_{\tilde{\nu}}=$
$\displaystyle\frac{1}{b\gamma}\left[\langle\tilde{\nu}+\eta,\tilde{\nu}+\eta\rangle-\langle\eta,\eta\rangle\right]$
$\displaystyle=$
$\displaystyle\frac{1}{24\gamma}\left[(\tilde{\nu}_{2}+1)^{2}+(\tilde{\nu}_{1}+\tilde{\nu}_{2}+2)^{2}+(\tilde{\nu}_{1}+2\tilde{\nu}_{2}+3)^{2}-((-1)^{2}+(-2)^{2}+3^{2})\right]$
$\displaystyle=$
$\displaystyle\frac{1}{24\gamma}\left[\nu_{2}^{2}+(\nu_{1}+\nu_{2})^{2}+(\nu_{1}+2\nu_{2})^{2}-14\right],$

with $\nu_{j}:=\tilde{\nu}_{j},\ j=1,2.$ As before, we define
$\nu:=(\nu_{1},\nu_{2})\in\mathbb{N}^{2}$ and
$\lambda_{\nu}:=\lambda_{\tilde{\nu}}$ and obtain the result. ∎

The following important theorem summarises all the previous results.

######  Theorem 2.12.

Let $G$ be a connected compact simple Lie group of rank $\ell\geq 2$ and
$\tilde{G}$ its universal cover. Then there exist $b,R_{0}\in\mathbb{N}$ and
$\Omega\subseteq\mathbb{N}^{\ell}$ such that the spectrum of the Laplace
operator on $G$ is

$\textnormal{Spec($G$)}=\\{\lambda_{\nu}:\nu=(\nu_{1},...,\nu_{\ell})\in\Omega\\},$
(2.14)

where $\lambda_{\nu}$ is given by the formula

$\displaystyle\lambda_{\nu}=\frac{1}{b\gamma}(n_{1}^{2}+\cdots+n_{m}^{2}-R_{0}),$
(2.15)

with

$m=\left\\{\begin{array}[]{lll}\ell+1,&&\textnormal{if}\
\tilde{G}\cong\textnormal{SU($\ell+1$)}\ \textnormal{or}\ \tilde{G}\cong
G_{2},\\\ \ell,&&\textnormal{otherwise},\end{array}\right.$

and

$[n_{1}\ \cdots\ n_{m}]^{T}=A[\nu_{1}\ \cdots\ \nu_{\ell}]^{T}$

for some integer $m\times\ell$ matrix $A$ with maximal rank equal to $\ell.$

###### Proof.

By Remark 2.4 we may assume that $G$ is simply-connected. As a consequence of
Theorems 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, and 2.11 we can conclude that the
spectrum of the Laplace operator on $G$ has, in fact, the form of (2.14),
where $b$ and $R_{0}$ are as in the following table

Type of the Lie algebra | $b$ | $R_{0}$
---|---|---
$A_{\ell}$ | $2(\ell+1)^{3}$ | $\frac{1}{4}\sum\limits_{k=1}^{\ell+1}(\ell-2(k-1))^{2}(\ell+1)^{2}$
$B_{\ell}$ | $16\ell-8$ | $\sum\limits_{k=1}^{\ell}\left(2(\ell-k)+1\right)^{2}$
$C_{\ell}$ | $4(\ell+1)$ | $\frac{\ell(\ell+1)(2\ell+1)}{6}$
$D_{\ell}$ | $16\ell-16$ | $\sum\limits_{k=1}^{\ell}\left(2(\ell-k)\right)^{2}$
$\textnormal{E}_{8}$ | 240 | 2480
$\textnormal{F}_{4}$ | 72 | 156
$\textnormal{G}_{2}$ | 24 | 14
Table 1. Values of $b$ and $R_{0}.$

and in the formula $[n_{1}\ \cdots\ n_{m}]^{T}=A[\nu_{1}\ \cdots\
\nu_{\ell}]^{T}$ we have that

$\displaystyle A=\begin{bmatrix}\ell&\ell-1&\ell-2&\ell-3&\cdots&2&1\\\
-1&\ell-1&\ell-2&\ell-3&\cdots&2&1\\\ -1&-2&\ell-2&\ell-3&\cdots&2&1\\\
-1&-2&-3&\ell-3&\cdots&2&1\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
-1&-2&-3&-4&\cdots&-(\ell-1)&1\\\ 1&2&3&4&\cdots&\ell-1&\ell\\\
\end{bmatrix}_{(\ell+1)\times\ell},\ \textnormal{if $G$ is of type
$A_{\ell}$},$ $\displaystyle A=\begin{bmatrix}2&2&\cdots&2&1\\\
0&2&\cdots&2&1\\\ 0&0&\cdots&2&1\\\ 0&0&\cdots&2&1\\\ 0&0&\cdots&0&1\\\
\end{bmatrix}_{\ell\times\ell},\ \textnormal{if $G$ is of type $B_{\ell}$,}$
$\displaystyle A=\begin{bmatrix}1&1&1&1&\cdots&1&1\\\ 0&1&1&1&\cdots&1&1\\\
0&0&1&1&\cdots&1&1\\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&0&\cdots&1&1\\\ 0&0&0&0&\cdots&0&1\\\ \end{bmatrix}_{\ell\times\ell},\
\textnormal{if $G$ is of type $C_{\ell}$,}$ $\displaystyle
A=\begin{bmatrix}2&2&2&\cdots&2&1&1\\\ 0&2&2&\cdots&2&1&1\\\
0&0&2&\cdots&2&1&1\\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&2&1&1\\\ 0&0&0&\cdots&0&1&1\\\ 0&0&0&\cdots&0&-1&1\\\
\end{bmatrix}_{\ell\times\ell},\ \textnormal{if $G$ is of type $D_{\ell}$},$
$\displaystyle A=\begin{bmatrix}0&1&-1&0&0&0&0&0\\\ 0&1&1&0&0&0&0&0\\\
0&1&1&2&0&0&0&0\\\ 0&1&1&2&2&0&0&0\\\ 0&1&1&2&2&2&0&0\\\ 0&1&1&2&2&2&2&0\\\
0&1&1&2&2&2&2&2\\\ 4&5&7&10&8&6&4&2\\\ \end{bmatrix},\ \textnormal{if $G\cong
E_{8}$},$ $\displaystyle A=\begin{bmatrix}2&4&3&2\\\ 2&2&1&0\\\ 0&2&1&0\\\
0&0&1&0\\\ \end{bmatrix},\ \textnormal{if $G\cong$F}_{4},\ \textnormal{and}\
A=\begin{bmatrix}1&1\\\ 1&2\\\ 0&1\\\ \end{bmatrix},\ \textnormal{if $G\cong
G_{2}.$}$

It is easy to see that all these matrices have all their columns linearly
independent which completes the proof. ∎

###  2.2. Tools from analytic number theory

In this section we record some results from the analytic number theory applied
in this manuscript related to the Waring problem. For our further analysis, we
are interested in the problem of analysing the number of solutions in positive
integers $n_{1},n_{2},\cdots,n_{s}$ of the equation

$n_{1}^{k}+\cdots n_{s}^{k}=R,\quad R\in\mathbb{N},$ (2.16)

which is denoted by $r_{s,k}(R).$ This problem has a long history. The case
$k=2,$ that is counting representations of a number by sums of $s$-squares is
well understood in view of the circle method due to Hardy, Littlewood and
Ramanujan [12, 13, 14, 15, 16, 17, 18, 19, 26, 27], with other developements
e.g. by Bateman [1] and Kloosterman [23]. The case $k\geq 3$ is extremely
exciting. With earlier fundamental contributions from Hardy and Littlewood and
further developments by Vinogradov [31] the problem of estimating $r_{s,k}(N)$
when $k\geq 3,$ is related to both fields, analytic number theory and harmonic
analysis. It is related to the Vinogradov mean value theorem proved in 2015 by
Bourgain, Demeter and Guth [7], using the decoupling theorem of Bourgain and
Demeter [6]. The work [6] and further developments have shown deep connections
between analytic number theory and other fundamental problems in Fourier
analysis, like the Stein restriction conjecture and the Kakeya problem. We
refer to Pierce [25] for a complete historical perspective on the subject.

By the structure of the spectrum of the Laplace operator on a compact Lie
group, we are interested in the case $k=2,$ and in the case were $s\geq 2.$ In
this case the estimates for $r_{s,k}(R)$ are essentially deduced from the
Hardy-Littlewood-Ramanujan circle method. We record the following estimates of
$r_{s,2}(R)$ for $s\geq 2,$ for our further analysis.

######  Theorem 2.13 (Representing a number as sums of squares).

Let $R\in\mathbb{N}.$ For $k=2$ and $s\in\mathbb{N},$ with $s\geq 2,$ the
following statements hold.

  * •

Let $s=2.$ Then, for any $\varepsilon>0,$ there exists $C_{\varepsilon}>0,$
such that

$r_{2,2}(R)\leq C_{\varepsilon}R^{\varepsilon},\,R\geq 1.$ (2.17)

  * •

Let $s=3.$ Then,

$r_{3,2}(R)=2\pi
R^{\frac{1}{2}}\mathfrak{S}_{3}(R),\,\,|\mathfrak{S}_{3}(R)|\leq
C|\log(4R)(\log(\log(4R)))|,\,\,R\geq e^{e}/4.$ (2.18)

  * •

Let $s=4.$ Then,

$r_{4,2}(R)=\pi^{2}R\mathfrak{S}_{4}(R),\,\,R\geq
e^{e},\,\,\,\limsup_{n\rightarrow\infty}\frac{\mathfrak{S}_{4}(R)}{\log(\log(R))}=6\pi^{-2}e^{\gamma_{*}},$
(2.19)

where

$\gamma_{*}=0.57721566490153286060651209008240243104215933593992\cdots,$

is the Euler constant.

  * •

For $s\geq 5,$ there are constants $c_{1}(s)$ and $c_{2}(s)$ such that

$c_{1}(s)R^{\frac{s-2}{2}}\leq r_{s,2}(R)\leq c_{2}(s)R^{\frac{s-2}{2}}.$
(2.20)

######  Remark 2.14.

For (2.17) and a historical conjecture about the asymptotic behaviour of the
terms $r_{k,k}(R)$ (the Conjecture K) we refer the reader to Mahler [24] as
well as the discussion there about the circle method. For (2.18) we refer the
reader to Bateman [1, Page 84] and for (2.19) we refer the reader to Bateman
[1, Page 79]. For the historical aspects about the estimate in (2.20) we refer
the reader to Wainger [32]. In general, the circle method establishes that for
$s$ large enough with respect to $k,$ one has the expression

$r_{s,k}(R)=\frac{\Gamma(1+k/s)}{\Gamma(s/k)}\mathfrak{S}_{s}(R)+O(R^{s/k-1-\delta}),$
(2.21)

where $\delta>0,$ and $\mathfrak{S}_{s}(R)$ being a singular series. In
particular, for $s>2^{k},$ the precise statement about (2.21) can be found in
[21, Page 456].

##  3\. Strichartz estimates

In this section, we analyse the Strichartz estimates on compact connected
simple Lie groups. It was mentioned in the introduction, we exclude the case
$\ell=1$ since
$\textnormal{Spin(3)}\cong\textnormal{SU}(2)\cong\mathbb{S}^{3}$ and the
Strichartz estimates in the case of the spheres have been analysed in the work
of Burq, Gerard, and Tzvetkov [8]. For this, we need several preparatory
useful lemmas. We start our analysis with the following result due to Sogge
and Hörmander about the $L^{p}$-norm for the spectral projections of the
Laplacian, see [28, 29] and Zhang [36, Theorem 2] for the improved estimate in
the case $\frac{4d}{d-\ell}<q\leq\infty.$

######  Lemma 3.1 (Sogge-Hörmander).

Let $G$ be a compact Lie group of rank $\ell.$ Let $p\in[2,\infty]$ and
$P_{\lambda_{\nu}}:L^{2}(G)\rightarrow H_{\lambda_{\nu}}$ be the orthogonal
projection onto the eigenspace $H_{\lambda_{\nu}}$ corresponding to
$\lambda_{\nu}$. Then

$\left|\left|P_{\lambda_{\nu}}u_{0}\right|\right|_{L^{p}(G)}\leq
C\lambda_{\nu}^{s(p)}\left|\left|P_{\lambda_{\nu}}u_{0}\right|\right|_{L^{2}(G)},$
(3.1)

where

$s(p):=\left\\{\begin{array}[]{lll}\frac{d-1}{4}\left(\frac{1}{2}-\frac{1}{p}\right),&&\textnormal{if}\
2\leq p\leq\frac{2(d+1)}{d-1},\\\ \\\
\frac{d-1}{4}-\frac{d}{2p},&&\textnormal{if}\
\frac{2(d+1)}{d-1}<p\leq\frac{4d}{d-\ell},\\\ \\\
\frac{d-\ell}{4}-\frac{d}{2p},&&\textnormal{if}\
\frac{4d}{d-\ell}<p\leq\infty,\end{array}\right.$

and with the constant $C$ in (3.1) depending on geometric data of $G.$

######  Remark 3.2.

Note that we always consider $d\geq\ell\geq 2.$ So, observe that for all
$p\in[2,\infty],$ we have that $s(p)\geq 0.$ This is clear when $p$ belongs to
the interval $[2,\frac{2(d+1)}{d-1}].$ Indeed, in this case we have that
$s(p)=\frac{d-1}{4}(\frac{1}{2}-\frac{1}{p})$ since the inequality $p\geq 2,$
implies that

$\frac{1}{2}-\frac{1}{p}\geq 0.$

On the other hand, when $p$ belongs to the interval
$(\frac{2(d+1)}{d-1},\frac{4d}{d-\ell}],$ one can use that $p>2(d+1)/(d-1),$
in order to estimate $1/2p<(d-1)/4(d+1).$ Then one can deduce that

$s(p)=\frac{d-1}{4}-\frac{d}{2p}>\frac{d-1}{4}-\frac{d-1}{4}\times\frac{d}{d-1}>0.$

Finally, note that for $p>4d/(d-\ell),$ we have that $1/p<(d-\ell)/4d,$ and
then $d/2p<(d-\ell)/8.$ In this case

$s(p)=\frac{d-\ell}{4}-\frac{d}{2p}>\frac{d-\ell}{4}-\frac{d-\ell}{8}>0.$

The following lemma will be crucial in our further analysis. We provide some
crude Strichartz estimates that for a range of values of $(p,q)$ that could be
non-sharp with respect to the values provided by Burq, Gerard and Tzvetkov
[8], specially if the pair $(p,q)$ is admissible. However, in the next
sections we will clarify, for which values of $(p,q)$ the following lemma
provides improvements of the regularity orders in [8], in our setting. As we
will analyse, all our results give improvements in the case of Bourgain's
exponents $p=q,$ see Remark 1.4. We observe that these improvements are with
respect to the established regularity orders for the Lie groups that we
analyse below, but not in the setting of the manifolds considered in [8],
where their result is sharp on any manifold having closed geodesics with a
common period. As it was mentioned in the introduction, we conclude that the
geometry of the manifold has a strong incidence in the improvement of
Strichartz estimates.

######  Lemma 3.3.

Let $p\in[2,\infty),$ $q\in[2,\infty],$ and let $b$ be as in Table 1. Let
$T=2\pi b\gamma.$ Then

$\left|\left|u\right|\right|_{L^{q}(G,L^{p}[0,T])}\lesssim\left[\sum\limits_{R=0}^{\infty}N_{R}R^{2(\alpha+s(q))}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right]^{\frac{1}{2}},$

where, for any $R\in\mathbb{N},$
$N_{R}=\\#\left\\{\nu:\lambda_{\nu}=\frac{R}{b\gamma}\right\\}$ is the number
of solutions of the equation $\lambda_{\nu}=\frac{R}{b\gamma},$ and
$\alpha=1/2-1/p$ is a loss of regularity order of the $L^{p}$-estimate with
respect to the $L^{2}$-norm in the time variable. Moreover, for $p=2,$ we have
the identity

$\forall x\in
G,\,||u(\cdot,x)||_{L^{2}([0,T])}=\left[\sum\limits_{R=0}^{\infty}\
2TN_{R}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}\right]^{\frac{1}{2}}.$
(3.2)

###### Proof of Lemma 3.3.

Since $u_{0}\in C^{\infty}(G)$, the solution of the linear Schrödinger
equation is given by

$\forall t\in[0,T],\,\forall x\in
G,\,\,u(t,x)=\displaystyle\sum\limits_{\nu}e^{i\lambda_{\nu}t}P_{\lambda_{\nu}}u_{0}(x).$

In order to estimate the $L^{p}([0,T])$-norm of $u$, we shall use the Wainger-
Sobolev embedding theorem, which states that for $1<r<p<\infty,$ we have that

$\left|\left|\sum\limits_{n\in\mathbb{Z}}a_{n}e^{itn}\right|\right|_{L^{p}([0,2\pi])}\lesssim\left|\left|\sum_{n\in\mathbb{Z}}|n|^{\alpha}a_{n}e^{itn}\right|\right|_{L^{r}([0,2\pi])},\
\alpha=\frac{1}{r}-\frac{1}{p}.$

In particular, when $r=2,$ and for any fixed $x\in G,$ let us consider the
function

$f(t)=\sum_{R=0}^{\infty}a_{R}e^{itR},$

defined by the Fourier coefficients

$a_{R}=\sum_{\nu\in\mathbb{N}^{n}:b\gamma\lambda_{\nu}=R}P_{\lambda_{\nu}}u_{0}(x),$

or in other words, defined by

$f(t)=\sum_{R=0}^{\infty}a_{R}e^{itR}=\sum_{\nu}e^{itb\gamma\lambda_{\nu}}P_{\lambda_{\nu}}u_{0}(x).$

Then the Wainger Sobolev embedding theorem implies that

$\displaystyle\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}b\gamma\tau}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{p}([0,2\pi])}$
$\displaystyle=\left|\left|\sum\limits_{R=0}^{\infty}a_{R}e^{iR\tau}\right|\right|_{L^{p}([0,2\pi])}$
$\displaystyle\lesssim\left|\left|\sum\limits_{R=0}^{\infty}R^{\alpha}a_{R}e^{iR\tau}\right|\right|_{L^{2}([0,2\pi])}$
$\displaystyle=\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}b\gamma\tau}(\lambda_{\nu}b\gamma)^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{2}([0,2\pi])}.$

To estimate the $L^{p}$-norm $||u(t,x)||_{L^{p}([0,2\pi b\gamma])},$ let us
apply the change of variables $t\mapsto\tau,$ $t:=b\gamma\tau,$ and the
estimate $L^{p}$-$L^{2}$-estimate above as follows,

$\displaystyle||u(t,x)||_{L^{p}([0,2\pi b\gamma])}$
$\displaystyle=\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}t}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{p}([0,2\pi
b\gamma])}$
$\displaystyle=(b\gamma)^{\frac{1}{p}}\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}b\gamma\tau}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{p}([0,2\pi])}$
$\displaystyle\lesssim(b\gamma)^{\frac{1}{p}}\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}b\gamma\tau}(\lambda_{\nu}b\gamma)^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{2}([0,2\pi])}.$

The change of variables $\tau\mapsto s,$ $s=b\gamma\tau,$ gives

$\displaystyle(b\gamma)^{\frac{1}{p}}\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}b\gamma\tau}(\lambda_{\nu}b\gamma)^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{2}([0,2\pi])}$
$\displaystyle=(b\gamma)^{\frac{1}{p}-\frac{1}{2}}\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}s}(\lambda_{\nu}b\gamma)^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{2}([0,2\pi
b\gamma])}$
$\displaystyle=(b\gamma)^{\frac{1}{p}-\frac{1}{2}}(b\gamma)^{\alpha}\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}s}\lambda_{\nu}^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{2}([0,2\pi
b\gamma])}$
$\displaystyle=\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}s}\lambda_{\nu}^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|_{L^{2}([0,2\pi
b\gamma])},$

where $\alpha=1/2-1/p.$ The last expression above can be computed as follows:

$\displaystyle\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}s}\lambda_{\nu}^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|^{2}_{L^{2}([0,T])}$
$\displaystyle=\int_{0}^{T}\left|\sum\limits_{\nu}e^{i\lambda_{\nu}s}\lambda_{\nu}^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|^{2}ds$
$\displaystyle=\int_{0}^{T}\left(\sum\limits_{\nu}e^{i\lambda_{\nu}s}\lambda_{\nu}^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right)\left(\sum\limits_{\nu^{\prime}}e^{-i\lambda_{\nu^{\prime}}s}\lambda_{\nu^{\prime}}^{\alpha}\overline{P_{\lambda_{\nu^{\prime}}}u_{0}(x)}\right)ds$
$\displaystyle=\int_{0}^{T}\left(\sum\limits_{\nu}\sum\limits_{\nu^{\prime}}e^{i(\lambda_{\nu}-\lambda_{\nu^{\prime}})s}(\lambda_{\nu}\lambda_{\nu^{\prime}})^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\overline{P_{\lambda_{\nu^{\prime}}}u_{0}(x)}\right)ds$
$\displaystyle=\sum\limits_{\begin{subarray}{c}\nu,\nu^{\prime}\\\
\lambda_{\nu}=\lambda_{\nu^{\prime}}\end{subarray}}\left[\int_{0}^{T}e^{i(\lambda_{\nu}-\lambda_{\nu^{\prime}})s}ds\right](\lambda_{\nu}\lambda_{\nu^{\prime}})^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\overline{P_{\lambda_{\nu^{\prime}}}u_{0}(x)}$
$\displaystyle\ \ \ +\sum\limits_{\begin{subarray}{c}\nu,\nu^{\prime}\\\
\lambda_{\nu}\neq\lambda_{\nu^{\prime}}\end{subarray}}\left[\int_{0}^{T}e^{i(\lambda_{\nu}-\lambda_{\nu^{\prime}})s}ds\right](\lambda_{\nu}\lambda_{\nu^{\prime}})^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\overline{P_{\lambda_{\nu^{\prime}}}u_{0}(x)}.$

In the last equality, we can interchange the integration and summation symbols
since $u_{0}\in C^{\infty}(G).$ Recalling that $T=2\pi b\gamma,$ we have that

$\int_{0}^{T}e^{i(\lambda_{\nu}-\lambda_{\nu^{\prime}})s}ds=\left\\{\begin{array}[]{lll}T,&&\textnormal{if}\
\lambda_{\nu}=\lambda_{\nu^{\prime}},\\\ 0,&&\textnormal{if}\
\lambda_{\nu}\neq\lambda_{\nu^{\prime}}.\end{array}\right.$

Hence, these orthogonality relations allow us to write the equalities

$\displaystyle\left|\left|\sum\limits_{\nu}e^{i\lambda_{\nu}s}\lambda_{\nu}^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\right|\right|^{2}_{L^{2}([0,T])}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}\nu,\nu^{\prime}\\\
\lambda_{\nu}=\lambda_{\nu^{\prime}}\end{subarray}}T(\lambda_{\nu}\lambda_{\nu^{\prime}})^{\alpha}P_{\lambda_{\nu}}u_{0}(x)\overline{P_{\lambda_{\nu^{\prime}}}u_{0}(x)}$
$\displaystyle=\sum\limits_{R=0}^{\infty}\
\sum\limits_{\nu:\lambda_{\nu}=\frac{R}{b\gamma}}\
\sum\limits_{\nu^{\prime}:\lambda_{\nu^{\prime}}=\frac{R}{b\gamma}}T\left(\frac{R}{b\gamma}\right)^{2\alpha}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}$
$\displaystyle=\sum\limits_{R=0}^{\infty}\
2T\left(\frac{R}{b\gamma}\right)^{2\alpha}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}\sum\limits_{\nu:\lambda_{\nu}=\frac{R}{b\gamma}}1$
$\displaystyle=\sum\limits_{R=0}^{\infty}\
2T\left(\frac{R}{b\gamma}\right)^{2\alpha}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}N_{R},$

where

$N_{R}:=\sum\limits_{\nu:\lambda_{\nu}=\frac{R}{b\gamma}}1=\\#\left\\{\nu:\lambda_{\nu}=\frac{R}{b\gamma}\right\\}.$

So far, we have shown that

$||u(t,x)||^{2}_{L^{p}([0,T])}\lesssim\sum\limits_{R=0}^{\infty}\
2T\left(\frac{R}{b\gamma}\right)^{2\alpha}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}N_{R},$

or, equivalently,

$||u(t,x)||_{L^{p}([0,T])}\lesssim\left[\sum\limits_{R=0}^{\infty}\
2T\left(\frac{R}{b\gamma}\right)^{2\alpha}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}N_{R}\right]^{\frac{1}{2}},$

and also the following equality when $p=2,$

$||u(t,x)||_{L^{2}([0,T])}=\left[\sum\limits_{R=0}^{\infty}\
2T\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}N_{R}\right]^{\frac{1}{2}}.$

By computing the $L^{q}(G)$-norm of $||u(t,x)||_{L^{p}([0,T])}$ we have that

$\displaystyle||u(t,x)||_{L^{q}(G,L^{p}[0,T])}$
$\displaystyle=\left\\{\int_{G}||u(t,x)||^{q}_{L^{p}([0,T])}dx\right\\}^{\frac{1}{q}}$
$\displaystyle\lesssim\left\\{\int_{G}\left[\sum\limits_{R=0}^{\infty}\
2T\left(\frac{R}{b\gamma}\right)^{2\alpha}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}N_{R}\right]^{\frac{q}{2}}dx\right\\}^{\frac{1}{q}}$
$\displaystyle\leq\left\\{\sum\limits_{R=0}^{\infty}\left[\int_{G}\left(2T\left(\frac{R}{b\gamma}\right)^{2\alpha}N_{R}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}\right)^{\frac{q}{2}}dx\right]^{\frac{2}{q}}\right\\}^{\frac{1}{2}},$

where the last inequality is nothing but the Minkowski integral inequality.
Note that

$\displaystyle\left\\{\sum\limits_{R=0}^{\infty}\left[\int_{G}\left(2T\left(\frac{R}{b\gamma}\right)^{2\alpha}N_{R}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{2}\right)^{\frac{q}{2}}dx\right]^{\frac{2}{q}}\right\\}^{\frac{1}{2}}$
$\displaystyle=\left\\{\sum_{R=0}^{\infty}2T\left(\frac{R}{b\gamma}\right)^{2\alpha}N_{R}\left[\int_{G}\left|P_{\frac{R}{b\gamma}}u_{0}(x)\right|^{q}dx\right]^{\frac{2}{q}}\right\\}^{\frac{1}{2}}$
$\displaystyle=\left\\{\sum_{R=0}^{\infty}2T\left(\frac{R}{b\gamma}\right)^{2\alpha}N_{R}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{q}(G)}\right\\}^{\frac{1}{2}},$

and, by Lemma 3.1, we know that
$\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{q}(G)}\lesssim\left(\frac{R}{b\gamma}\right)^{2s(q)}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}$
so that

$\displaystyle\left\\{\sum_{R=0}^{\infty}2T\left(\frac{R}{b\gamma}\right)^{2\alpha}N_{R}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{q}(G)}\right\\}^{\frac{1}{2}}$
$\displaystyle\lesssim\left\\{\sum_{R=0}^{\infty}2T\left(\frac{R}{b\gamma}\right)^{2\alpha}N_{R}\left(\frac{R}{b\gamma}\right)^{2s(q)}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right\\}^{\frac{1}{2}}$
$\displaystyle=\frac{\sqrt{2T}}{(b\gamma)^{(\alpha+s(q))}}\left[\sum\limits_{R=0}^{\infty}N_{R}R^{2(\alpha+s(q))}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right]^{\frac{1}{2}}.$

Hence

$||u(t,x)||_{L^{q}(G,L^{p}[0,T])}\lesssim\frac{\sqrt{2T}}{(b\gamma)^{(\alpha+s(q))}}\left[\sum\limits_{R=0}^{\infty}N_{R}R^{2(\alpha+s(q))}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right]^{\frac{1}{2}},$

and the proof is complete. ∎

######  Theorem 3.4 (Strichartz estimates for compact simple Lie groups).

Let $2\leq p<\infty.$ Let $G$ be a compact connected simple Lie group of rank
$\ell\geq 2$, let $b$ as in Theorem 2.12, and let $m$ be defined by

$m=\left\\{\begin{array}[]{lll}\ell+1,&&\textnormal{if}\
\tilde{G}\cong\textnormal{SU($\ell+1$)}\ \textnormal{or}\ \tilde{G}\cong
G_{2},\\\ \ell,&&\textnormal{otherwise},\end{array}\right.$

where $\tilde{G}$ is the universal cover of $G$. Let $T=2\pi b\gamma.$ Then

$\left|\left|u\right|\right|_{L^{q}(G,L^{p}[0,2\pi
b\gamma])}\lesssim_{s}\|u_{0}\|_{H^{s}(G)},$ (3.3)

where $s\geq\frac{m}{4}-\frac{1}{p}+s(q)$ if $m\geq 5,$
$s>\frac{m}{4}-\frac{1}{p}+s(q)$ if $m=2,3,4$ and

$s(q):=\left\\{\begin{array}[]{lll}\frac{d-1}{4}\left(\frac{1}{2}-\frac{1}{q}\right),&&\textnormal{if}\
2\leq q\leq\frac{2(d+1)}{d-1},\\\ \\\
\frac{d-1}{4}-\frac{d}{2q},&&\textnormal{if}\
\frac{2(d+1)}{d-1}<q\leq\frac{4d}{d-\ell},\\\ \\\
\frac{d-\ell}{4}-\frac{d}{2q},&&\textnormal{if}\
\frac{4d}{d-\ell}<q\leq\infty.\end{array}\right.$

###### Proof.

Let $G$ be a compact connected simple Lie group of rank $\ell\geq 2$. By
Theorem 2.12, there exist integers $b,R_{0}$ and an integer matrix of rank
$\ell$ such that every eigenvalue of the Laplacian $\mathcal{L}_{G}$ can be
written as

$\lambda_{\nu}=\frac{1}{b\gamma}\left(n_{1}^{2}+\cdots+n_{m}^{2}-R_{0}\right),$
(3.4)

where $[n_{1}\ \cdots\ n_{m}]^{T}=A[\nu_{1}\ \cdots\nu_{\ell}]^{T}$ for some
$(\nu_{1},...,\nu_{\ell})\in\mathbb{N}^{\ell}.$ On the other hand, in view of
Lemma 3.3 we have the following estimate

$\left|\left|u\right|\right|_{L^{q}(G,L^{p}[0,2\pi
b\gamma])}\lesssim\left[\sum\limits_{R=0}^{\infty}N_{R}R^{2(\alpha+s(q))}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right]^{\frac{1}{2}},$
(3.5)

with

$N_{R}=\\#\left\\{\nu:\lambda_{\nu}=\frac{R}{b\gamma}\right\\}$

and $\alpha=1/2-1/p$. To estimate $N_{R}$ observe that since $A$ has maximal
rank equal to $\ell$, then the linear map $T_{A}:\nu\mapsto A\nu,$ from
$\mathbb{R}^{\ell}$ to $\mathbb{R}^{m}$ is injective. Define the exceptional
set

$\mathscr{N}_{R}=\\{Y=(n_{i})_{1\leq i\leq
m}\in\textnormal{Im}(T_{A}):n_{1}^{2}+\cdots+n_{m}^{2}=R+R_{0},\,\,\textnormal{
and }A\nu=Y\Longrightarrow\nu\in\mathbb{N}^{\ell}\\}.$ (3.6)

Since there is a bijective correspondence between elements
$Y=(n_{1},n_{2},\cdots,n_{m})\in\mathscr{N}_{R}$ with solutions of the
equation $\lambda_{\nu}=\frac{R}{b\gamma},$ that indeed is provided by the
identity $A\nu=Y$, we have that

$N_{R}=\\#\left\\{\nu:\lambda_{\nu}=\frac{R}{b\gamma}\right\\}=\\#\mathscr{N}_{R}\leq
r_{m,2}(R+R_{0}).$ (3.7)

Recalling Theorem 2.13 we obtain the following inequalities for any
$\varepsilon>0$:

$\displaystyle\bullet\ r_{2,2}(R+R_{0})\lesssim_{\varepsilon}$
$\displaystyle(R+R_{0})^{\varepsilon}\lesssim_{\varepsilon}R^{\epsilon},$
$\displaystyle\bullet\ r_{3,2}(R+R_{0})\lesssim\ $
$\displaystyle(R+R_{0})^{\frac{1}{2}}[\log(4(R+R_{0}))(\log(\log(4(R+R_{0})))]$
$\displaystyle\lesssim_{\varepsilon}$
$\displaystyle(R+R_{0})^{\frac{1}{2}+\varepsilon}$
$\displaystyle\lesssim_{\varepsilon}$ $\displaystyle
R^{\frac{1}{2}+\varepsilon},$ $\displaystyle\bullet\ r_{4,2}(R+R_{0})\lesssim\
$
$\displaystyle(R+R_{0})\log(\log(R+R_{0}))\lesssim_{\varepsilon}(R+R_{0})^{1+\varepsilon}\lesssim_{\varepsilon}R^{1+\varepsilon},$
$\displaystyle\bullet\ r_{m,2}(R+R_{0})\lesssim\ $
$\displaystyle(R+R_{0})^{\frac{m-2}{2}}\lesssim R^{\frac{m-2}{2}}$

or, equivalently, $r_{m,2}(R+R_{0})\lesssim R^{\frac{m-2}{2}+\varepsilon}$ for
all $\varepsilon>0$ if $m=2,3,4$ and for all $\varepsilon\geq 0$ if $m\geq 5.$

Now, we can estimate the right hand side of (3.5) as follows

$\displaystyle\left[\sum\limits_{R=0}^{\infty}N_{R}R^{2(\alpha+s(q))}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right]^{\frac{1}{2}}$
$\displaystyle\lesssim\left[\sum\limits_{R=0}^{\infty}R^{2\left(\frac{\varepsilon}{2}+\frac{m-2}{4}+\alpha+s(q)\right)}\left|\left|P_{\frac{R}{b\gamma}}u_{0}\right|\right|^{2}_{L^{2}(G)}\right]^{\frac{1}{2}}$
$\displaystyle\asymp\|u_{0}\|_{H^{\frac{\varepsilon}{2}+\frac{m-2}{4}+\alpha+s(q)}}.$

Observing that $\alpha=1/2-1/p,$ and that

$\frac{m-2}{4}+\alpha+s(q)=\frac{m-2}{4}+\frac{1}{2}-\frac{1}{p}+s(q)=\frac{m}{4}-\frac{1}{p}+s(q),$
(3.8)

we have proved (3.3). The proof is complete. ∎

######  Remark 3.5.

Note that $m$ always can be written in terms of the dimension $d$ of $G.$ For
instance, when $G=\textnormal{Spin}(2\ell+1)$ we have that $d=\ell(2\ell+1),$
thus

$m=\ell=\frac{-1+\sqrt{1+8d}}{4},$

and the inequality for the regularity order $s$ in (3.3) can be written as

$s\geq\frac{\sqrt{8d+1}-1}{16}-\frac{1}{p}+s(q).$ (3.9)

In terms of the dimension $d=\ell(\ell+2)$ of $\textnormal{SU}(\ell+1),$ we
can write the inequality for the regularity order in (3.3) as follows

$s\geq\frac{-1+\sqrt{1+d}+1}{4}-\frac{1}{p}+s(q)=\frac{\sqrt{1+d}}{4}-\frac{1}{p}+s(q).$
(3.10)

As for the case of $\textnormal{Spin}(2\ell)$ whose dimension is
$d=\ell(2\ell-1),$ we have that

$\ell=\frac{1+\sqrt{1+8d}}{4},$

and the inequality for the regularity order $s$ in (3.3) can be written as

$s\geq\frac{1+\sqrt{8d+1}}{16}-\frac{1}{p}+s(q).$ (3.11)

Note that in the case of $\textnormal{Spin}(2\ell)$ which has dimension
$d=\ell(2\ell-1),$ we have that

$\ell=\frac{1+\sqrt{1+8d}}{4},$

and the inequality for the regularity order $s$ in (3.3) can be written as

$s\geq\frac{1+\sqrt{8d+1}}{16}-\frac{1}{p}+s(q).$ (3.12)

Also, in terms of the dimension $d=\ell(2\ell+1)$ of $\textnormal{Sp}(\ell),$
we have that

$\ell=\frac{-1+\sqrt{1+8d}}{4},$

and then, we can write the inequality for the regularity order in (3.3) as
follows

$s\geq\frac{-1+\sqrt{1+8d}}{16}-\frac{1}{p}+s(q).$ (3.13)

###  3.1. Data statement

No data were obtained during the realization of this work.

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