\section{Introduction}

Obtaining a theory of Quantum Gravity (QG) is still an unsolved problem in modern physics \cite{carlip01,kiefer05}. The objective one has in mind when making this statement is obtaining a theory of gravity which is valid at all scales. Since gravity is a rather weak force, its quantum effects are not expected to be observable at scales which we can currently probe. The criteria by which a candidate theory for QG may be assessed are limited to mathematical/internal consistency, reproducing the classical descriptions of General Relativity (GR) and leading to nontrivial predictions that can eventually be tested.
As it is standard, we try to implement quantisation methods to GR. However, GR is perturbatively non-renormalizable and new methods have to be found.
One option is to follow a minimal approach and obtain a perturbatively renormalizable UV completion of GR as a Quantum Field Theory (QFT) for the metric field in 3+1 dimensions. Although, this proposal requires giving up some properties of GR, as its original formulation does not satisfy the perturbative renormalisability requirement, at least not at the Gaussian fixed point.
This line of thought can be viewed as the motivation for a theory of QG with anisotropic scaling in the UV, the so-called Ho\v rava-Lifshitz gravity (HLG) \cite{hor09a,hor09b}, where the treatment of the UV fixed point is akin to what is done in critical systems.

In a recent work \cite{Luongo:2018oil}, HLG was tested in the context of cosmography. Essentially, cosmography assumes only the validity of the Cosmological Principle, which fixes the geometry of spacetime, without invoking any particular dynamical theory. Therefore, it is a powerful tool to test the theoretical predictions of dynamical models and to compare different models concerning observations. In this analysis, HLG turns out to be disfavored when compared with the standard cosmological model, which follows from Einstein gravity.
For our proposal, we perform an alternative analysis by considering HLG as
the classical limit of a not yet formulated  model of quantum gravity and search for quantum signatures at the cosmological level. Certainly, quantum effects are expected to be dominant in the very early universe and very small in the present time. Nevertheless, we use in this work current time observations to search for detectable effects in a way that resembles the analysis of the cosmic microwave background radiation as the present signature of the initial Big Bang singularity. As long as no realistic quantum model for the origin of the Universe is known, we must use classical models such as HLG, from which we expect to contain footprints of quantum effects due to its property of being renormalizable.

\color{black}
 To investigate cosmology in HLG,  we use the new formalism called \textit{inverse cosmography} \cite{Munoz:2020gok} with recent and simulated observational data, including a higher redshift range, from which we do not exclude {\it a priori} the possibility of having quantum prints. Then, we analyse HLG to establish signatures that could follow from the quantum nature of the model. In these lines of thought, the inverse cosmography was proposed to test the cosmodynamical parameters and use them to analyse the kinematics via its generic cosmographic parameter, this without experimenting with the standard problems of truncation of the cosmographic series.

\section{Standard cosmography description}

The cosmographic method can be described as an independent technique to obtain viable limits on the cosmic accelerated universe history at late times, considering by default the validation of the Cosmological Principle \cite{wei72,visser05}. Its requirements are
homogeneity and isotropy with spatial curvature given by a Friedmann-Lema{\^i}tre -Robertson-Walker (FLRW) metric
 ${\rm d} s^2={\rm d} t^2-a(t)^2\left({\rm d} r^2+r^2{\rm d}\Omega^2\right),$
where we employ ${\rm d}\Omega^2\equiv {\rm d}\theta^2+\sin^2\theta {\rm d}\phi^2$.
Usually, the cosmological scenario provides a whole energy density dominated by a constant $\Lambda$,
or by some dark energy density, with cold dark matter in second place and baryons as a small fraction on the rest. The corresponding
spatial curvature in the case of time-independent dark energy density is actually constrained to be negligible.
This methodology can give us numerical results which do not
depend on the particular choice of the cosmological model, since only Taylor expansions are compared with observational samplers.
The cosmographic expansions can be obtained by determining the scale factor $a(t)$ as a Taylor series
around present time $t_0(z=0)$:
\begin{eqnarray}
a(t)  & \sim & a(t_0)+ a'(t_0) \Delta t + \frac{a''(t_0)}{2} \Delta t^2+
\frac{a'''(t_0)}{6} \Delta t^3
+   \frac{a^{(iv)}(t_0)}{24} \Delta t^4 +\frac{a^{(v)}(t_0)}{120} \Delta t^5+\ldots\,,
\end{eqnarray}
which recovers signal causality if one assumes $\Delta t\equiv t-t_0>0$. Therefore, from the latter expansion of $a(t)$, we can define
\begin{eqnarray}
H &\equiv& \frac{1}{a} \frac{{\rm d} a}{{\rm d} t}\,,\quad
q \equiv -\frac{1}{a H^2} \frac{{\rm d}^2a}{{\rm d} t^2}\,,\quad
j  \equiv \frac{1}{a H^3} \frac{{\rm d}^3a}{{\rm d} t^3}\,,
s \equiv \frac{1}{a H^4} \frac{{\rm d}^4a}{{\rm d} t^4}\,, \quad
l  \equiv \frac{1}{a H^5} \frac{{\rm d}^5a}{{\rm d} t^5}\,.
\end{eqnarray}

These equations are model-independent, i.e., they do not depend on the form of the dark energy fluid present in Friedmann equations
since they can be directly constrained by observations. They are the so-called: Hubble rate ($H$), acceleration parameter ($q$),  jerk parameter ($j$),  snap parameter ($s$), and lerk parameter ($l$), respectively.
Rewriting $a(t)$ in terms of these cosmographic parameters with a normalised scale factor to $a(t_0) = 1$
\begin{eqnarray}
a(t)  & \sim & 1+  H_0 \Delta t - \frac{q_0}{2}  H_0^2\Delta t^2+
+\frac{j_0}{6} H_0^3 \Delta t^3
+   \frac{s_0}{24}  H_0^4\Delta t^4
+\frac{l_0}{120}  H_0^5\Delta t^5 +\ldots\,
\end{eqnarray}
We can re-scale the above cosmographic parameters in terms of the Hubble rate and its derivatives so that only three independent cosmographic coefficients appear explicitly
\begin{eqnarray}
\dot{H} &=& -H^2 (1 + q)\,,\\
\ddot{H} &=& H^3 (j + 3q + 2)\,,\\
\dddot{H} &=& H^4 \left [ s - 4j - 3q (q + 4) - 6 \right]\,,\\
\ddddot{H} &=& H^5 \left [ l - 5s + 10 (q + 2) j + 30 (q + 2) q + 24\right ]. \quad
\end{eqnarray}
Notice the correspondence degeneracy between the Hubble parameter derivatives and the cosmographic parameters
as a consequence that all these expressions are multiplied by $ H$.

\section{Background for inverse cosmography}

In \cite{Munoz:2020gok}, it was pointed out that the degeneracy problem in cosmological models can be modulated using an \textit{inverse cosmography} methodology, which lies on the idea that the kinematics of the universe can be understood by setting limits of the current standard cosmography using not only Taylor-series like parameterisations, but also high order polynomials. This proposal also solves the problem of the error propagation over the statistical test, making it possible to set a cutoff on the cosmographic parameters directly from the dynamics of the cosmological ones. Therefore, we are directly seeing the dynamics of a model through its cosmographic parameters.

We can set the equations to be used in the case of a specific model or, in our case, a model that can relax the UV limit. We derive the inverse cosmography considering a spatial flatness hypothesis on the Hubble function as
\begin{equation}\label{eq:friedmann_GR}
\left(\frac{H(z)}{H_0}\right)^2 =\Omega_k (1+z)^2 + \Omega_{m}(1+z)^3 + \Omega_r (1+z)^4 + \Omega_{i} f_{i}(z),
\end{equation}
where the curvature and radiation terms are $\Omega_k$ and $\Omega_r$, respectively. The term $\Omega_i$ is related with the standard description of the current universe evolution once we know the form of $f(z)$. Moreover, this approach requires a fiducial model, then we can write
a \textit{generic} expression for the cosmological equation of state (EoS)  \cite{Escamilla-Rivera:2019aol,Munoz:2020gok}
 \begin{equation}\label{eq:genericEoS}
w(z)= -1+\frac{1}{3} (1+z)\frac{f_{i}(z)^{\prime}}{f_{i}(z)},
\end{equation}
where the prime denotes $d/dz$. We note that the \textit{generic EoS} depends only on the form of $f_{i}(z)$. Following the inverse cosmography idea, we can use (\ref{eq:genericEoS}) with the chain rule
$\dot{}=d/dt =-(1+z)H(z)d/dz$, to obtain the set of cosmographic parameters in terms of $H(z)$ and its derivatives:
\begin{eqnarray}
q(z)= -1 +\frac{1}{2}(1+z)\frac{[H(z)^2]'}{H(z)^2}, \label{eq:q}
\end{eqnarray}
\begin{eqnarray}
j(z) &=&\frac{1}{2}(1+z)^2 \frac{[H(z)^2]''}{H(z)^2}
 -(1+z)\frac{[H(z)^2]'}{H(z)^2} +1, \label{eq:j}\quad
\end{eqnarray}
\begin{eqnarray}
s(z) &=& -\frac{1}{6} (1+z)^3 \frac{[H(z)^2]'''}{H(z)^2} +\frac{1}{2}(1+z)^2 \frac{[H(z)^2]''}{H(z)^2}
+(1+z)\frac{[H(z)^2]'}{H(z)^2} -1. \label{eq:s}
\end{eqnarray}
By solving and evaluating these expressions at $z=0$, we get the usual cosmographic series
\begin{eqnarray}\label{eq:Hcosmo}
H(z)&=& H_0 +\frac{dH}{dz}\bigg |_{z=0} z +\frac{1}{2!}\frac{d^2 H}{dz^2}\bigg |_{z=0} z^2
+\frac{1}{3!} \frac{d^3 H}{dz^3}\bigg |_{z=0} z^3 +\ldots %
\end{eqnarray}
Similar expressions can be calculated by expressing everything in terms of the function normalised by the Hubble constant, $E(z)=H(z)/H_0$ and its derivatives. Nonetheless, the information we can obtain is exactly equivalent.

\section{Ho\v rava-Lifshitz dynamics}

In HLG, it is  assumed that Lorentz invariance is violated at the UV regime and that the theory reduces to GR with  cosmological constant in the IR limit.
The idea consists in introducing Lorentz breaking terms in the action such that the possibility appears that the resulting theory becomes renormalisable in the UV limit.
Indeed, it was established that
if space and time re-scale differently as $x^{i}\rightarrow lx^{i},\quad t\rightarrow l^{z}t$, where $z$ is the scaling exponent, then in the case $z=3$ one can add non-relativistic terms to the Einstein-Hilbert action in such a way that  the resulting theory is renormalisable by power counting.
The problem appears that the most general action for such a theory contains about 100 terms, which makes the problem quite difficult to handle and drastically reduces the predictability of the theory due to the large number of arbitrary coupling constants entering the action \cite{wang17}. It is, therefore, convenient to consider particular cases with a reduced number of coupling constants. To this end, additional symmetries are imposed at the level of the action, which often lead to new difficulties that affect either the energy limits or even the renormalisability of the theory.
Several actions have been proposed so far in order to solve different conceptual problems that appear at the UV and IR energy levels.
In the original HL model, two major simplifying conditions were assumed, namely, projectability and  detailed balance which, however, lead to difficulties \cite{cnps09,bps09}. The first assumption led to inconsistencies in the IR limit, which can be
treated by imposing strong conditions on the free parameters of the general action
\cite{Sotiriou:2009gy}. In addition, instabilities at the UV and the IR energetic limits
were found \cite{Mukohyama:2010xz,Izumi:2011eh,Gumrukcuoglu:2011ef}  that make it difficult to handle these limits from a physical point of view, even at the level of particular gravitational configurations \cite{Mukohyama:2010xz,Izumi:2011eh,Gumrukcuoglu:2011ef}. Although, several results concerning instabilities are controversial \cite{wang17}, the simplest solution to this problem is to consider only non-projectable configurations  \cite{Fukushima:2018xgv}.

Despite these difficulties and  ambiguities, HLG has been applied in many frameworks, especially in black hole physics and cosmology \cite{ Ramos:2018oku,Ma:2016lwr,Gao:2020ddm,Nilsson:2019bxv}.
In this work, we will consider one of the simplest versions of HLG, in which the conditions of projectability and detailed balance are imposed \cite{hor09a}.
The original version of this minimal model had the problem that the Schwarzschild-AdS black hole metric was not a solution of the field equations in the corresponding IR limit.  This problem was solved by including ad additional term in the action that leads to IR field equations with the Schwarzschild-AdS spacetime as a particular solution \cite{park09}, implying at the same time to a redefinition of the IR limit.
As a result, only six coupling constants enter the action  that can be written as
\color{black}
\begin{eqnarray}\label{eq:action_Horava}
S_g& =& \int d^4 x N \sqrt{g}  \left[\frac{2}{\kappa^2}\left(K_{ij}K^{ij}-\lambda K^2\right)-\frac{\kappa^2}{2\nu^4}C_{ij}C^{ij}
+
\frac{\kappa^2\mu}{2\nu^2}\epsilon^{ijk} R^{ }_{i\ell} \nabla_{j}R^{ \ell}{}_k -\frac{\kappa^2\mu^2}{8} R^{ }_{ij} R^{ ij} +
\right. \nonumber \\
&&  \left.
 \frac{\kappa^2 \mu^2}{8(3\lambda-1)}
\left(\frac{4\lambda-1}{4}R^2-\Lambda_W R+3\Lambda_W^2\right)+
 \frac{\kappa^2 \mu^2\omega}{8(3\lambda-1)} R\right],
\end{eqnarray}
where
\begin{eqnarray}
K_{ij}&=&\frac{1}{2N} (\dot g_{ij}-\nabla_i N_j -\nabla_j N_i)\,, \\
C^{ij}&=& \epsilon^{ikl} \nabla_k \left(R^j_l -\frac{1}{4} R \delta^j_l\right)\,,
\end{eqnarray}
\noindent are the extrinsic curvature and the Cotton tensor, respectively. The dot represents the derivative with respect to the time coordinate, and $R$ is the scalar curvature. Further, $N$ and $N_i$ are the lapse and shift in the 3+1 decomposition, i.e.
$ds^2 = -N^2 c^2 dt^2 + g_{ij}(dx^i + N^idt)(dx^j+N^jdt)$.

This model presents a significant degeneracy since it depends upon six parameters:
$\kappa$, $\lambda$, $\mu$, $\nu$, $\Lambda_W$ and $\omega$, which are not \textit{completely} free. Some of them
determine the speed of light $c$, the gravitational constant $G$ and the cosmological constant $\Lambda$ according to
\begin{eqnarray}
\mu^2 &=& \frac{16 c^2 (3\lambda -1)^2 |\Lambda|}{3\kappa^4 \Lambda^{2}_{W}},  \label{eq:mu}
\quad G=\frac{\kappa^2c^2}{16\pi(3\lambda-1)}\ , \label{eq:G}
\end{eqnarray}
where $\Omega_{\Lambda_W} = \frac{2}{3}\Omega_\Lambda\,$, and if $\lambda<1/3$ implies the presence of repulsive gravitational effects. The contrary case ($\lambda>1/3$) leads to attractive gravity.
In the IR limit, the HL action reduces to
\begin{equation}
S_{IR} = \frac{c^2}{8\pi G (3\lambda -1)} \int d^4 x \sqrt{g} N \Bigg[ K_{ij} K^{ij} - \lambda K^2
+ c^2 R \left(1 + \frac{3\omega c^2 (3\lambda -1)}{ 4 |\Lambda|} \right)\Bigg]\ ,
\end{equation}
which is expected to be equivalent to the GR action. Then, it follows that the Einstein-Hilbert action with cosmological constant is obtained in the limit $\lambda\to 1$ and $\omega\to 0$. The limit $\lambda \to 1$ is also consistent with the expected values of the universal constants $G$, $c$, and $\Lambda$. We will see below, in Sec. 7, that the constant $\omega$ encodes completely the quantum contribution of the HL gravity model at the cosmological level. Consequently, the IR limit $\omega\to 0$ is also consistent with this interpretation.

In this work, we will study the large-scale cosmological model described by the Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric
\begin{equation}
ds^2 = - c^2 dt^2 + a^2(t)\left[ \frac{ dr^2}{1-k\frac{r^2}{r_0^2}} + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)\right] \ .
\end{equation}
The use of this metric could arouse certain concerns. Indeed,
when considering perturbations
around a background metric, as mentioned above, it has been argued that HLG does not have an IR limit corresponding to GR because of the presence of strong coupled gravity fluctuations. This would be a major problem for the consistency of the theory. Nevertheless, a detailed discussion of this problem is beyond the scope of the present
work. In practice, however, this problem might not be quite relevant to our case since we will consider only cosmological solutions with a non-vanishing but tiny cosmological constant, as favored by current observations. In fact, when considering perturbations around such cosmological solutions a natural cut-off appears due to the tiny value of the cosmological constant. Of course, a more detailed analysis will be necessary to confirm this intuitive idea.

Under the ansatz $\Lambda_W > 0$, the only scenario to be in agreement with local scale limits, and considering a
perfect-fluid source with energy density and pressure $\rho$ and $p=w\rho$, respectively, we can compute the new Friedmann equations for this theory and obtain
\begin{eqnarray}\label{eq:friedmann}
\left(\frac{\dot{a}}{a}\right)^2&=&b_1\Bigg\{\rho \pm b_2 \left[-\Lambda^2_W + \frac{2k(\Lambda_W-\omega)}{r^2_0a^2}-\frac{k^2}{r^4_0a^4}\right]\Bigg\},\quad\quad \\
\frac{\ddot a}{a}&=&b_1\left[-\frac{1}{2}(\rho+3p) \pm b_2 \left(-\Lambda^2_W+\frac{k^2}{r^4_0a^4}\right)\right]\,,
\end{eqnarray}
with $b_1=\kappa^2/[6(3\lambda-1)]$ and $b_2=3\kappa^2\mu^2/[8(3\lambda-1)]$.  We can combine the latter expression and write a differential equation in terms of the redshift $a=(1+z)^{-1}$ as
the corresponding differential equation, obtained by combining the above two expressions, can be recast as
\begin{eqnarray}
&&(1+z)\frac{dH^2}{dz}-3(1+w)H^2+\Lambda(1+w)c^2
-\frac{\Lambda kc^2}{3\Lambda_{W}^{2}r_{0}^{2}}
\Big[(1+3w)(\Lambda_W-\omega)(1+z)^2
\nonumber \\ &&
+\frac{k}{2r_{0}^{2}}(1-3w)(1+z)^4\Big]=0\,. \quad
\label{feq}
\end{eqnarray}
We defined the Hubble rate $H\equiv \dot{a}/a$ and $|\Lambda| = 3\mu^2\kappa^4\Lambda^2_W /16c^2(3\lambda-1)^2$.
Following, we need to characterise the type of fluid in the HLG to study its dynamics.

Equation (\ref{feq}) shows that the term proportional to $a^{-4}$ does not appear in the corresponding model in Einstein gravity and so it represents the contribution of HLG. This term vanishes for $k=0$ or $w=1/3$. This indicates that in the case of a flat universe with arbitrary EoS or a radiation dominated universe with arbitrary topology, there is no difference between Einstein gravity and HLG. Since in this work we are interested in analyzing HLG in the lowest redshift regime, i.e., the recently observed universe, we can neglect the contribution of radiation and consider a fluid with $w=0$ as the next interesting case. Indeed, this pressureless fluid can be used to describe either baryonic matter or cold dark matter, which together with the cosmological constant constitute the main components of the Universe.
Accordingly, we will consider now the case of dust, which is also the simplest case of a matter term with EoS: $p=0 \ (w=0)$.

In \cite{Luongo:2018oil}, the Ho\v rava corrections to the Friedmann equations were found, showing that they become relevant for different regimes, e.g.,  at the low redshift regime, the term $\omega$ does not contribute so that it is difficult to bound it in a FLRW scenario. If we assume a constant barotropic factor $w$ for a given cosmological fluid, the Hubble rate $H(z)$ can be written as
\begin{eqnarray}\label{eos_horava}
E^2(z)&=&\sum_i\Omega_i(1+z)^{3(1+w_i)}+\Omega_k\left(1-\frac{\Omega_\omega}{\Omega_\Lambda}\right)(1+z)^2
+\Omega_{r}^{*}(1+z)^4+\Omega_\Lambda\,,
\end{eqnarray}
where $E^2(z)=H^2/H_0^2$ and $|\Omega_\omega|\equiv\omega c^2/(2H_0^2)$.
\\

\section{Ho\v rava-Lifshitz inverse cosmography}

To compute the \textit{generic} cosmograhic parameters for the HLG theory, we use its EoS (\ref{eos_horava}) in the inverse cosmography equations (\ref{eq:q})-(\ref{eq:j})-(\ref{eq:s}). The simplest scenario to be considered is the HLG dust case:
\begin{eqnarray}
q&=&-1+ \frac{(z+1)^2 \left(\Omega _k \left(2-\frac{2 \Omega _w}{\Omega _{\Lambda }}\right)+(z+1) \left(3 \Omega
   _m+4 (z+1) \Omega^{*}_{r}\right)\right)}{2 \left((z+1)^2 \Omega _k \left(1-\frac{\Omega
   _w}{\Omega _{\Lambda }}\right)+\Omega _{\Lambda }+(z+1)^3 \Omega _m+(z+1)^4 \Omega^{*}_{r}\right)} \biggr |_{z=0}
   \nonumber\\
   &=&  -1+ \frac{\Omega _k \left(2-\frac{2 \Omega _w}{\Omega _{\Lambda }}\right)+3 \Omega _m+4 \Omega^{*}_{r}}{2 \left(\Omega _k \left(1-\frac{\Omega _w}{\Omega _{\Lambda }}\right)+\Omega _{\Lambda
   }+\Omega _m+\Omega^{*}_{r}\right)},  \quad \quad
  \\
 j&=& \frac{\Omega _{\Lambda } \left(\Omega _{\Lambda }+(z+1)^3 \Omega _m+3 (z+1)^4 \Omega^{*}_{r}\right)}{\Omega _{\Lambda } \left(\Omega _{\Lambda }+(z+1)^3 \Omega _m+(z+1)^4 \Omega^{*}_{r}\right)-(z+1)^2 \Omega _k \left(\Omega _w-\Omega _{\Lambda }\right)} \biggr |_{z=0}
  \nonumber\\
& =& \frac{\Omega _{\Lambda } \left(\Omega _{\Lambda }+\Omega _m+3\Omega^{*}_{r}\right)}{\Omega
   _{\Lambda } \left(\Omega _{\Lambda }+\Omega _m+\Omega^{*}_{r}\right)-\Omega _k \left(\Omega
   _w-\Omega _{\Lambda }\right)},
 \\
   s&=& \frac{\Omega _{\Lambda } \left(\Omega _{\Lambda }+z (z+1)^2 \Omega _m+(3 z-1) (z+1)^3 \Omega^{*}_{r}\right)}{\Omega _{\Lambda } \left(\Omega _{\Lambda }+(z+1)^3 \Omega _m+(z+1)^4 \Omega^{*}_{r}\right)-(z+1)^2 \Omega _k \left(\Omega _w-\Omega _{\Lambda }\right)},  \biggr |_{z=0}
    \nonumber\\
&= & \frac{\Omega _{\Lambda } \left(\Omega _{\Lambda }-\Omega^{*}_{r}\right)}{\Omega _{\Lambda }
   \left(\Omega _{\Lambda }+\Omega _m+\Omega^{*}_{r}\right)-\Omega _k \left(\Omega _w-\Omega
   _{\Lambda }\right)},
   \\
   l&=& \frac{\Omega _{\Lambda } \left(\Omega _{\Lambda }+z (z+1)^2 \Omega _m+(z (3 z+2)+1) (z+1)^2\Omega^{*}_{r}\right)}{\Omega _{\Lambda } \left(\Omega _{\Lambda }+(z+1)^3 \Omega _m+(z+1)^4 \Omega^{*}_{r}\right)-(z+1)^2 \Omega _k \left(\Omega _w-\Omega _{\Lambda }\right)} \biggr |_{z=0}
    \nonumber\\
   &=& \frac{\Omega _{\Lambda } \left(\Omega _{\Lambda }+\Omega^{*}_{r}\right)}{\Omega _{\Lambda }
   \left(\Omega _{\Lambda }+\Omega _m+\Omega^{*}_{r}\right)-\Omega _k \left(\Omega _w-\Omega
   _{\Lambda }\right)}.
\end{eqnarray}
\\
We can derive expression of HLG parameter $\Omega_W$ in terms of the cosmographic parameters as at $z=0$ denoted by the subindex \textit{0}:
\begin{eqnarray}
\Omega_W (q_0) &=& \frac{\Omega _{\Lambda } \left(2 q_0 \Omega _k+2 \Omega _{\Lambda }+2 q_0 \Omega _m-\Omega
   _m+2 q_0 \Omega _{\Lambda }\right)}{2 q_0 \Omega _k}, ~\quad \\
   \Omega_W (j_0) &=& \frac{\Omega _{\Lambda } \left(j_0 \Omega _k+j_0 \Omega _{\Lambda }+j_0 \Omega
   _m-\Omega _{\Lambda }-\Omega _m\right)}{j_0 \Omega _k}, \\
     \Omega_W (s_0) &=& \frac{\Omega _{\Lambda } \left(s_0 \Omega _k-\Omega _{\Lambda }+s_0 \Omega _m+s_0
   \Omega _{\Lambda }\right)}{s_0 \Omega _k}.
\end{eqnarray}
In Figure 1, we illustrate the cosmographic parameters for Ho\v rava in comparison to standard dust case in an observational redshift range of interest.

\begin{figure*}
\centering
    \includegraphics[width=0.45\textwidth,origin=c,angle=0]{qhorava.pdf}
          \includegraphics[width=0.47\textwidth,origin=c,angle=0]{jhorava.pdf}
              \includegraphics[width=0.45\textwidth,origin=c,angle=0]{shorava.pdf}
    \caption{Cosmographic parameters for Ho\v rava (dashed line) in comparison to standard dust case (solid line).}
    \label{fig:cosmography_Horava}
\end{figure*}

\begin{figure}
\centering
    \includegraphics[width=0.7\textwidth,origin=c,angle=0]{hlgtestfull.png}
    \caption{C.L Cosmographic parameters space for Ho\v rava dust model at $z=0$ using observational samplers: Pantheon, GW, H0liCOW and CC.}
    \label{fig:cosmography_Horava}
\end{figure}

\section{Observational and forecast surveys}

\begin{enumerate}
\item Pantheon SNeIa sampler: Consist
of 1048 SN Ia in a redshift region $z \in [0.01, 2.3]$ \cite{Scolnic:2017caz}, whose distance moduli are standardized through the SALT-2 light-curve fitter.
Under the only assumption of a flat universe, the full Pantheon catalog can be compressed into six model-independent $E^{-1}(z)$ measurements. Therefore, we use these measurements correlated among them according to the covariance matrix $C_{ij}$. In this case, the likelihood probability function is described by
\begin{equation}
\mathcal{L}_{SNeIa} \propto \exp \left( -\frac{1}{2}\bm{V}^{T} C_{ij}^{-1}\bm{V} \right) ,
\end{equation}
where $\bm{V}=E_{obs}^{-1}-E_{th}^{-1}$ is the differences between the observed and the theoretical expectation values.

\item Cosmic Chronometers: The late cosmic expansion can be studied in a model-independent way by measuring the age difference of Cosmic Chronometers (CC).
\cite{Jimenez:2019onw}.
From the spectroscopic measurements of the redshifts between pairs of  old and passively evolving galaxies and their differential age, one can obtain the Hubble parameter
$H(z)=-(1+z)^{-1}dz/dt $.
In this work we consider 31 uncorrelated measurements of $H(z)$ in a redshift range $z: [0,2]$. Confronting these values with the corresponding Hubble expansion rates predicted by the theoretical scenarios, one can write the likelihood function as
\begin{equation}
\mathcal{L}_{CC}\propto \exp{\left[-\frac{1}{2}\sum_{i=1}^{31}\left(\frac{H^{obs}_i-H^{th}_i}{\sigma_{H,i}}\right)^2\right]}.
\end{equation}

\item H0LiCOW data: For this sampler, we employ the time-delay distance given by
\begin{equation}
D_{\Delta t}=(1+z_l)\frac{D_l D_s}{D_{ls}}\	 ,
\end{equation}
where $z_l$ is the redshift of the lens and
\begin{eqnarray}
D_l&=& \frac{H_0^{-1}}{1+z_l}\int_0^{z_l}\frac{dz}{E(z)}\ , \\
D_s&= & \frac{H_0^{-1}}{1+z_s}\int_0^{z_s}\frac{dz}{E(z)}\ , \\
D_{ls}&=&\frac{H_0^{-1}}{1+z_s}\int_{z_l}^{z_s}\frac{dz}{E(z)}\ ,
\end{eqnarray}
and $z_s$ are the angular diameter distances to the lens, to the source, between the lens and the source and the source redshift, respectively.
In this work, we use the six systems of strongly lensed quasars analysed by the H0LiCOW collaboration \cite{Rusu:2019xrq}, in where the
likelihood function for the $D_{\Delta t}$ is described by
\begin{equation}
\mathcal{L}_{H0LiCOW}\propto \exp\left[-\frac{1}{2}\sum_{i=1}^{6}\left(\frac{D_{\Delta t, i}^{obs}-D_{\Delta t,i}^{th}}{\sigma_{D_{\Delta t},i}}\right)^2\right] .
\end{equation}

\item Standard Sirens ET simulations: We use here simulations from a third-generation
ground-based GW detector, Einstein Telescope (ET), since it will allow to have a better statistical sensitivity
detection in comparison to the current LIGO detectors. The advantage of this observation is that it can be used
to relax (or even break) degeneracies generated by other surveys due to their definition in terms of the absolute luminosity distance \cite{Kyutoku:2016zxn}, and
its redshift range will reach $z\approx 2-3$.
To perform the simulation, we first consider the best fits parameters for the HLG dust model obtained from H0LICOW + Pantheon + CC, then this will be the fiducial model to generate the GW standard sirens simulations from the ET\footnote{The full method is described in \cite{Zhang:2018byx}.}. The simulation consist in 1000 standard sirens from the ET, where each point consist of a vector given by redshift, the luminosity distance and its error as: $(z_i, d_{L}, \sigma_i )$. The set of simulated GW data will consist of $N$ data points and the standard likelihood can be described by
\begin{equation}
\mathcal{L}_{GW}\propto \exp \left(-\frac{1}{2}\sum^{N}_{i=1} \left[\frac{d^{i}_{L}- d_{L}(z_{i};\Theta)}{\sigma^{i}_{d_L}}\right]^2\right) ,
\end{equation}
where $z_i$, $d^{i}_{L}$ and $\sigma^{i}_{d_L}$ are the $i$th redshift, luminosity distance and its error, respectively. $\Theta$ denotes the set of HLG cosmographic parameters obtained by the above samplers.
\end{enumerate}

{\renewcommand{\tabcolsep}{0.03mm}
{\renewcommand{\arraystretch}{0.6}
\begin{table}
\caption{Cosmographic parameters results for HLG theory dust case with SNeIa Pantheon, CC, H0LICOW, GW and the combinations of the samplers.}
\begin{tabular}{c|ccccc}
\hline
&& &  Best fits         \\
\small{Parameters} & Pantheon & CC & GW & \small{H0LiCOW} & \small{SN + CC + GW + H0LiCOW} \\ \hline \\
$H_0$ & $   71.84_{-    0.708 }^{+    0.706}$ & $69.010_{-    1.332}^{+    1.329}$  & $   72.574_{-    0.757}^{+    0.747}$ & $   71.367_{-    0.689}^{+    0.6840}$ &$   71.904_{-    1.347 }^{+    1.406}$\\ \\
$q_0$ & $   -0.424_{-    0.060}^{+    0.077}$ &  $   -0.533_{-    0.119}^{+    0.142}$ & $   -0.450_{-    0.152}^{+    0.152}$& $   -0.385_{-    0.045}^{+    0.053}$ &$   -0.423_{-    0.141}^{+    0.133}$\\ \\
$j_0$ & $   1.199_{-    1.003}^{+    0.907}$ &  $  0.114_{-    1.459}^{+    1.322}$ &$   0.912_{-    1.832}^{+    1.518}$  & $   1.539_{-    0.834}^{+    0.704}$ &$   1.210_{-    1.863}^{+    1.891}$\\ \\
$s_0$ & $  -18.484_{-    8.825}^{+    7.945}$  & $  -11.171_{-   13.405}^{+   11.101}$ & $  -19.428_{-   12.790}^{+    7.509}$ & $  -21.387_{-    7.255}^{+    6.164 }$ &  $  -18.510_{-    16.514}^{+    16.746}$\\ \\
\hline
\end{tabular}
\label{tab:results2}
\end{table}}}

{\renewcommand{\tabcolsep}{0.03mm}
{\renewcommand{\arraystretch}{0.6}
\begin{table}
\caption{Best fits values for HLG theory parameters SNeIa Pantheon, CC, H0LICOW, GW and the join sampler.}
\begin{tabular}{c|ccccc}
\hline
&& & &Best fits         \\
Parameters & Pantheon & CC & GW & H0LiCOW &SN + CC + GW + H0LiCOW \\ \hline \\
$\Omega_m$ & $0.292^{+0.032}_{-0.132}$   & $0.311^{+0.022}_{-0.013}$   & $0.325^{+0.072}_{-0.252}$	& $0.308^{+0.023}_{-0.013}$ & $0.331^{+0.101}_{-0.101}$\\ \\
$\Omega_{\Lambda}$ 	&$0.718^{+0.706}_{-0.708}$	&$0.690_{-    1.332}^{+    1.329}$ 	&$   0.726_{-    0.757}^{+    0.747}$ &$   0.714_{-    0.689}^{+    0.6840}$ & $   0.719_{-    1.347 }^{+    1.406}$  \\ \\
$\Omega_k$ & $-0.012^{+0.12}_{-0.18}$ & $-0.012^{+0.02}_{-0.02}$ & $-0.020^{+0.013}_{-0.019}$& $-0.01^{+0.021}_{-0.018}$ & $-0.021^{+0.023}_{-0.029}$\\ \\
$\Omega_{\omega}$ & $0.090^{+4.01}_{-4.03}$ & $0.081^{+3.11}_{-3.13}$ & $0.120^{+3.18}_{-3.18}$& $0.122^{+5.01}_{-4.12}$ & $0.090^{+3.241}_{-3.241}$\\ \\
\hline
\end{tabular}
\label{tab:results1}
\end{table}}}

{\renewcommand{\tabcolsep}{0.03mm}
{\renewcommand{\arraystretch}{0.6}
\begin{table}
\caption{Derived quantities values for HLG theory using the results given in Table \ref{tab:results2}.}
\begin{tabular}{c|ccccc}
\hline
Parameters & Pantheon & CC & GW & H0LiCOW & SN+CC+GW+H0LiCOW \\ \hline \\
$\Omega_{\Lambda_\omega}$ 	& $0.195^{+0.021}_{-0.008}$ 	& $0.207^{+0.015}_{-0.009}$ 	&$0.217^{+0.048}_{-0.168}$ &$0.205^{+0.015}_{-0.009}$ &$0.220^{+0.067}_{-0.067}$\\ \\
$\omega \times 10^{-14} L^{-2}$ & $-3.349^{+331.128}_{-1363.33}$ &$-3.291^{+211.198}_{-125.479}$ &$-3.804^{+759.417}_{-2655.54}$& $-3.386^{+235.127}_{-133.273}$ & $-3.803^{+1046.24}_{-1046.17}$\\ \\
$\Omega_{W} (q_0)$ 	& $21.005^{+10.021}_{-10.008}$ 	& $0.766^{+1.015}_{-1.009}$ 	&$8.030^{+1.048}_{-0.148}$ &$31.598^{+12.115}_{-12.009}$ &$9.600^{+2.167}_{-2.167}$\\ \\
$\Omega_{W} (j_0)$ 	& $9.311^{+2.021}_{-1.008}$ 	& $454.279^{+322.115}_{-309.109}$ 	&$4.40^{+3.148}_{-3.168}$ &$24.842^{+11.115}_{-8.933}$ &$5.520^{+2.667}_{-2.667}$\\ \\
$\Omega_{W} (s_0)$ 	& $62.038^{+32.021}_{-28.008}$ 	& $61.224^{+23.015}_{-19.449}$ 	&$38.789^{+12.148}_{-12.148}$ &$74.641^{+13.015}_{-7.009}$ &$36.561^{+10.133}_{-5.121}$\\ \\
\hline
\end{tabular}
\label{tab:results3}
\end{table}}}

\section{HLG quantum signatures}

To perform the cosmographic statistics, we employ combinations from the observations described and
denote the total $\chi^{2}_{tot}$ function as
\begin{equation}
\chi^2_{tot} = \chi^{2}_{Pantheon} +\chi^{2}_{CC} +\chi^{2}_{H0LICOW} +\chi^{2}_{GW}.
\end{equation}
Our best fits values for the HLG theory parameters are reported in Table \ref{tab:results1}. For the cosmographic parameters derived from the HLG theory in the dust case scenario are reported in Table \ref{tab:results2}. Finally, in Table \ref{tab:results3} we present the values for the derived quantities for the HLG theory using the results reported in the latter Tables. In Figure \ref{fig:cosmography_Horava} we present the cosmographic parameters space for Ho\v rava dust model at $z=0$ using observational samplers described above.

From (\ref{eq:friedmann}), we notice that $a^{-4}$ denotes the contribution from the higher-derivatives terms that are present in the Ho\v rava action and are expected to play a relevant role in the UV limit. This term vanishes in the case of a spatially flat universe $(k=0)$, implying that UV contributions can be detected only if $k\neq 0$. We interpret this result as an indication that the quantum effects in HLG cosmology necessarily lead to a spatially curved universe. This is probably the reason why a comparison of HLG cosmology with the flat $\Lambda$CDM model leads to inconsistencies \cite{Luongo:2018oil}. To evaluate the contribution of the quantum effects in the HLG cosmology as given by (\ref{eq:friedmann}), we introduce the notation
\begin{equation}
\rho_k \equiv \frac{2 k (\Lambda_W-\omega)}{r^2_0a^2},\, \quad \rho_Q\equiv \frac{k^2}{r^4_0a^4}.
\end{equation}
Then, the quantum contribution $\rho_Q$ is related to the curvature contribution $\rho_k$ by
\begin{equation}
\frac{\rho_k^2}{\rho_Q}= 4 (\Lambda_W - \omega)^2\ .
\end{equation}
Afterward, we can compute the constant $\Lambda_W-\omega$ by using our samplers and this will allow us to estimate the contribution of the quantum effects at the cosmological level. Recall that $\Lambda_W$ represents in this theory the cosmological constant. Therefore, the quantum contribution is entirely contained in the constant $\omega$.

In particular, for the best fits obtained using the join samplers Pantheon+CC+GW+H0LiCOW, we can set current bounds at cosmic scales in order to found a quantum signature.
To this end, we evaluate the quantum contribution
\begin{equation}\label{eq:qc}
 \frac{\rho^{2}_{k}}{\rho_Q}= 5.785 \times 10^{-27} m^{-2},
\end{equation}
where we used the value for the cosmological constant $\Lambda = 1.1056\times 10^{-52} m^{-2}$ given by Planck 2018 and our best fit for $\omega$ for the join sampler reported in Table \ref{tab:results3}. This result is crucial since we are obtaining a direct value of the HLG parameter $\omega \approx -3.8\times 10^{-14}$, from late time observations that have effect in the density quantum contribution (\ref{eq:qc}). These effects will have negative sign contributions if we deal with the modified Friedmann constriction $\Omega_{m} +\Omega_{\Lambda} +\Omega_{Q} +\Omega^{*}_{r}=1$, with
$\Omega_{Q}\equiv \Omega_k \left(1- \frac{\Omega_\omega}{\Omega_{\Lambda}}\right), \quad \Omega^{*}_{r}\equiv \frac{\Omega^{2}_{k}}{4\Omega_{\Lambda}}+\Omega_{r}, $ since the HLG terms are related to the curvature density parameter that need to be compensated by the total matter/energy contributions with the values of the parameters $b_1$ and $b_2$.

It is interesting to note that in the IR limit $\omega \to 0$, we obtain values for the cosmographic parameters which agree with the standard $GR$ values for a non-flat cosmology. This works as a smocking gun analysis in order to check the viability of our proposal.

\section{Discussion}

In this paper we present a proposal of inverse cosmography in order to set quantum signatures on Ho\v rava gravity via its cosmographic parameters, without experimenting truncation problems over the Taylor series. With this approach we obtain a HLG generic cosmography scenario, where it is possible to study parameters that came from quantum signatures when a non-flat universe is considered. Notice that this kind of non-flat scenario also has been considered in \cite{Handley:2019tkm,DiValentino:2019qzk}, to explain the presence of a lensing amplitude in CMB linear power spectra in comparison to the standard $\Lambda$CDM, which report a value of $\Omega_k = 0.0007\pm 0.0019$ at $68\%$ CL. Even though, this latter result relaxes the $H_0$ tension problem by estimated a value of $\Omega_k =-0.091\pm 0.037$ at $68\%$ C.L., i.e a closed universe. The problem with this result lies in adding curvature as an additional free parameter to the standard six from $\Lambda$CDM (so-called $\Lambda$CDM+$\Omega_k$) and a deviation of around 3-$\sigma$ arise with respect to Planck 2018 prediction and the BAO observations in different galactic catalogs. With our HLG proposal, we obtain \textit{naturally} a non-flat scenario with $\Omega_k =-0.021^{+0.023}_{-0.029}$ with $H_0 = 71.904^{+1.406}_{-1.347}$, without showing a $3.4$-$\sigma$ inconsistency as it was reported in the latter reference. Also, we employ forecast GW data from Einstein Telescope to enrich the observational redshift range of the other samplers. Moreover, according to \cite{Efstathiou:2020wem}, including BAO sample leads to a strong preference for a flat universe, wherever the CMB likelihood is used. Therefore, to relax this bias, we consider only distance ladder measurements in this work.

The main result of the present work is that by applying the novel method of inverse cosmography in the HLG cosmological model, we can explicitly analyse the contribution of the terms that are relevant in the UV regime, i.e., that are related to the quantum nature of the model.
Notice, however, that we are not testing the HLG model in the UV regime. Indeed, the additional terms  of the HL action lead to the modified Friedman equations, which are then evaluated and confronted with observations in the IR regime. That is, although the additional HL terms are expected to be essential in the UV regime and small in the IR limit, we investigate the possibility that they lead to measurable effects in the modified Friedman equations, according to current observations.
 We obtain as a first result that our universe must be  spatially curved due to the presence of the additional HL terms in the IR limit. This  implies that a different inflationary scenario would be necessary to be in agreement with a spatially curved universe.

  Since in Einstein gravity, the $\Lambda$CDM model predicts a flat universe,
{ we can conclude that classically our universe is flat, but from our results we notice that in this HLG scenario can be curved
due to the presence of additional terms in the HL action.}
Moreover, by using recent results of cosmological and Gravitational Wave observations, we were able to obtain a concrete value for the Ho\v rava parameter  $\omega \approx -3.8\times 10^{-14}$. This is a concrete result of our analysis that  could be compared with results obtained in other scenarios in order to establish the observational validity of HLG at the quantum level.

To perform the cosmographic analysis presented in this work, we have chosen  an explicit  cosmological scenario in a particularly simple HLG model, which has been the subject of intensive criticism regarding stability. We argue that the presence of a tiny cosmological constant could be used as a natural cut-off that prevents instabilities. However, the confirmation of this idea would imply an extensive analysis, which is beyond the scope of the present work. Indeed, the IR limit at the level of the HL action is consistently obtained if the running parameters as chosen as $\lambda\to 1$ and $\omega\to 0$. Stability problems have been found in the limit $\lambda\to 1$. If the additional parameter $\omega$ could be interpreted as corresponding to a supplementary degree of freedom, it would necessary to perform a detailed analysis to see if this could meliorate the stability problem. Another possibility to avoid from the very beginning the problem of instabilities would be to consider the general non-projectable action and probably also without the detailed balance condition. Again, this task is beyond the scope of the present work, but we expect to explore in future investigations.

\section{Acknowledgments}\label{sec:acknowledgements}

CE-R acknowledges the Royal Astronomical Society as FRAS 10147 and the support
by PAPIIT Project IA100220 and would like to acknowledge networking support by the COST Action CA18108.
This work was partially supported  by UNAM-DGAPA-PAPIIT, Grant No. 114520,
Conacyt-Mexico, Grant No. A1-S-31269,
and by the Ministry of Education and Science (MES) of the Republic of Kazakhstan (RK), Grant No.
BR05236730 and AP05133630.

\section*{References}


\section{Introduction}
The dynamics of self-gravitating objects is the topic of great
interest in general theory of relativity (GR), which is modern
theory of gravity. This problem becomes source of inspiration for
researchers when the stellar objects remain in stable for the most
of time against the perturbation caused by the self gravitational
force of the massive objects. This process provides the information
to study structure formation of the gravitationally collapsing
objects. In the relativistic gravitational physics the dynamics of
the stars studied by the Chandrasekhar \cite{1} first time in 1964,
since then there has been growing interest to study the dynamics of
stars in this research direction.  This work was extended by the
Herrera et al.\cite{2,3} explicitly for spherically symmetric heat
conducting, isotropic/anisotropic and viscous fluids in the
framework of GR. Recently, Herrera et al.\cite{5} have investigated
the dynamics of the expansion-free fluids using first order
perturbation of metric components as well as matter variables.

Several properties of the fluid play a dominant role in dynamical
process of the gravitationally collapsing objects. Herrera et
al.\cite{7} have studied the expansion free condition for the
collapsing sphere. Herrera and his collaborators \cite{7f,7a}
discussed the dynamical process of gravitational collapse using
Misner and Sharp's formulation. They considered the matter
distribution with shear-free spherically symmetric. The realistic
model of heat conducting star which shows dissipation in the form of
heat flux in radial direction and shear viscosity was studied by
Chan \cite{7b}. Herrera et al. \cite{7c} also formulated the
dynamical equations of the fluids which contains heat flux,
radiation and bulk viscosity and then coupled these equations with
causal transport equations. The inertia of heat flux and its
significance in the dynamics of dissipative collapse was studied by
Herrera \cite{7d}. The present paper is the particular case
(adiabatic case) of this work in $5D$ Einstein Gauss-Bonnet gravity.
Sharif and Azam \cite{11a}-\cite{13} have studied the effects of
electromagnetic field on the dynamical stability of the collapsing
dissipative and non dissipative fluids in spherical and cylindrical
geometries. This work has been further extended by Sharif and his
collaborators \cite{15}-\cite{22} in modified theories of gravity,
like $f(R)$ and $f(T)$ and $f(R,T)$. Recently \cite{23}, Abbas and
Sarwar have studied the dynamical stability of collapsing star in
Gauss-Bonnet gravity.

The dynamical system dealing with the dimensions greater or equal to
5 are usually discussed in the Gauss-Bonnet gravity theory. The
natural appearance of this theory occurs in the low energy effective
action of the modern string theory. Boulware and Deser \cite{44}
investigated the black hole (BH) solutions in $N$ dimensional string
theory with four dimensional Gauss-Bonnet invariant. This work is
the extension of $N$ dimensional solutions formulated by Tangherili
\cite{45}, Merys and Perry \cite{46}. Wheeller \cite{47} discussed
the spherically symmetric BH solutions with their physical
properties in detail. The topological structure of nontrivial BHs
has been explored by Cai \cite{48}. Kobayashi \cite{49} and Maeda
\cite{50} have formulated the structure of Vaidya BH in Gauss-Bonnet
gravity. All investigations show that the presence of the
Gauss-Bonnet term in the field equations would effect the final
state of the gravitational collapse. Recently \cite{51}, Jhinag and
Ghosh have consider the $5D$ action with the Gauss-Bonnet terms in
Tolman-Bondi model and give an exact model of the gravitational
collapse of a inhomogeneous dust. Motivated by these studies, we
have studies, we have explored the dynamics of the gravitationally
collapsing spheres in Einstein Gauss-Bonnet gravity. This paper is
extension of Herrera work\cite{7d} in Einstein Gauss-Bonnet gravity.
We would like to mention that the objective of this paper is to
study the effects of Gauss-Bonnet term on dynamics of collapsing
system. As heat flux is absent in the source equation, so transport
equations and their coupling with the dynamical equations is not the
objective of this paper. Hopefully it will be discussed explicitly
elsewhere.

This paper is organized as follow: In section \textbf{{2}} the
Einstein Gauss-Bonnet field equations and matching conditions have
been discussed. The dynamical equations have been formulated in
section \textbf{3}. We summaries the results of the paper in the
last section.

\section{Einstein Gauss-Bonnet Field Equations }

Consider the following action in $5D$
\begin{equation}\label{1}
S=\int d^{5}x\sqrt{-g}\left[ \frac{1}{2k_{5}^{2}}\left( R+\alpha
L_{GB}\right) \right] +S_{matter}
\end{equation
where $R$ ia the Ricci scalar in $5D$ and $k^2_{5}={8\pi G_{5}}$ is
coupling constant in $5D$. Also, the Gauss-Bonnet Lagrangian has the
form
\begin{equation}\label{2}
L_{GB}=R^{2}-4R_{\alpha\beta}R^{\alpha\beta}+R_{\alpha\beta
cd}R^{\alpha\beta cd}
\end{equation}
where coefficient $\alpha$ is coupling constant in Einstein
Gauss-Bonnet gravity. Such action is derivable in the low-energy
limiting case of super-string theory. Here, $\alpha $ is treated as
the inverse of string tension which is positive definite and $\alpha
\geq 0$ in this paper. For the $4D$ manifold, Gauss-Bonnet terms do
not contribute to field equations. The variation of action (\ref{1})
with respect to $5D$ metric tensor yields the following set of field
equations
\begin{equation}\label{3}
{G}_{\alpha\beta}=G_{\alpha\beta}+\alpha
H_{\alpha\beta}=T_{\alpha\beta},
\end{equation}
where
\begin{equation}\label{4}
G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R
\end{equation}
is the Einstein tensor and
\begin{equation}\label{5}
H_{\alpha\beta}=2\left[ RR_{\alpha\beta}-2R_{a\alpha }R_{b}^{a
}-2R^{a b
}R_{a\alpha b\beta }+R_{a}^{a b \gamma }R_{b\alpha \beta \gamma
\right] -\frac{1}{2}g_{\alpha\beta}L_{GB},
\end{equation}
is the Lanczos tensor.

A spacelike $4D$ hypersurface $\Sigma^{(e)}$ is taken such that it
divides a $5D$ spacetime into two 5D manifolds, $M^-$ and $M^+$,
respectively. The $5D$ TB spacetime is taken as an interior manifold
$M^-$  which is inner region of a collapsing inhomogeneous and
anisotropic star is given by \cite{51}
\begin{equation}\label{6}
ds_{-}^2=-dt^2+B^2dr^2+C^2(d\theta^2+\sin^2{\theta}d\phi^2
+\sin^2{\theta}\sin^2{\phi}d\psi^2),
\end{equation}
where $B$ and $C$ are functions of $t$ and $r$. The energy-momentum
tensor $T_{\alpha \beta }^{-}$ for anisotropic fluid has the form
\begin{equation}\label{7}
T_{\alpha \beta }^{-}=(\mu +P_{\perp })V_{\alpha }V_{\beta
}+P_{\perp }g_{\alpha \beta }+(P_{r}-P_{\perp })\chi _{\alpha }\chi
_{\beta },
\end{equation}
where $\mu $ is the energy density, $P_{r}$ the radial pressure,
$P_{\perp }$the tangential pressure, $V^{\alpha }$ the four velocity
of the fluid and $ \chi _{\alpha }$ a unit four vector along the
radial direction. These
quantities satisfy
\begin{equation}
V^{\alpha }V_{\alpha }=-1\ \ ,\ \ \ \ \ \chi ^{\alpha }\chi _{\alpha
}=1\ \ ,\ \ \ \ \ \chi ^{\alpha }V_{\alpha }=0  \label{N8}
\end{equation
The expansion scalar $\Theta $ for the fluid is given by
\begin{equation}\label{8}
\Theta =V_{\ ;\ \alpha ,}^{\alpha }.
\end{equation
Since we assumed the metric (6) comoving, then
\begin{equation}\label{9}
V^{\alpha }=\delta _{0}^{\alpha }\ ,\ \ \ \ \ \chi ^{\alpha
}=B^{-1}\delta _{1}^{\alpha }\
\end{equation
and for the expansion scalar, we get
\begin{equation}\label{10}
\Theta =\frac{\dot{B}}{B}+\frac{3\dot{C}}{C}.
\end{equation}
Hence, Einstein Gauss-Bonnet field equations take the for
\begin{eqnarray}\nonumber
{\kappa}^2_{5}\mu &&=\frac{12\left( C^{\prime 2}-B^{2}\left(
1+\dot{C}^{2}\right) \right) }{C^{3}B^{5}}\left[ C^{\prime
}B^{\prime }+B^{2}\dot{C}\dot{B}-BC^{\prime \prime }\right] \alpha
\\\label{11} &&\ \ \ \ \ -\frac{3}{B^{3}C^{2}}\left[ B^{3}\left(
1+\dot{C}^{2}\right) +B^{2}C\dot{C}\dot{B}+CC^{\prime }B^{\prime
}-B(CC^{\prime \prime }+C^{\prime 2})\right]\\\label{12a}
{\kappa}^2_{5}p_{r} &&=-12\alpha \left( \frac{1}{C^{3}}-\frac{C^{^{\prime }2}}{B^{2}C^{3}
+\frac{\dot{C}^{2}}{C^{3}}\right) \ddot{C}+3\frac{C^{^{\prime }2}}{B^{2}C^{2
}  -3\Big(\frac{1+\dot{C}^{2}+C\ddot{C}}{C^{2}}\Big)\\\nonumber
{\kappa}^2_{5}p_{\perp } &&=\frac{4\alpha }{B^{4}C^{2}}\Big[
-2B\left( B^{^{\prime
}}C^{^{\prime }}+B^{2}\dot{B}\dot{C}-BC^{^{\prime \prime }}\right) \ddot{C
 +B\left( C^{^{\prime
}2}-B^{2}\left( 1+\dot{C}^{2}\right) \right)
\ddot{B}\\\nonumber&&+2\Big( \dot{B}C^{^{\prime
}}-B\dot{C}^{^{\prime }}\Big] -\frac{1}{B^{3}C^{2}}\Big[ B^{3}\Big(
1+\dot{C}^{2}+2C\ddot{C}\Big) +B^{2}C\left(
2\dot{C}\dot{B}+C\ddot{B}\right)\\&&+2CC^{^{\prime }}B^{^{\prime
}}-2B\left( CC^{^{\prime \prime }}+C^{^{\prime }2}\right)\Big]
\label{13}\\
 &&\frac{12\alpha }{B^{5}C^{3}}\left( \dot{B}C^{^{\prime
}}-B\dot{C}^{^{\prime
}}\right) \left( B^{2}\left( 1+\dot{C}^{2}\right) -C^{^{\prime }2}\right) -
\frac{B\dot{C}^{^{\prime }}-\dot{B}C^{^{\prime }}}{B^{3}C}=0.
\label{14}
\end{eqnarray}
The mass function $m(t,r)$ analogous to Misner-Sharp mass in $n$
manifold without ${\Lambda}$ is given by \cite{51a}
\begin{equation}\label{15}
m(t,r)=\frac{(n-2)}{2k_{n}^{2}}{V^k}_{n-2}\left[ R^{n-3}\left(
k-g^{ab}R,_{a}R,_{b}\right) +(n-3)(n-4)\alpha \left(
k-g^{ab}R,_{a}R,_{b}\right) ^{2} \right],
\end{equation}
where a comma denotes partial differentiation and ${V^k}_{n-2}$ is
the surface of $(n-2)$ dimensional unit space. For $k=1$,
${V^1}_{n-2}=\frac{2{\pi}^{(n-1)/2}}{\Gamma((n-1)/2)}$, using this
relation with $n=5$ and Eq.(\ref{6}), the mass function (\ref{15})
reduces to
\begin{equation}\label{16}
m(r,t)=\frac{3}{2}\left[ C^{2}\left( 1-\frac{C^{^{\prime }2}}{B^{2
}+\dot{C}^{2}\right) +2\alpha \left( 1-\frac{C^{^{\prime }2}}{B^{2}}+\dot{C
^{2}\right) ^{2}\right]
\end{equation}

In the exterior region to $\Sigma^{(e)}$, we consider Einstein
Gauss-Bonnet Schwarzschild solution which is given by

\begin{equation}\label{c1}
ds_{+}^2=-F(R)d{\nu}^2-2d\nu dR+R^2(d\theta^2+\sin^2{\theta}d\phi^2
+\sin^2{\theta}\sin^2{\phi}d\psi^2),
\end{equation}
where
$F(R)=1+\frac{{R}^2}{4\alpha}-\frac{{R}^2}{4\alpha}\sqrt{1+\frac{16\alpha
M}{\pi {R}^4}}$.

The smooth matching of the $5D$ anisotropic fluid sphere (\ref{6})
to GB Schwarzschild BH solution (\ref{c1}), across the interface at
$r = {r_{\Sigma}}^{(e)}$ = constant, demands the continuity of the
line elements and extrinsic curvature components (i.e., Darmois
matching conditions \cite{53}), implying
\begin{eqnarray}\label{c2}
dt \overset{\Sigma^{(e)}}{=}\sqrt{F(R)}d\nu,\\
R \overset{\Sigma^{(e)}}{=}R, \\\label{cm}
m(r,t)\overset{\Sigma^{(e)}}{=}M,
\end{eqnarray}
\begin{eqnarray}\nonumber
 &&-12\alpha \left( \frac{1}{C^{3}}-\frac{C^{^{\prime }2}}{B^{2}C^{3}
+\frac{\dot{C}^{2}}{C^{3}}\right) \ddot{C}+3\frac{C^{^{\prime }2}}{B^{2}C^{2
}  -3\Big(\frac{1+\dot{C}^{2}+C\ddot{C}}{C^{2}}\Big)\\
 &&\overset{\Sigma^{(e)}}{=}\frac{12\alpha }{B^{5}C^{3}}\left( \dot{B}C^{^{\prime
}}-B\dot{C}^{^{\prime
}}\right) \left( B^{2}\left( 1+\dot{C}^{2}\right) -C^{^{\prime }2}\right) -
\frac{B\dot{C}^{^{\prime }}-\dot{B}C^{^{\prime }}}{B^{3}C}.
\label{c3}
\end{eqnarray}
Comparing Eq.(\ref{c3}) with (\ref{12a}) and (\ref{14}) (for detail
see \cite{12}), we get
\begin{equation}\label{c4}
p_r\overset{\Sigma^{(e)}}{=}0.
\end{equation}
Hence, the matching of the interior inhomogeneous anisotropic fluid
sphere (\ref{6}) with the exterior vacuum Einstein Gauss-Bonnet
spactime (\ref{c1}) produces Eqs.(\ref{6}) and (\ref{cm}).

\section{Dynamical Equations}
In this section, we formulate the equations that deal with the
dynamics of collapsing process in Einstein Gauss-Bonnet gravity.
Following Misner and Sharp formalism \cite{52}, we discuss the
dynamics of the collapsing system. We introduce proper time
derivative as well as the proper radial as follows:
\begin{equation}\label{D1}
D_{T}=\frac{\partial}{\partial{t}},\quad
D_{R}=\frac{1}{R'}\frac{\partial}{\partial{r}},\quad R=C.
\end{equation}
The velocity of the collapsing fluid is the proper time derivative
of $R$ defined as
\begin{equation}\label{D2}
U=D_{T}(R)\equiv \dot{C}.
\end{equation}
Using above result in the mass function given by Eq.(\ref{16})
\begin{equation}\label{D3}
m(r,t)=\frac{3}{2}\left[ C^{2}\left( 1-\frac{C^{^{\prime }2}}{B^{2
}+U^{2}\right) +2\alpha \left( 1-\frac{C^{^{\prime }2}}{B^{2}}+
^{2}\right) ^{2}\right].
\end{equation}
Solving above equation for $\frac{C^{^{\prime }}}{B}$, we get the
positive and negative roots, the positive roots are given by
\begin{equation}\label{D3}
E=\frac{C^{^{\prime
}}}{B}=\sqrt{1+U^2+\frac{R^2}{4\alpha}\pm\frac{\sqrt{3R^2+16m\alpha}}{4\sqrt{3}\alpha}}.
\end{equation}
The rate of change of mass (Eq.(\ref{16})) with respect proper time
is given by
\begin{equation}\label{D3}
D_{{T}}m(t,r)=-{\kappa_5}^2P_{r}U{R}^{3},
\end{equation}
where we have used Einstein Gauss-Bonnet field equations
Eqs.(\ref{12a}) and (\ref{14}). The right hand side of this equation
has a single term . The this term is due to effective pressure
(means the pressure is effected by the Gauss-Bonnet term) in
$r$-direction. This term is positive in case of collapse ($U<0$).
This implies that as effective pressure in $r$-direction increases,
mass (energy) also increases with the same amount. Similarly, we can
calculate
\begin{equation}\label{D4}
D_{R}m(t,r)=\frac{2}{3}k^2_{5}\mu R^{3},
\end{equation}
where we have used Einstein Gauss-Bonnet field equations
Eqs.(\ref{11}) and (\ref{14}). This equation explain how effective
energy density affects the mass between neighboring hypersurfaces in
the interior fluid distribution. Integration of Eq.(\ref{D4}) yields
\begin{equation}\label{D5}
m(t,r)=\frac{2}{3}k^2_{5}\int^{{R}}_{0}({R}^{3}\mu)d{R}.
\end{equation}
The dynamical equations can be obtained from the contracted Bianchi
identities ${T^{ab}}_{;b}=0$. Consider the following two equations
\begin{eqnarray}
{T^{\alpha\beta}}_{;\beta}V_{\alpha}&&=\left[ \dot{\mu}+\left( \mu
+P_{r}\right) \frac{\dot{B}}{B}+3\left( \mu +P_{\perp }\right)
\frac{\dot{C}}{C}\right] =0 , \label{D6}\\ T_{;\beta }^{\ \alpha
\beta }\chi _{\alpha }&&=\frac{1}{B}\left[
P_{r}^{^{\prime }}+3\left( P_{r}-P_{\perp }\right) \frac{C^{^{\prime }}}{C
\right] =0. \label{D7}
\end{eqnarray}
The acceleration of the collapsing fluid is defined as
\begin{equation}\label{D8}
D_{{T}}U=\ddot{C}.
\end{equation}
Using Eqs.(\ref{D7}), (\ref{D8}) and (\ref{12a}), we get
\begin{eqnarray}
&&\Big[12\alpha
C\Big(\frac{9(p_r-p_{\bot})^2}{C^3}(1+U^2)+(\frac{{p_r}'}{
B^2}+1)\Big)\Big]D_T
U\\\nonumber&&=-9(p_r-p_{\bot})^2\Big[\kappa^2_5p_rC+3(\frac{1}{C}+\frac{U^2}{C})\Big].\label{D9}
\end{eqnarray}
This equation yields the effect of different forces on the
collapsing process. It can be interpreted in the form of Newton's
second law of motion i.e., Force = mass density $\times$
acceleration. The term within square bracket on left side of above
equation represent the inertial or passive gravitational mass. All
the quantities on right side in square bracket are positive, hence
this side is consequently negative and implies the retardation of
dynamical system giving rise to the collapse of the system.

\section{Outlook}

This paper investigates the effects of the Gauss-Bonnet term on the
dynamics of anisotropic fluid collapse in the $5D$ Einstein
Gauss-Bonnet gravity. We have extended the work of Herrera \cite{7d}
to $5D$ Einstein Gauss-Bonnet gravity. To this end, the
non-conducting anisotropic fluid with $5D$ spherical symmetry has
been taken as the source of gravitation in Einstein Gauss-Bonnet
gravity. The Misner-Sharp mass has been calculated in the present
scenario. The smooth matching of the interior source has been
carried out with $5D$ Schwarzschild BH solution in Einstein
Gauss-Bonnet gravity by using the Darmois \cite{53} junction
conditions. The matching of the two regions implies the vanishing of
radial pressure over the boundary of the star and continuity of the
gravitational masses in the interior and exterior regions. By using
the Misner and Sharp approach for the proper time and radial
derivatives, we have formulated the velocity as well as acceleration
of the system. These definitions have been also applied to formulate
the general dynamical equations in Gauss-Bonnet gravity.

The analysis of the dynamical equations predicts the following
consequences:
\begin{itemize}
\item Mass of the collapsing spheres increases with the passage of
time.
\item Effective energy density of the system would effects the mass of the system during the different stages of the
collapse
\item The system under consideration goes to retardation implying the gravitational
collapse.
\end{itemize}

We would like to mention that transport equations and their coupling
with dynamical equations are not the objective of this paper as heat
flux is absent in anisotropic fluid. This will be done in an other
investigation with the \textbf{inclusion of charge term in the
interior source }in future.
 \vspace{0.25cm}


\section*{Introduction}

Let $F$ be a finite field extension of $\mathbb Q_p$ and $L$ be the completion of the maximal unramified extension of $F$. Let $G$ be a connected reductive algebraic group over $F$ and $\sigma$ be the Frobenius morphism. We denote by $B(G)$ the set of $\sigma$-conjugacy classes of $G(L)$. The set $B(G)$ is classified by Kottwitz in \cite{Ko1} and \cite{Ko2}. This classification generalize the Dieudonn\'e-Manin classification of isocrystals by their Newton polygons.

Let $\tilde W$ be the Iwahori-Weyl group of $G$. Let $\{\mu\}$ be a conjugacy class of characters of $G$ defined over $L$.
Let $\Adm(\mu) \subset \tilde W$ be the admissible subset of $\tilde W$ (\cite{KR1}) and $B(G, \mu)$ be the finite subset of $B(G)$ defined by the group-theoretic version of Mazur's theorem \cite[\S 6]{Ko2}.

The main result of this paper is as follows.

\begin{thm}\label{main}
The set $B(G, \mu)$ contains a unique maximal element and this element is represented by an element in $\Adm(\mu)$.
\end{thm}

For quasi-split groups, this is obvious as the unique maximal element of $B(G, \mu)$ is represented by $t^\mu$. However, it is much more complicated for non quasi-split groups.

This result is an important ingredient in the proof \cite{He00} of Kottwitz-Rapoport conjecture \cite[Conjecture 3.1]{KR} and \cite[Conjecture 5.2]{R:guide} on the union of affine Deligne-Lusztig varieties. The knowledge of the explicit description of the maximal element of $B(G, \mu)$ is also useful in the study of $\mu$-ordinary locus of Shimura varieties.

\section{Preliminaries}

\subsection{} We first recall the classification of $B(G)$ obtained by Kottwitz in \cite{Ko1} and \cite{Ko2}.

For any $b \in G(L)$, we denote by $[b]$ the $\sigma$-conjugacy class of $G(L)$ that contains $b$. Let $\Gamma_F=\Gal(\bar L/F)$ be the absolute Galois group of $F$. Let $\kappa_G: B(G) \to \pi_1(G)_{\Gamma_F}$ be the Kottwitz map \cite[\S 7]{Ko2}. This gives one invariant.

Another invariant is given by the Newton map.

Let $S$ be a maximal $L$-split torus that is defined over $F$ and let $T$ be its centralizer. Since $G$ is quasi-split over $L$, $T$ is a maximal torus. We also fix a $\sigma$-invariant alcove $\mathbf a$ in the apartment of $G_L$ corresponding to $S$.

To an element $b \in G(L)$, we associate its Newton point $\nu_b$. It is a $\sigma$-invariant element in the closed dominant chamber $X_*(T)_\mathbb Q^+$.

By \cite[\S 4.13]{Ko2}, the map $$B(G) \to X_*(T)_\mathbb Q^+ \times \pi_1(G)_{\Gamma_F}, \qquad b \mapsto (\nu_b, \kappa_G(b))$$ is injective.

The partial order on $B(G)$ is defined as follows. Let $b, b' \in G(L)$, then $[b] \leqslant [b']$ if $\kappa_G(b)=\kappa_G(b')$ and $\nu_b \leqslant \nu_{b'}$, i.e., $\nu_{b'}-\nu_b$ is a non-negative $\mathbb Q$-linear combination of positive relative coroots.

\subsection{} We follow \cite{HR}. Let $N$ be the normalizer of $T$. The {\it finite Weyl group} assoicated to $S$ is $W_0=N(L)/T(L).$ The Iwahori-Weyl group associated to $S$ is $\tilde W=N(L)/T(L)_1$, where $T(L)_1$ denotes the unique Iwahori subgroup of $T(L)$. The Frobenius morphism $\sigma$ induces actions on $W$ and $\tilde W$, which we still denote by $\sigma$.

Let $\Gamma=\Gal(\bar L/L)$. The Iwahori-Weyl group $\tilde W$ contains the affine Weyl group $W_a$ as a normal subgroup and $$\tilde W=W_a \rtimes \Omega,$$ where $\Omega \cong \pi_1(G)_{\Gamma}$ is the normalizer of the alcove $\mathbf a$. The Bruhat order on $W_a$ extend in a natural way to $\tilde W$.

Let $G_{sc}$ be the simply connected cover of the derived group of $G$. Denote by $T_{sc}$ the maximal torus of $G_{sc}$ given by the choice of $T$. Then we have a natural injective map $X_*(T_{sc})_\Gamma \to X_*(T)_\Gamma$. We fix a special vertex of $\mathbf a$ and represent $\tilde W$ and $W_a$ as \begin{gather*} \tilde W=X_*(T)_\Gamma \rtimes W_0=\{t^\lambda w; \lambda \in X_*(T)_\Gamma, w \in W_0\}, \\ W_a=X_*(T_{sc})_\Gamma \rtimes W_0=\{t^\lambda w; \lambda \in X_*(T_{sc})_\Gamma, w \in W_0\}. \end{gather*}

\subsection{} For any $w \in \tilde W$, we choose a representative in $N(L)$ and also write it as $w$. By \cite[\S 3]{He99}, any $\sigma$-conjugacy class of $G(L)$ contains an element in $\tilde W$. The restriction of Kottwitz map and Newton map on $\tilde W \subset G(L)$ can be described explicitly as follows.

The map $N(L) \to G(L)$ induces a map $\tilde W \to B(G)$. Here $\kappa_G(w)$ is the image of $w$ under the projection $\tilde W \to \Omega \cong \pi_1(G)_\Gamma \to \pi_1(G)_{\Gamma_F}$.

For any $w \in \tilde W$, we consider the element $w \sigma \in \tilde W \rtimes \<\sigma\>$. There exists $n \in \mathbb N$ such that $(w \sigma)^n=t^\lambda$ for some $\lambda \in X_*(T)_\Gamma$. Let $\nu_{w, \sigma}=\lambda/n$ and $\bar \nu_{w, \sigma}$ the unique dominant element in the $W_0$-orbit of $\nu_{w, \sigma}$. It is known that $\nu_{w, \sigma}$ is independent of the choice of $n$ and is $\Gamma$-invariant. Moreover, $\bar \nu_{w, \sigma}$ is the Newton point of $w$ when regarding $w$ as an element in $G(L)$.

\subsection{}\label{set-up} Let $\mathbb S$ and $\tilde{\mathbb S}$ be the set of simple reflections of $W_0$ and $W_a$ respectively. Then $\sigma(\tilde{\mathbb S})=\tilde{\mathbb S}$. In general $\mathbb S$ is not $\sigma$-stable since the special vertex of $\mathbf a$ may not be $\sigma$-invariant. However, we may write $\sigma$ as $\sigma=\tau \circ \sigma_0$, where $\sigma_0$ is a diagram automorphism of $W_0$ and the induced action of $\tau$ on the adjoint group $G_{ad}$ is inner.

Let $N$ be the order of $\sigma_0$. For $\mu \in X_*(T)$, we set $$\mu^\diamond=\frac{1}{N} \sum_{i=0}^{N-1} \sigma_0^i(\mu) \in X_*(T)_\mathbb Q.$$ Let $\mu^\sharp$ be the image of $\mu$ in $\pi_1(G)_{\Gamma_F}$. Set $$B(G, \mu)=\{[b] \in B(G); \kappa_G(b)=\mu^\sharp, \nu_b \leqslant \mu^\diamond\}.$$ The elements in $B(G, \mu)$ are called the (neutral) acceptable elements for $\mu$.

Let $\underline \mu$ be the image of $\mu$ in $X_*(T)_\Gamma$. The $\mu$-admissible set is defined as $$\Adm(\mu)=\{w \in \tilde W; w \leqslant t^{x(\underline \mu)} \text{ for some }x \in W_0\}.$$

Now we may reformulate the main theorem \ref{main} as follows.

\begin{thm}\label{main'}
We keep the notation as in \S \ref{set-up}. Set $B(\tilde W, \mu, \sigma)=\{\bar \nu_{w, \sigma}; w \in t^\mu W_a, \bar \nu_{w, \sigma} \leqslant \mu^\diamond\}$. Then

(1) The set $B(\tilde W, \mu, \sigma)$ contains a unique maximal element $\nu$.

(2) There exists an element $w \in \Adm(\mu)$ with $\bar \nu_{w, \sigma}=\nu$.
\end{thm}

\section{The maximal element in $B(G, \mu)$}

\subsection{}\label{adj} Let $G_{ad}$ be the adjoint group of $G$, i.e., the quotient of $G$ by its center. Since the buildings of $G$ and $G_{ad}$ coincide, the choice of an alcove $\mathbf a$ in the building of $G$ determines an alcove of $G_{ad}$. Then the Iwahori-Weyl group $\tilde W_{ad}$ of $G_{ad}$ is $X_*(T_{ad})_\Gamma \rtimes W_0$. Let $\pi: G \to G_{ad}$ be the projection map. Set $T_{ad}=\pi(T)$. Then $\pi$ induces maps $\tilde W \to \tilde W_{ad}$ and $X_*(T)_\mathbb Q^+ \to X_*(T_{ad})_\mathbb Q^+$, which we still denote by $\pi$.

It is easy to see that $\pi(\nu_{w, \sigma})=\nu_{\pi(w), \sigma}$ for $w \in \tilde W$ and $\pi$ induces a bijection of posets from $B(\tilde W, \mu, \sigma)$ to $B(\tilde W_{ad}, \pi(\mu), \sigma)$. Thus Theorem \ref{main'} holds for $B(\tilde W, \mu, \sigma)$ if and only if it holds for $B(\tilde W_{ad}, \pi(\mu), \sigma)$.

\subsection{}\label{adjoint}
In the rest of this section, we assume that $G$ is adjoint. We write $\sigma$ as $\sigma=\text{Ad}(\tau) \circ \sigma_0$, where $\tau \in \tilde W$ is a length zero element and $s_0$ is a diagram automorphism of $W_0$. Then $\nu_{w, \sigma}=\nu_{w \tau, \sigma_0}$ for all $w \in \tilde W$.

Set $V=X_*(T)_\Gamma \otimes_\mathbb Z \mathbb R$. For any $i \in \mathbb S$, let $\omega^\vee_i \in V$ be the fundamental coweight and $\alpha_i^\vee \in V$ be the simple coroot. We denote by $\omega_i, \alpha_i \in V^*$ the fundamental weight and simple root, respectively.

We also fix $\lambda \in X_*(T)_+$ such that $\tau \in t^\lambda W_0$. For each $\sigma_0$-orbit $c$ of $\mathbb S$, we set $\omega_c=\sum_{i \in c} \omega_i$, where $\omega_i$ is the fundamental weight for $i$. For any $v \in X_*(T)_\mathbb Q^+$, we set $J(v)=\{s \in \mathbb S; s(v)=v\}$ and $I(v)=\mathbb S \backslash J(v)$. If $v=\sigma_0(v)$, then both $J(v)$ and $I(v)$ are $\sigma_0$-stable.

The follow lemma is essentially contained in \cite{Chai}. Due to its importance, we provide a proof for completeness.
\begin{lem}\label{Newton}
Let $v \in X_*(T)_\mathbb Q^+$ with $\sigma_0(v)=v$. Then $v =\nu_{w, \sigma}$ for some $w \in t^\mu W_a$ if and only if $\<\omega_c, \mu^\diamond + \lambda^\diamond-v\> \in\mathbb Z$ for any $\sigma_0$-orbit $c$ of $I(v)$.
\end{lem}
\begin{proof}

Since $\nu_{w, \sigma}=v$, we have $w \tau=t^\gamma x$ for some $\gamma \in X_*(T)_\Gamma$ and $x \in W_{J(v)}$. Let $N_0$ be the order of $W_0 \rtimes \<\sigma_0\>$. Then
\begin{align*} \nu_{w, \sigma}=\nu_{w \tau, \sigma_0} &=\frac{1}{N_0} \sum_{k=0}^{N_0-1} (x \sigma_0(x) \cdots \sigma_0^{k-1}(x)) \sigma_0^k(\gamma) \\ & \in \frac{1}{N_0} \sum_{k=0}^{N_0-1} \sigma_0^k(\gamma)+ \sum_{j \in J(v)} \mathbb Q \alpha_j^\vee \\ &=\gamma^\diamond + \sum_{j \in J(v)} \mathbb Q \alpha_j^\vee.\end{align*}

If $w \in t^\mu W_a$, then $w \tau \in t^{\mu+\lambda} W_a$ and $\mu+\lambda-\gamma \in X_*(T_{sc})_\Gamma$. Hence $\<\omega_c, \mu^\diamond+\lambda^\diamond-\gamma^\diamond\>=\<\omega_c, \mu+\lambda-\gamma\> \in \mathbb Z$.

On the other hand, let $a_c=\<\omega_c, \mu^\diamond + \lambda^\diamond-v\> \in \mathbb Z$ for each $\sigma_0$-orbit $c$ of $I(v)$. We construct an element $w \in t^\mu W_a$ such that $\nu_{w,\sigma}=v$.

For each $\sigma_0$-orbit of $J(v)$, we choose a representative. Let $x$ be the product of these representatives (in some order). Then $x$ is a $\sigma_0$-twisted Coxeter element of $W_{J(v)}$ in the sense of \cite[7.3]{Spr}. For each $\sigma_0$-orbit $c$ of $I(v)$, we choose a representative $i_c$. Let $\alpha^\vee_{i_c}$ be the corresponding simple coroot in $X_*(T_{sc})_\Gamma$. Set $\beta=\mu+\lambda-\sum_c a_c \alpha^\vee_{i_c}$ and $w=t^{\beta} x \tau ^{-1} \in t^\mu W_a$. Write $\beta=h+r$ with $r \in \sum_{j \in J(v)} \mathbb Q \alpha_j^\vee$ and $h \in \sum_{i \in I(v)} \mathbb Q \omega_j^\vee$. Then
\begin{align*} \nu_{w, \sigma}=\nu_{w \tau, \sigma_0} &=\frac{1}{N_0} \sum_{i=0}^{N_0-1} (x \sigma_0)^i(\beta) \\ &=h^\diamond + \frac{1}{N_0} \sum_{k=0}^{N_0-1}(x \sigma_0)^k(r) \\ &=h^\diamond=\mu^\diamond+\lambda^\diamond-\sum_c a_c (\alpha^\vee_{i_c})^\diamond - r^\diamond,\end{align*} where the fourth equality is due to the fact that $x$ is $\sigma_0$-elliptic in $W_{J(v)}$.

Hence for any $\sigma_0$-orbit $c$ of $I(v)$ and any $j \in J(v)$, we have $$\<\omega_c, \mu^\diamond+\lambda^\diamond-\nu_{w, \sigma}\>=\<\omega_c, \sum_{c'} a_{c'} (\alpha^\vee_{i_{c'}})^\diamond\>=a_c=\<\omega_c, \mu^\diamond+\lambda^\diamond-v\>$$ and $$\<\alpha_j, \mu^\diamond+\lambda^\diamond-\nu_{w,\sigma}\>=\<\alpha_j, \mu^\diamond+\lambda^\diamond\>=\<\alpha_j, \mu^\diamond+\lambda^\diamond-v\>,$$ which means $\nu_{w,\sigma}=v$ as desired.
\end{proof}

\begin{cor}
$\mu^\diamond \in B(G,\mu)$ if and only if $\<\omega_c, \lambda^\diamond\> \in \mathbb Z$ for any $\sigma_0$-orbit $c$ of $I(\mu^\diamond)$. In this case, $\mu^\diamond$ is a priori the maximal Newton polygon of $B(G,\mu)$.
\end{cor}

\subsection{} We follow \cite[\S 6]{Chai}. For any $\sigma_0$-stable subset $B$ of $X_*(T)_\mathbb Q^+$, we define
$$C_{\geqslant B}=\{v \in X_*(T)_\mathbb Q^+; \sigma_0(v)=v \text{ and } v \geqslant b, \forall b \in B\}.$$ We say $B$ is {\it reduced} if $C_{\geqslant B'} \subsetneqq C_{\geqslant B}$ for any $\sigma_0$-stable proper subset $B' \subsetneqq B$.

For any $i \in \mathbb S$, let $$pr_{(i)}: V=\mathbb R \omega^\vee_i \oplus \sum_{j \neq i} \mathbb R \alpha^\vee_j \to \mathbb R \omega^\vee_i$$ be the projection map.

Now we prove part (1) of Theorem \ref{main'}.

\subsection{Proof of Theorem \ref{main'} (1)}
For any $\sigma_0$-orbit $c$ of $\mathbb S$ and $i \in c$, we define $e_i \in \mathbb Q \omega_i^\vee$ by $$\<\omega_i, e_i\>=\frac{1}{\sharp c}\max(\{t \in \<\omega_c, \mu^\diamond+\lambda^\diamond\> + \mathbb Z; t \leqslant \<\omega_c, \mu^\diamond\> \} \cup \{0\}).$$ Let $E_0=\{e_i; i \in \mathbb S\}$ and $E \subset E_0$ be a $\sigma_0$-stable subset which is reduced and satisfies $C_{\geqslant E}=C_{\geqslant E_0}$. Let $I(E)=\{i \in \mathbb S; e_i \in E\}$. By \cite[Theorem 6.5]{Chai}, there exists an element $\nu \in C_{\geqslant E}$ defined by $\mu^\diamond + \lambda^\diamond -\nu \in X_*(T_{sc})_\Gamma \otimes_\mathbb Z \mathbb R$, $I(\nu)=I(E)$ and $\<\omega_j, \nu\>=\<\omega_j, e_j\>$ for $j \in I(E)$, which satisfies $C_{\geqslant \nu}=C_{\geqslant E}=C_{\geqslant E_0}$. Since $\mu^\diamond \in C_{\geqslant E}$, we have $\nu \leqslant \mu^\diamond$. By Lemma \ref{Newton}, $\nu \in B(\tilde W, \mu, \sigma)$.

Let $\nu' \in B(\tilde W,\mu, \sigma)$. Set $E(\nu')=\{pr_{(j)}(\nu'); j \in I(\nu')\}$. By Lemma \ref{Newton} and the inequality $\nu' \leqslant \mu^\diamond$, we have, for any $\sigma_0$-orbit $c$ of $I(\nu')$ and $j \in c$, that $$\sharp{c} \cdot \<\omega_j, pr_{(j)}(\nu')\>=\sharp{c} \cdot \<\omega_j, \nu'\>=\<\omega_c, \nu'\> \in \<\omega_c, \mu^\diamond +\lambda^\diamond\> + \mathbb Z$$ and $$\sharp c \cdot \<\omega_j, pr_{(j)}(\nu')\> \leqslant \sharp c \cdot \<\omega_j, \mu^\diamond\>=\<\omega_c, \mu^\diamond\>.$$ So $\<\omega_j, pr_{(j)}(\nu')\> \leqslant \<\omega_j, e_j\>$, that is, $pr_{(j)}(\nu') \leqslant e_j \leqslant \nu$ for $j \in I(\nu')$. By \cite[Lemma 6.2 (i)]{Chai}, we deduce that $\nu' \leqslant \nu$. Therefore $\nu$ is the unique maximal element of $B(\tilde W,\mu, \sigma)$.

\section{Reduction to the irreducible case}

\begin{lem}\label{change}
Let $\tau \in \Omega$. Then Theorem \ref{main'} holds for $(\tilde W, \mu, \sigma)$ if and only if it holds for $(\tilde W, \mu, \tau \sigma \tau ^{-1})$.
\end{lem}
\begin{proof}
For any $v \in V$, $\bar \tau(v)=\bar v$. Thus $\tau \Adm(\mu) \tau ^{-1}=\Adm(\mu)$ and $\bar \nu_{w, \tau \sigma \tau ^{-1}}=\overline{\tau(\nu_{\tau ^{-1} w \tau, \sigma})}=\bar \nu_{\tau ^{-1} w \tau, \sigma}$. Therefore $B(\tilde W, \mu, \tau \sigma \tau ^{-1})=B(\tilde W, \mu, \sigma)$. Since conjugation by $\tau$ preserves the Bruhat order, Theorem \ref{main'} (2) holds for $(\tilde W, \mu, \sigma)$ if and only if it holds for $(\tilde W, \mu, \tau \sigma \tau ^{-1})$.
\end{proof}

\subsection{}\label{3.1} In the rest of this section, we assume that $G$ is adjoint and $\sigma$ acts transitively on the set of connected components of the affine Dynkin diagram of $W_a$. In other words, $\tilde W=\tilde W_1 \times \cdots \tilde W_m$, where $\tilde W_1 \cong \cdots \cong \tilde W_m$ are extended affine Weyl groups of adjoint type with connected affine Dynkin diagram and $\sigma(\tilde W_1)=\tilde W_2, \cdots, \sigma(\tilde W_m)=\tilde W_1$. After conjugating a suitable element in $\Omega$, we may assume that $\sigma=\text{Ad}(\tau) \circ \sigma_0$ with $\tau \in \tilde W_m$. We may write $\mu$ as $\mu=(\mu_1, \cdots, \mu_m)$, where $\mu_i$ is a dominant coweight for $\tilde W_i$. Set $\gamma=\sum_{i=1}^m \sigma_0^{m-i} (\mu_i)$. Then the natural projection induces a bijection from $B(\tilde W, \mu, \sigma)$ to $B(\tilde W_m, \gamma, \sigma^m)$.

\begin{lem}\label{irreducible}
We keep the notations in \S \ref{3.1}. If Theorem \ref{main'} (2) holds for $(\tilde W_m, \underline \mu, \sigma^m)$, then it holds for $(\tilde W, \mu, \sigma)$.
\end{lem}
\begin{proof}
Let $\nu$ be the maximal element in $B(\tilde W_m, \gamma, \sigma^m)$. By assumption, there exists $w \in \Adm(\gamma)$ such that $\bar \nu_{w, \sigma^m}=\nu$. By definition, there exists $x$ in the finite Weyl group associated to $\tilde W_m$ such that $w \leqslant t^{x(\underline \gamma)}$. Since $\ell(t^{x(\underline \gamma)})=\sum_{i=1}^m \ell(t^{x(\sigma_0^{m-i}(\underline{\mu_i)})})$, there exists $w_i \in \tilde W_m$ for each $i$ such that $w=w_1 \cdots w_m$ and $w_i \leqslant t^{x(\sigma_0^{m-i}(\underline{\mu_i)})}$ for all $i$. Hence $\sigma_0^{i-m}(w_i) \leqslant t^{x(\underline{\mu_i})}$. Set $y=(\sigma_0^{1-m}(w_i), \cdots, w_m) \in \tilde W$. Then $y \in \Adm(\mu)$ and $\nu_{y, \sigma}=(\sigma \nu_{w, \sigma^m}, \cdots, \nu_{w, \sigma^m})$. Hence $\bar \nu_{y, \sigma}=(\sigma_0 \nu, \cdots, \nu)$ is the maximal element in $B(\tilde W, \mu, \sigma)$.
\end{proof}

\section{Reduction to the superbasic case}

\subsection{} Let $\epsilon \in \tilde W \rtimes\<\sigma\>$. We say that $\epsilon$ is {\it superbasic} if $\ell(\epsilon)=0$ and each $\epsilon$-orbit on $\tilde{\mathbb S}$ is a union of the connected components of the affine Dynkin diagram of $\tilde W$. By \cite[3.5]{HN}, $\epsilon$ is superbasic if and only if $W_a=W_1^{m_1} \times \cdots W_l^{m_l}$, where $W_i$ is an affine Weyl group of type $\tilde A_{n_i-1}$ and $\epsilon$ gives an order $n_i m_i$ permutation on the set of simple reflections of $W_i^{m_i}$.

\subsection{}\label{4.2} The main purpose of this section is to reduce to the case where $\sigma$ is superbasic. We keep the assumption in $\S$\ref{adjoint}.

For any $J \subset \mathbb S$, let $W_J$ be the subgroup of $W_0$ generated by $s_j$ for $j \in J$ and ${}^J W_0$ be the set of minimal coset representatives in $W_J \backslash W_0$. Let $\tilde W_J=X_*(T)_\Gamma \rtimes W_J$.

We regard $\sigma$ as an element in $\tilde W \rtimes \<\sigma\>$. We will construct a superbasic element in $\tilde W_J \rtimes \<\sigma_0\>$ for a suitable subset $J \subset \mathbb S$ with $\sigma_0(J)=J$. We follow the approach in \cite[\S 5]{HN2}.

Let $V^\sigma$ be the fixed point set of $\sigma$. Since $\sigma$ is an affine transformation on $V$ of finite order, $V^\sigma$ is a nonempty affine subspace. Set $V'=\{v-e; v \in V^\sigma\}$, where $e$ is an arbitrary point of $V^\sigma$. Then $V'$ is the (linear) subspace of $V$ parallel to $V^\sigma$. We choose a generic point $v_0$ of $V'$, i.e., for any root $\alpha$, $\<\alpha, v\>=0$ implies that $\<\alpha, v'\>=0$ for all $v' \in V'$. Let $\bar v_0$ be the unique dominant element of the $W_0$-orbit of $v_0$. We set $I=I(\bar v_0)$ and $J=J(\bar v_0)$. Let $z \in {}^J W_0$ be the unique element with $\bar v_0=z(v_0)$. Set $\sigma^J=z \sigma z ^{-1}$.

\begin{lem}
The element $\sigma^J$ is a superbasic element in $\tilde W_J \rtimes \<\sigma_0\>$.
\end{lem}

\begin{proof}
Since $\sigma(0)=\lambda$, $\sigma(v_0)=v_0+\lambda$ and $\sigma^J(\bar v_0)=\bar v_0+z(\lambda)$. Write $\sigma^J$ as $\sigma^J=t^{z(\lambda)}u \sigma_0$ for some $u \in W_0$. Then $u\sigma_0(\bar v_0)=\bar v_0$. Therefore $\sigma(\bar v_0)=u ^{-1} \bar v_0$ is the unique dominant element in the $W_0$-orbit of $v_0$. Hence $\bar v_0=\sigma(\bar v_0)=u ^{-1} \bar v_0$. Therefore $u \in W_J$ and $\sigma_0(J)=J$.

Let $\ell_J$ be the length function on $\tilde W_J \rtimes \<\sigma_0\>$. By \cite[Proposition 3.2]{HN2}, $\ell_J(\sigma^J)=0$.
Since $v_0$ is generic, by \cite[\S 5.5]{HN2}, $V^{u \sigma_0} \subset V^{W_J}$. Therefore there is no nonempty subset of $J$ that is stable under $\sigma^J$. Hence each orbit of $\sigma^J$ on the set of simple reflections of $\tilde W_J$ is a union of connected components of the affine Dynkin diagram of $\tilde W_J$. Hence $\sigma^J$ is superbasic.
\end{proof}

\begin{lem}\label{integer} We keep the notations in \S \ref{4.2}. Then

(1) $z(\lambda)^\diamond \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$.

(2) Let $c$ be a $\sigma_0$-orbit of $\mathbb S$. Then $\<\omega_c, \lambda^\diamond\> \in \mathbb Z$ if and only if $c \subset I$.
\end{lem}
\begin{proof}
Assume $z(\lambda) \in r+h$ with $r \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$ and $h \in \sum_{i \in I} \mathbb Q \omega_i^\vee$. Since $\sigma$ is of length zero, we have $\nu_{t^{z(\lambda)}u, \sigma_0}=0$, which implies $h^\diamond=0$. Hence $z(\lambda)^\diamond \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$ and (1) is proved.

Write $\lambda=z(\lambda)+\theta$ for some $\theta \in X_*(T_{sc})_\Gamma$. We have $$\<\omega_c, \lambda^\diamond\>=\<\omega_c, z(\lambda)^\diamond\> + \<\omega_c, \theta\> = \<\omega_c, r^\diamond\>+ \<\omega_c, \theta\> \equiv \<\omega_c, r\> \mod \mathbb Z.$$ Hence $\<\omega_c, \lambda^\diamond\> \in \mathbb Z$ if $c \subset I$. On the other hand, since $\sigma^J$ is a super-basic element of $\tilde W_J$, $\tilde W_J$ has only type $A$ factors. One may check directly that $\<\omega_c, r\>=\<\omega_c^J, r\> \notin \mathbb Z$ for any $\sigma_0$-orbit $c$ of $I$ and (2) is proved.
\end{proof}

\begin{prop}\label{image}
The maximal Newton point of $B(\tilde W, \mu, \sigma)$ is contained in the natural inclusion $B(\tilde W_J, \mu, \sigma^J) \hookrightarrow B(\tilde W, \mu, \sigma)$.
\end{prop}
\begin{proof}
We denote by $\omega_j^J \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$ the corresponding fundamental coweight of $\Phi_J$ and set $\omega_c^J=\sum_{j \in c} \omega_j^J$ for any $\sigma_0$-orbit of $c$ of $J$. Let $\nu$ be the maximal Newton point of $B(\tilde W, \mu, \sigma)$. By the proof of Theorem \ref{main'} (1), for each $\sigma_0$-orbit $c$ of $\mathbb S$, $$\<\omega_c, \mu^\diamond\> \geqslant \<\omega_c, \nu\> \geqslant \<\omega_c, \mu^\diamond+\lambda^\diamond\>-\lceil \<\omega_c, \lambda^\diamond\> \rceil.$$

Let $c$ be a $\sigma_0$-orbit of $I$. By Lemma \ref{integer} (2), $\<\omega_c, \lambda^\diamond\> \in \mathbb Z$. Hence $\<\omega_c, \mu^\diamond\>=\<\omega_c, \nu\>$ and $\mu^\diamond - \nu \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$.

By Lemma \ref{integer} (1), $z(\lambda)^\diamond \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$. Thus $\mu^\diamond+z(\lambda)^\diamond-\nu \in \sum_{j \in J} \mathbb Q \alpha_j^\vee$. Now $c'$ be a $\sigma_0$-orbit in $I(\nu) \cap J$. Then
$$\<\omega_{c'}^J, \mu^\diamond+z(\lambda)^\diamond-\nu\> = \<\omega_{c'}, \mu^\diamond+z(\lambda)^\diamond-\nu\>=\<\omega_c, \mu^\diamond+\lambda^\diamond-\nu \> - \<\omega_{c'}, \theta\>,$$ where $\theta=\lambda-z(\lambda) \in X_*(T_{sc})_\Gamma$. by Lemma \ref{Newton} $\<\omega_{c'}, \mu^\diamond+\lambda^\diamond-\nu\> \in \mathbb Z$. Hence $\<\omega_{c'}^J, \mu^\diamond+z(\lambda)^\diamond-\nu\> \in \mathbb Z$. Again by Lemma \ref{Newton}, $\nu \in B(\tilde W_J, \mu, \sigma^J)$.
\end{proof}

\begin{lem}
Let $J \subset \mathbb S$ and $x \in {}^J W_0$. If $w, w' \in \tilde W_J$ with $w \leqslant_J w'$ for the Bruhat order of $\tilde W_J$, then $z ^{-1} w z \leqslant z ^{-1} w' z$ for the Bruhat order of $\tilde W$.
\end{lem}

\begin{proof}
It suffices to consider the case where $w'=w s_\alpha$ for some positive affine root $\alpha$ of $\tilde W_J$. Since $w' \geqslant w$, $w(\alpha)$ is again a positive affine root of $\tilde W_J$. Since $x ^{-1}$ sends positive affine roots of $\tilde W_J$ to positive affine roots of $\tilde W$, $z ^{-1} w' z=z ^{-1} w z s_{z ^{-1}(\alpha)}$ and $z ^{-1} w(\alpha)$ is a positive affine root of $\tilde W$. Hence $z ^{-1} w' z \geqslant z ^{-1} w z$.
\end{proof}

\begin{cor}\label{super}
If Theorem \ref{main'} (2) holds for $B(\tilde W_J, \mu, \sigma^J)$, then it it holds for $B(\tilde W, \mu, \sigma)$.
\end{cor}
\begin{proof}
Let $\nu$ be the maximal Newton point of $B(\tilde W, \mu, \sigma)$, which is also the maximal Newton point of $B(\tilde W_J, \mu, \sigma^J)$ by Proposition \ref{image}. By assumption, there exist $w_1 \in t^\mu (W_a \cap \tilde W_J)$ and $x_1 \in W_J$ such that $\bar \nu_{w_1, \sigma^J}^J=\nu$ and $w_1 \leqslant_J t^{x_1(\mu)}$. Here $\leqslant_J$ is the Bruhat order on $\tilde W_J$ defined with respect to $J$. Let $w=z^{-1} w_1 z$ and $x=z^{-1} x_1$. Then we have $\bar \nu_{w, \sigma}=\nu$, $w \in t^\mu W_a$ and $w \leqslant t^{x(\mu)}$ as desired.
\end{proof}

\section{The superbasic case}

\subsection{} In this section, we consider the extended affine Weyl group $\tilde W$ of $G=GL_n$. Then $\tilde W \cong \mathbb Z^n \rtimes \mathfrak{S}_n$, where $\mathfrak{S}_n$ is the permutation group of $\{1, 2, \dots, n\}$ which acts on $\mathbb Z^n \cong \oplus_{i=1}^n \mathbb Z e_i^\vee$ by $w(e_i^\vee)=e_{w(i)}^\vee$ for $w \in \mathfrak{S}_n$. Let $\{e_i\}_{i=1, \cdots, n}$ be the dual basis. Set $d=\sum_{i=1}^n e_i$ and $d^\vee=\sum_{i=1}^n e_i^\vee$. The simple roots and fundamental weights are given by $\alpha_i=e_i-e_{i+1}$ and $\omega_i=-\frac{i}{n}d+\sum_{j=1}^i e_j$ respectively for $i \in [1, n-1]$. Set $\varpi_{m,n}=\sum_{j=1}^i e_j^\vee$.

For any positive integer $m<n$, let $\sigma_{m, n}=t^{\varpi_{m,n}} u_{m,n} \in t^{\varpi_{m,n}} W_0$ be the unique length zero element with $u_m \in W_0$. Then any superbasic element in $\tilde W$ is of the form $t^{\varpi_{m,n}} \sigma_{m, n}$ for some $m$ coprime to $n$.

The main purpose of this section is to prove the following result.

\begin{prop}\label{superbasic}
Let $m<n$ be a positive integer coprime to $n$. Let $\mu \in \mathbb Z^n$ be a dominant coweight of $G=GL_n$. Then there exists $\tilde w \in \tilde W$ and $x \in W_0$ such that $\tilde w < t^{x(\mu)}$ and $\bar \nu_{\tilde w, \sigma_{m, n}}-\frac{m}{n}d^\vee$ equals the unique maximal Newton point $\nu$ of $B(\tilde W, \mu, t^{-\frac{m}{n}d^\vee}\sigma_{m,n})$.

\end{prop}

The proof will be given in \S \ref{proof-5.1}.

\subsection{} We first show that Proposition \ref{superbasic} implies Theorem \ref{main'} (ii).

By \S\ref{adj}, we may assume that $\tilde W$ is the Iwahori-Weyl group of an adjoint $p$-adic group $G$. Then by Lemma \ref{irreducible} and Lemma \ref{super}, it suffices to prove the case where $G=PGL_n$ and $\sigma$ is superbasic, which follows form Proposition \ref{superbasic} and \S\ref{adj}.

\subsection{} Now we give an algorithm to construct the maximal element in $B(\tilde W, \mu, \sigma_{m, n})$.

We recall the definition of $\underline{\mathbf{a}}$-sequence and $\chi_{m, n}$ in \cite[\S 3 \& \S 5]{He10}.

Let $r \in \mathbb N$ and $\chi \in \mathbb Z^r$. For each $j \in [1, r]$ we define $\underline{\mathbf{a}}_\chi^j: \mathbb Z_{\geqslant 0} \to \mathbb Z$ by $\underline{\mathbf{a}}_\chi^j(k)=\chi(j-k)$. Here we identify $l$ with $l+r$ for $l \in \mathbb Z$. We say $i \geqslant_\chi j$ if $\underline{\mathbf{a}}_\chi^i \geq \underline{\mathbf{a}}_\chi^j$ in the sense of lexicographic order. If $\geqslant_\chi$ is a linear order, we define $\epsilon_\chi \in \mathfrak{S}_r$ such that $\epsilon_\chi(i) < \epsilon_\chi(j)$ if and only if $i >_\chi j$.

Define $\chi_{m,n} \in \mathbb Z^n$ by $\chi_{m,n}(i)=\lfloor im \rfloor-\lfloor (i-1)m \rfloor $ for $i \in [1,n]$. Set $\epsilon_{m,n}=\epsilon_{\chi_{m,n}} \in \mathfrak{S}_n$. Since $m$ and $n$ are co-prime, it is well defined. Note that $\epsilon_{m,n}(\chi_{m,n})=\varpi_{m,n}$.

\subsection{}\label{polygon} Let $\mathcal S=\cup_{1 \leqslant i \leqslant j \leqslant n}\mathbb Z^{[i, j]}$, whose elements are called {\it segments}. Let $\eta \subset \mathcal S$ be a segment. Assume $\eta \in \mathbb Z^{[i, j]}$. We call $\text{h}(\eta)=i$ and $\text{t}(\eta)=j$ the {\it head} and the {\it tail} of $\eta$ respectively. We call the nonnegative integer $j-i+1$ the {\it size} of $\eta$. We set $$|\eta|=\sum_{k=\text{h}(\eta)}^{\text{t}(\eta)} \eta(k), \qquad \text{av}(\eta)=\frac{1}{\text{t}(\eta)-\text{h}(\eta)+1} |\eta|.$$ Let $[i', j'] \subset [i, j]$ be a sub-interval, we call the restriction $\eta|_{[i',j']}$ of $\eta$ to $[i', j']$ a {\it subsegment} of $\eta$ and write $\eta|_i=\eta|_{[i,i]}$. Let $\theta$ be another segment such that $\text{h}(\theta)=\text{t}(\eta)+1$. We denote by $\eta \vee \theta \in \mathbb Z^{[\text{h}(\eta), \text{t}(\theta)]}$ the natural union of $\eta$ and $\theta$. For $k \in \mathbb Z$, we denote by $\eta[k]$ the {\it $k$-shift} of $\eta$ defined by $\eta[k](i)=\eta(i+k)$. We say two segments are of the same {\it type} if they can be identified with each other up to some shift.

For $\eta \in \mathbb Q^n \cong \mathbb Q^{[1,n]}$ we denote by $\text{Con}(\eta) \in \mathbb Q^2$ the convex hull of the points $(0,0)$ and $(k, |\eta|_{[1, k]}|)$ for $k \in [1,n]$. We say a subsegment $\gamma$ of $\eta$ is {\it sharp} if $\text{av}(\gamma)$ is maximal/minimal among all subsegments of $\eta$ with the same head/tail. If $\eta=\gamma^1 \vee \gamma^2 \vee \cdots \vee \gamma^s$ with each $\gamma^k$ a sharp subsegments, then the points $(0,0)$ and $(\text{t}(\gamma^i), |\gamma^1 \vee \cdots \vee \gamma^j|)$ in $\mathbb R^2$ for $i \in [1,s]$ lie on the boundary of $\text{Con}(\eta)$ and their convex hall is just $\text{Con}(\eta)$. We call the dominant vector $$\text{sl}(\text{Con}(\eta))=(\textbf{av}(\gamma^1) \vee \cdots \vee \textbf{av}(\gamma^s)) \in \mathbb Q^s$$ the {\it slope sequence} of $\text{Con}(\eta)$. Here for any $\gamma \in \mathcal S$, we define $\textbf{av}(\gamma) \in \mathbb Z^{[\text{h}(\gamma),\text{t}(\gamma)]}$ by $\textbf{av}(\gamma)(i)=\text{av}(\gamma)$ for $i \in [\text{h}(\gamma),\text{t}(\gamma)]$.

Let $\mu \in \mathbb Z^n$ be a dominant coweight. Define $\mu_{m,n}=\mu+\chi_{m,n}$. Then \begin{align*} \<\omega_i, \mu_{m,n}\>&=\<\omega_i,\mu\>-(\frac{m i}{n}-\lfloor \frac{m i}{n} \rfloor)\\ &=\<\omega_i, \mu+\varpi_{m,n}\>-\lceil \<\omega_i, \varpi_{m,n}\> \rceil. \end{align*} According to the proof of Theorem \ref{main'} (1), the slope sequence $\nu=\text{sl}(\text{Con}(\mu_{m,n}))$ is the unique maximal Newton point of $B(G, \mu, \sigma_{m,n})$.

\begin{example} Now we provide an example.

For a sequence of (distinct) elements $i_1, i_2, \dots, i_r$ in $[1, n]$, we denote by $\text{cyc}(i_1, i_2, \dots, i_r) \in \mathfrak{S}_n$ the cyclic permutation $i_1 \mapsto i_2 \mapsto \cdots \mapsto i_r \mapsto i_1$, which acts trivially on the remaining elements of $[1,n]$.

Let $n=8$, $m=5$ and $\mu=(1, 1, 1, 0, 0, 0, 0, 0)$. Then $\chi_{m,n}=(0, 1, 0, 1, 1, 0, 1, 1) \in \mathbb Z^8$, $\epsilon_{m,n}=\text{cyc}(1, 6, 7, 4, 5, 2, 3, 8)$ and $u_{m, n}=\text{cyc}(6, 3, 8, 5, 2, 7, 4, 1)$.

We have the following sharp decomposition: $$\mu_{m,n}=(1, 2, 1, 1, 1, 0, 1, 1)=(1, 2) \vee(1) \vee (1, 1) \vee (0, 1, 1).$$ Hence $\nu=(\frac{3}{2}, \frac{3}{2}, 1, \frac{1}{2}, \frac{1}{2}, \frac{2}{3}, \frac{2}{3}, \frac{2}{3})$. Moreover, one checks that \begin{align*} t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} &> t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} \text{cyc}(8, 3)\\ &> t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} \text{cyc}(8, 3) \text{cyc}(1, 3)\\ &> t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} \text{cyc}(8, 3) \text{cyc}(1, 3) \text{cyc}(1, 2).\end{align*} Set $\tilde w=t^{\epsilon_{m,n}(\mu)} \sigma_{m,n} \text{cyc}(8, 3) \text{cyc}(1, 3) \text{cyc}(1, 2) \sigma_{m, n} ^{-1}$. Then $\tilde w<t^{\epsilon_{m, n}(\mu)}$ and $\bar \nu_{\tilde w, \sigma_{m,n}}=\nu$. This verifies Proposition \ref{superbasic} in this case.
\end{example}

\subsection{}\label{Eucliean} Similar to \cite[\S 5]{He10}, we use the Euclidean algorithm to give a recursive construction of $\chi_{m,n}$, which plays a crucial role in the proof of Proposition \ref{superbasic}.

Let $D=\{(i,j) \in \mathbb Z_{>0}^2; i<j \text{ are co-prime}\}$. We define $f: D \to D \sqcup \{(1,1)\}$ by \begin{align*}f(m,n)=\begin{cases} (m(\lfloor\frac{n}{m}\rfloor+1)-n, m), &  \text{ if } \frac{n}{m} \geqslant 2;\\ (n-(n-m)\lfloor\frac{n}{n-m}\rfloor, n-m), & \text{ otherwise.} \end{cases}\end{align*}

Define two types of segments $1_{m,n}$ and $0_{m,n}$ in $\mathcal S$ by \begin{align*} (1_{m,n}, 0_{m,n})&=\begin{cases} ((0^{(\lfloor\frac{n}{m}\rfloor-1)}, 1), (0^{(\lfloor\frac{n}{m}\rfloor)},1)), &  \text{ if } \frac{n}{m} \geqslant 2; \\ ((0, 1^{(\lfloor\frac{n}{n-m}\rfloor)}), (0, 1^{(\lfloor\frac{n}{n-m}\rfloor-1)})), & \text{ otherwise.}\end{cases}\end{align*}

For $\eta \in \mathcal S$ and $k \in [\text{h}(\eta), \text{t}(\eta)]$, set \begin{align*}\eta(k)_{m,n}=\begin{cases} 1_{m,n}, &\text{ if } \eta(k)=1;\\ 0_{m,n}, & \text{ otherwise.} \end{cases}\end{align*}

For $k \in [\text{h}(\eta), \text{t}(\eta)]$, let $\eta_{m,n,k}$ be the shift of $\eta(k)_{m,n}$ whose head is determined recursively as follows: \begin{align*} \text{h}(\eta_{m,n,k})=\begin{cases} \text{h}(\eta), & \text{ if } k=\text{h}(\eta);\\ \text{t}(\eta_{m,n,k-1})+1, & \text{ if } k > \text{h}(\eta). \end{cases}\end{align*}
Now we define $\phi_{m,n}: \mathcal S \to \mathcal S$ by $\phi_{m,n}(\eta)=\eta_{m,n,\text{h}(\eta)} \vee \cdots \vee \eta_{m,n,\text{t}(\eta)}$ for $\eta \in \mathcal S$.

If $f^{h-1}(m,n)$ is defined, we set $\phi_{m,n,h}=\phi_{f^{h-1}(m,n)} \circ \cdots \circ \phi_{m,n}$. Using the Euclidean algorithm, one checks that $$\phi_{m,n,h}(\chi_{f^h(m,n)})=\chi_{m,n}.$$ We say a subsegment $\gamma$ of $\chi_{m,n}$ is of {\it level} $h$ if it is the image of some subsegment $\gamma^h$ of $\chi_{f^h(m,n)}$ under the map $\phi_{m,n,h}$. When $h=1$ and $\gamma^h$ is of size one, we say $\gamma$ is an {\it elementary} subsegment of $\chi_{m,n}$.

Let $\beta^1$ and $\gamma^1$ be two segments of $\chi^1=\chi_{f(m,n)}$ and let $\gamma$ be a level one subsegment of $\chi=\chi_{m,n}$. Using the Euclidean algorithm, we have the following basic facts:

(a) $\text{av}(\beta^1) \geqslant \text{av}(\gamma^1)$ if and only if $\text{av}(\phi_{m,n}(\beta^1)) \geqslant \text{av}(\phi_{m,n}(\gamma^1))$.

(b) Each sharp subsegment of $\gamma$ with the same head is of level one.

(c) If moreover $\gamma$ is an elementary subsegment of $\chi$, then $\underline{\mathbf{a}}_\chi^{j} < \underline{\mathbf{a}}_\chi^{\text{h}(\gamma)-1}$ and $\underline{\mathbf{a}}_\chi^{j} < \underline{\mathbf{a}}_\chi^{\text{t}(\gamma)}$ for $j \in [\text{h}(\gamma), \text{t}(\gamma)-1]$.

(d) $\underline{\mathbf{a}}_{\chi^1}^i < \underline{\mathbf{a}}_{\chi^1}^j$  if and only if $\underline{\mathbf{a}}_\chi^{\text{t}(\phi_{m,n}(\chi^1|_{[i]}))} < \underline{\mathbf{a}}_\chi^{\text{t}(\phi_{m,n}(\chi^1|_{[i]}))}$.

(e) $\epsilon_{m,n}(n)=1$.

\subsection{Proof of Proposition \ref{superbasic}}\label{proof-5.1}
For $\eta \in \mathcal S$ we set $x_\eta=\text{cyc}(\text{h}(\eta), \text{h}(\eta)+1, \dots, \text{t}(\eta)) \in \mathfrak{S}=\cup_{i=1}^\infty \mathfrak{S}_i$. Similarly, for a sequence $\textbf{c}=(c^1, \dots, c^s)$ of segments, we set $x_{\textbf{c}}=x_{c^1, \dots, c^s}=x_{c^1} \cdots x_{c^s}$. If $\eta=c^1 \vee \cdots \vee c^s$, we say $\textbf{c}$ is {\it decomposition of $\eta$}. Now we are ready to prove Proposition \ref{superbasic}.

Write $\chi=\chi_{m,n}$, $\theta=\mu_{m,n}$ and $\epsilon=\epsilon_{m,n}$. For $h \in \mathbb Z_{>0}$ we set $\phi_h=\phi_{m,n,h}$ and $\chi^h=\chi_{f^h(m,n)}$. By \S\ref{polygon}, we have $\nu=\text{sl}(\text{Con}(\theta))$. The proof will proceed as follows. First we construct a suitable sharp decomposition $\textbf{c}$ of $\theta$. One checks directly $\nu_{w_{\textbf{c}}, \id}=\text{sl}(\text{Con}(\theta))=\nu$, where $w_{\textbf{c}}=t^\theta x_{\textbf{c}} \in \tilde W$. Then we show that $$\epsilon w_{\textbf{c}} \epsilon^{-1} < t^{\epsilon(\mu)} \sigma_{m,n}=\epsilon t^\theta x_\theta \epsilon^{-1}.$$ Set $\tilde w=\epsilon w_c \epsilon ^{-1} \sigma_{m, n} ^{-1}$. Then $\tilde w <t^{\epsilon(\mu)}$ and $\nu_{\tilde w, \sigma_{m, n}}=\nu_{\epsilon w_c \epsilon ^{-1}, id}=\epsilon(\nu)$. This completes our proof.

\

Assume $I(\mu)=\{j \in [1,n-1]; \<\alpha_j, \mu\> \neq 0\}=\{b_1, b_2, \dots, b_{r-1}\}$ with $b_1 < b_2 < \cdots < b_{r-1}$. We set $b_0=0$ and $b_r=n$. Set $\theta^i=\theta|_{[b_{i-1}+1,b_i]}$ for $i \in [1,r]$. Then $\theta=\theta^1 \vee \cdots \vee \theta^r$. Suppose we have a sharp decomposition $\textbf{c}_i$ of $\theta^i$ for each $i \in [1, r]$. Since $\chi \in \{0,1\}^{[1,n]}$ and $\theta=\mu+\chi$, for any subsegment $\eta^i$ (resp. $\eta^j$) of $\theta^i$ (resp. $\theta^j$) we have $\text{av}(\eta^i) \geqslant \text{av}(\eta^j)$ if $i<j$. Therefore the natural union $\textbf{c}=\textbf{c}_1 \vee \cdots \vee \textbf{c}_r$ forms a sharp decomposition of $\theta$.

\

Let $1 \leqslant i \leqslant r$. We will construct inductively the subsegments $\zeta_i^j, \gamma_i^j, \xi_i^j$ for $j \in [1, l_i]$ (some of them may be empty) such that

(a) $\gamma_i^0=\theta^i$ and $\gamma_i^{j-1}=\zeta_i^j \vee \gamma_i^j \vee \xi_i^j$ for $j \in [1, l_i]$;

(b) $\zeta_i^j$ and $\xi_i^j$ are sharp subsegments of $\gamma_i^{j-1}$; any sharp subsegment of $\gamma_i^j$ is also a sharp subsegment of $\gamma_i^{j-1}$; $\gamma_i^{l_i}$ is a sharp subsegments of itself ({\it self-sharp});

(c) For any $j$, $\epsilon z_{i,j-1} \epsilon^{-1} > \epsilon z_{i,j} \epsilon^{-1}$.

Here \begin{align*} z_{i,j}&=t^\theta y_{i-1} x_i^j v_{i,j}; \\ y_i &=x_{\textbf{c}_1} \cdots x_{\textbf{c}_{i-1}}; \\ x_i^j &=x_{\zeta_i^1, \dots, \zeta_i^j, \xi_i^j, \dots, \xi_i^1}; \\ v_{i,j}&=x_{\gamma_i^j} x_{\theta^{i+1} \vee \cdots \vee \theta^r} \text{cyc}(\text{t}(\gamma_i^j), n)) \\ &=\text{cyc}(\text{h}(\gamma_i^j), \dots, \text{t}(\gamma_i^j), b_i+1, \dots, n).\end{align*} Once we have (a), (b) and (c) for all $i$ and $j$, then $$\textbf{c}_i=(\zeta_i^1, \zeta_i^2, \dots, \zeta_i^{l_i}, \gamma_i^{l_i}, \xi_i^{l_i}, \dots, \xi_i^2, \xi_i^1)$$ forms a sharp decomposition of $\theta^i$, and $$\epsilon t^\theta x_\theta \epsilon^{-1} = \epsilon z_{1,0} \epsilon^{-1} > \cdots > \epsilon z_{1, l_1+1}\epsilon^{-1}=\epsilon z_{2, 0} \epsilon^{-1} > \cdots >\epsilon z_{r, l_r+1} \epsilon^{-1} =\epsilon w_{\textbf{c}} \epsilon^{-1}$$ as desired.

\

The construction is as follows. Suppose for $1 \leqslant k < i$ and $0 \leqslant l \leqslant j$, $\textbf{c}_k$, $z_i^l$, $\xi_i^l$, $\gamma_i^l$ are already constructed, and moreover $\epsilon z_{i,j-1} \epsilon^{-1} > \epsilon z_{i,j} \epsilon^{-1}$. We construct $\zeta_i^{j+1}, \gamma_i^{j+1}, \xi_i^{j+1}$ and show that $\epsilon z_{i,j} \epsilon^{-1} > \epsilon z_{i, j+1} \epsilon^{-1}$.

If $\gamma_i^j$ is empty, there is nothing to do. Otherwise, we assume $\gamma_i^j$ is of level $h$ but not of level $h+1$. Then $\gamma_i^j=\phi_h(\iota)$ for some subsegment $\iota$ of $\chi^h$.

Case (I): {\it $\iota$ is not a subsegment of any elementary subsegment of $\chi^h$.} Then there exist unique subsegments $\zeta$, $\gamma$ and $\xi$ of $\chi^h$ such that $\gamma$ is of level one, $\zeta$ (resp. $\xi$) is a proper subsegment of some elementary segment of $\chi^h$ with the same tail (resp. head), and $\iota=\zeta \vee \gamma \vee \xi$. Without loss of generality, we may assume that none of $\gamma$, $\zeta$ and $\xi$ is empty.

Define $\zeta_i^{j+1}=\phi_h(\zeta)$, $\gamma_i^{j+1}=\phi_h(\gamma)$ and $\xi_i^{j+1}=\phi_h(\xi)$. Note that $\text{av}(\chi^h|_{[\text{h}(\zeta), \text{t}(\zeta)]})$ (resp. $\text{av}(\chi^h|_{[\text{h}(\xi), \text{t}(\xi)]}$ is maximal (resp. minimal) among all subsegments of $\chi^h$ with the same head (resp. tail). Therefore, (b) follows from \S\ref{Eucliean} (a) $\&$ (b). For (c), it suffices to show that
\begin{gather*}
\tag{d} \epsilon z_{i, j+1} \epsilon ^{-1}<\epsilon\ z_{i,j}\ \text{cyc}(n, \text{t}(\zeta_i^{j+1}))\ \epsilon^{-1}; \\
\tag{e} \epsilon\ z_{i,j}\ \text{cyc}(n, \text{t}(\zeta_i^{j+1}))\ \epsilon^{-1}<\epsilon z_{i, j} \epsilon ^{-1}.
\end{gather*}

Note that $\epsilon z_{i,j+1} \epsilon^{-1}=\epsilon\ z_{i,j}\ \text{cyc}(n, \text{t}(\zeta_i^{j+1}))\ \text{cyc}(\text{h}(\xi_i^{j+1})-1, \text{t}(\xi_i^{j+1}))\ \epsilon^{-1}$. By \S\ref{Eucliean} (c), we have $\underline{\mathbf{a}}_{\chi^h}^{\text{h}(\xi)-1} > \underline{\mathbf{a}}_{\chi^h}^{\text{t}(\xi)}$. Hence by \S\ref{Eucliean} (d), $\epsilon(\text{h}(\xi_i^{j+1})-1) < \epsilon(\text{t}(\xi_i^{j+1}))$. If $\theta(\text{h}(\xi_i^{j+1})) > \theta(b_i+1)$, then (d) holds. If $\theta(\text{h}(\xi_i^{j+1})) = \theta(b_i+1)$, then $\chi(b_i+1)=1>0=\chi(\text{h}(\xi_i^{j+1}))$. Hence $\epsilon(b_i+1) < \epsilon(\text{h}(\xi_i^{j+1}))$. Then (d) still holds.

Since $\gamma \neq \emptyset$, $\text{t}(\zeta_i^{j+1}) \neq n$. By \S\ref{Eucliean}(e), $1=\epsilon(n) < \epsilon(\text{t}(\zeta_i^{j+1}))$. If $\theta(\text{t}(\zeta_i^{j+1})+1) < \theta(\text{h}(\zeta_i^{j+1}))$, then (e) holds. If $\theta(\text{t}(\zeta_i^{j+1})+1) = \theta(\text{h}(\zeta_i^{j+1}))$, then $\chi(\text{h}(\zeta_i^{j+1}))=\chi(\text{t}(\zeta_i^{j+1})+1)=0$. Since $\zeta$ is a proper subsegment of some elementary segment of $\chi^h$ and shares the same tail with it, by \S\ref{Eucliean} (c) we have that $\underline{\mathbf{a}}_{\chi^h}^{\text{t}(\zeta)} > \underline{\mathbf{a}}_{\chi^h}^{\text{h}(\zeta)-1}$, and by \S\ref{Eucliean} (d) we have that $\underline{\mathbf{a}}_\chi^{\text{t}(\zeta_i^{j+1})+1} > \underline{\mathbf{a}}_\chi^{\text{h}(\zeta_i^{j+1})}$. So $\epsilon(\text{h}(\zeta_i^{j+1})) > \epsilon(\text{t}(\zeta_i^{j+1})+1)$. Then (e) still holds.

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Case (II): {\it $\iota$ is a subsegment of some elementary subsegment of $\chi^h$.} We define $l_i=j$ and the construction of $\textbf{c}_i$ is finished. One checks directly that $\iota$ is self-sharp, hence so is $\gamma_i^{l_i}=\phi_h(\iota)$ by \S\ref{Eucliean} (a) $\&$ (b). If $\text{t}(\gamma_i)=n$, the induction step is finished. Otherwise, it remains to show \begin{align*} \tag{f} \epsilon z_{i,l_i} \epsilon^{-1} > \epsilon\ z_{i,l_i}\ \text{cyc}(\text{t}(\gamma_i^{l_i}), n)\ \epsilon^{-1}=\epsilon z_{i,l_i+1} \epsilon^{-1}. \end{align*} Note that $1=\epsilon(n) < \epsilon(\text{t}(\gamma_i))$. If $\theta(\text{h}(\gamma_i^{l_i})) > \theta(b_i+1)$, (f) holds. Otherwise, we have $\theta(\text{h}(\gamma_i^{l_i})) = \theta(b_i+1)$, $\chi(\text{h}(\gamma_i^{l_i}))=0$ and $\chi(b_i+1)=1$ since $b_i \in I(\mu)$. Hence $\epsilon(\text{h}(\gamma_i^{l_i})) > \epsilon(b_i+1)$ and (f) still holds.