Let $F=\left( F_{1}, \ldots , F_{m}\right)$ be a random vector whose components are differentiable in the Malliavin sense. Denote by $ \Gamma_{F}$ its Malliavin matrix, whose entries are given by
 $\Gamma_{F}(i,j)= \langle DF_{i}, DF_{j}\rangle _{\HHH}$, $1\leq i, j\leq m$,
where $D$ denotes the Malliavin derivative with respect to an isonormal process $(W(h), h\in \HHH)$ on a real and separable Hilbert space $\HHH$. A classical result in Malliavin calculus (the {\it Bouleau-Hirsch criterion}) states that if $\Gamma_{F}$ is almost surely invertible, then $F$ admits a density with respect to the Lebesgue measure on $\mathbb{R} ^{m}$.

The random vector $F$ is said to be nondegenerate if its components are infinitely Malliavin differentiable and if $\left( \det \Gamma_{F}\right) ^{-1}\in L ^{p}(\Omega)$ for every $p\geq 1$. In this case, its density is smooth and belongs to the Schwartz space of rapidly decreasing $C ^{\infty}$ functions on $ \mathbb{R} ^{m}$. Hence, obtaining a smooth density is tightly linked to the existence of negative moments of the Malliavin derivatives of the components.

When the components of $F$ lie in a fixed Wiener chaos, the above result can be strengthened. It was shown in \cite{NNP} that in this setting the almost sure invertibility of $\Gamma_F$ is actually equivalent to the absolute continuity of the law of $F$. Moreover, \cite{NT} proves that a pair of multiple stochastic integrals fails to admit a density if and only if its components are proportional, mirroring the Gaussian case.

In the recent work \cite{LNNT} we applied the Bouleau-Hirsch criterion to establish the absolute continuity of finite-dimensional distributions of Hermite processes. Let $q\geq 1$ be an integer and $ (Z ^{H, q}_{t}, t\in \mathbb{R})$ a Hermite process of order $q$ with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Recall that $Z^{H,q}$ is an $H$-self-similar process with stationary increments and long memory, living in the $q$th Wiener chaos. The class of Hermite processes includes fractional Brownian motion ($q=1$), which is Gaussian, and the Rosenblatt process ($q=2$), which belongs to the second Wiener chaos. For any $m\geq 1$ and any grid $0<t_{1},...<t _{m}$, \cite{LNNT} shows that the determinant of the Malliavin matrix of the Hermite vector $Z= \left( Z^{H, q}_{t_{1}}, \ldots , Z^{H, q}_{t_{m}}\right)$ is almost surely strictly positive; therefore, $Z$ admits a density.

The objective of the present work is to go a step further and investigate the smoothness of this density. As previously mentioned, smoothness is deeply linked to nondegeneracy, and in particular to the existence of negative moments of the determinant of the Malliavin matrix. While establishing these negative moments appears challenging for general $q$, we succeed in proving nondegeneracy for $q=2$, that is for Rosenblatt vectors. This relies crucially on the specific structure of random variables in the second Wiener chaos, which can be represented as weighted sums of independent centered chi-square random variables.

Our first main result concerns the smoothness of the density of increments of the Rosenblatt process.

Our first main result concerns the smoothness of the density of increments of the Rosenblatt process.

As a consequence,
for all $m\geq 1$ and for all $0\leq s_{1}<...< s_{m}$, the vector $(Z_{s_{1}},\ldots ,Z_{s_{m}})$ also has a density in $ \mathcal{S}(\mathbb{R} ^{m})$, see Corollary \ref{cor1}. The proof of Theorem \ref{tt3} relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.

\begin{theorem}\label{tt2}
 Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. For any $m\geq 1$ and $n=(n_{1},..., n_{m})\in \mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\pi} $ of the vector $Z_{\pi}$ defined in (\ref{zpi}) satisfies
 \begin{equation*}
 \left| \partial _{x} ^{n} p_{\pi }(x)\right| \leq C \prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},
 \end{equation*}
 with $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$
\end{theorem}

\begin{theorem}
 Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 2$ and $0=t_{0}<t_{1}<\dots<t_{m}$. Consider the random vector
 \begin{equation*}
 Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, Z_{t_{2}}-Z_{t_{1}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right).
 \end{equation*}
Then
\begin{equation*}
 P\left( \det \Gamma _{Z_{\pi} }>0\right)= 1.
\end{equation*}
In particular, the random vector $Z_{\pi}$ admits a density with respect to the Lebesgue measure on $\mathbb{R} ^{m}$.
\end{theorem}

\begin{lemma}\label{ll2}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 2$ and $0=t_{0}<t_{1}<\dots<t_{m}$. For $j\geq 1$, set
\begin{equation}
\label{21o-1}
\tilde{E}_{j-1}= \span \{ D(Z_{t_{k}}-Z_{t_{l}}): k,l=0,1,\dots, j-1\}.
\end{equation}
Then, for every $j \geq 1$,
\begin{equation*}
\left\| D(Z_{t_{j}}-Z_{t_{j-1}})- {\rm proj}_{\tilde{E}_{j-1}} \big(D(Z_{t_{j}}-Z_{t_{j-1}})\big)\right\| _{\HHH} ^{2} \overset{\rm law}{=} (t_{j}-t_{j-1})^{2H} F_{j},
\end{equation*}
where, for each $j\geq 1$, $F_{j}$ is a random variable such that $F_{j} \geq \Vert DZ_{1}\Vert ^{2}_{\HHH_{1}}$, with $\HHH_{1}= L ^{2}([0, 1])$.
\end{lemma}

\begin{lemma}\label{ll3}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0=t_{0}<t_{1}<\dots<t_{m}$. Consider the random vector
\begin{equation*}
Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
\end{equation*}
and let $\Gamma_{Z_{\pi}}$ be the Malliavin matrix of $Z _{\pi }$. Then, for any integer $k\geq 0$ and any $p\geq 1$,
\begin{equation}\label{kpnorm:invdet}
 \left\| \left( \det \Gamma_{Z_{\pi}}\right) ^{-1} \right\| _{k, p} \leq C \prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-2H}.
\end{equation}
\end{lemma}

\begin{corollary}\label{cor1}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0<s_{1}<\dots<s_{m}$. Then the vector $(Z_{s_{1}},\dots, Z_{s_{m}})$ admits a density which belongs to the Schwartz space $\mathcal{S}(\mathbb{R}^{m})$.
\end{corollary}

\begin{proposition}\label{pp2}
Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $n, m\geq 1$ be integers and fix $0=t_{0}<t_{1}<\dots<t_{m}$ and $\beta = \left( \beta_{1},\dots, \beta_{n} \right) \in \{1,\dots, m\}^{n}$. Let $Z_{\pi}$ be given by \eqref{zpi}. Then there exists a random variable $H ^{\beta} \in \mathbb{D} ^{\infty}$ such that, for every $\varphi \in C_{p} ^{\infty}(\mathbb{R} ^{m})$,
\begin{equation}\label{22o-1}
\mathbf{E} \left[ \partial _{\beta } \varphi (Z_{\pi})\right] = \mathbf{E}\left[ \varphi (Z_{\pi})H ^{\beta}\right].
\end{equation}
Moreover,
\begin{equation}\label{22o-2}
\mathbf{E}\left[ (H ^{\beta}) ^{2} \right]\leq C \prod_{j=1}^{n}( t_{\beta_{j}}-t_{\beta _{j-1}})^{-2H}.
\end{equation}
\end{proposition}

\begin{theorem}\label{tt3}
	Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector
	\begin{equation}\label{zpi}
		Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
	\end{equation}
	Then $Z_{\pi }$ admits a density belonging to the Schwartz space  $ \mathcal{S}(\mathbb{R} ^{m})$ of rapidly decreasing smooth functions.
\end{theorem}

\begin{equation}\label{zpi}
		Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
	\end{equation}

As a consequence,
for all  $m\geq 1$ and for all $0\leq s_{1}<...< s_{m}$, the vector $(Z_{s_{1}},\ldots ,Z_{s_{m}})$ also has a density in $ \mathcal{S}(\mathbb{R} ^{m})$, see Corollary \ref{cor1}. The proof of Theorem \ref{tt3} relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.

Our second main result provides exponential-type bounds for the partial derivatives of the density  of the  finite-dimensional distributions of the Rosenblatt process. For $m \in \mathbb{N}^{\ast}$  and  $n=(n_{1},..., n_{m})\in \mathbb{N} ^{m}$, set
\begin{equation*}
	\partial _{x} ^{n} =\prod_{j=1}^{m} \left( \frac{\partial}{\partial x_{j}}\right) ^{n_{j}}.
\end{equation*}

\begin{theorem}\label{tt2}
	Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$.  For any  $m\geq 1$ and $n=(n_{1},..., n_{m})\in \mathbb{N}^{m}$, there exists $C>0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}<t_{1}<...<t_{m}$, the density $p_{\pi} $ of the vector $Z_{\pi}$  defined in (\ref{zpi})  satisfies
	\begin{equation*}
		\left| \partial _{x} ^{n} p_{\pi }(x)\right| \leq C \prod_{j=1}^{m} (t_{j}- t_{j-1}) ^{-H(1+ n_{j})} e ^{-c\frac{x_{j}}{(t_{j}- t_{j-1}) ^{H}}},
	\end{equation*}
	with $c=c(m,q)>0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$
\end{theorem}

The proof of
Theorem \ref{tt2} is based on a Malliavin integration by parts formula expressing the partial derivatives of $p_{\pi}$ (see relation (\ref{22o-1}) in Proposition \ref{pp2} below)  and on a precise control of the resulting terms using Malliavin calculus and tail estimates for random variables in Wiener chaos.

The paper is organized  as follows. Section 2 provides the necessary preliminaries on Malliavin calculus, Wiener chaos and the Rosenblatt process. Section 3 contains the proof of the smoothness of the density of Rosenblatt vectors. Section 4 is devoted to the analysis of the regularity of the partial derivatives of this density.

The proof of Theorem \ref{tt3} "relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space."

The proof of the main theorem relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.

\label{tt3}
	Let $(Z_{t}, t\in \mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right)$. Let $m\geq 1$ and $0=t_{0}<t_{1}<...<t_{m}$. Consider the random vector
	\begin{equation}\label{zpi}
		Z_{\pi} =\left( Z_{t_{1}}-Z_{t_{0}}, \ldots , Z_{t_{m}}-Z_{t_{m-1}}\right),
	\end{equation}
	Then $Z_{\pi }$ admits a density belonging to the Schwartz space  $ \mathcal{S}(\mathbb{R} ^{m})$ of rapidly decreasing smooth functions.,

Question:
Let $(Z_t, t\in\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\in\left(\tfrac12,1\right)$. Let $m\ge 1$ and choose times $0=t_0<t_1<\cdots<t_m$. Define the increment vector
\[
Z_{\pi}=\bigl(Z_{t_1}-Z_{t_0},\; Z_{t_2}-Z_{t_1},\; \ldots,\; Z_{t_m}-Z_{t_{m-1}}\bigr)\in\mathbb{R}^m.
\]
Which statement holds for every such Rosenblatt process and every such grid? Here $\mathcal S(\mathbb R^m)$ denotes the Schwartz space of $C^\infty$ functions on $\mathbb R^m$ whose derivatives decay rapidly at infinity.

The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$, and that density belongs to $\mathcal S(\mathbb R^m)$.

Choices:
B. The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$, and this density is $C^{\infty}$ on $\mathbb R^m$ with all partial derivatives bounded by constants depending only on $m$ and $H$.
C. The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$.
D. If the Malliavin matrix $\Gamma_{Z_{\pi}}$ is almost surely invertible, then $Z_{\pi}$ admits a density belonging to $\mathcal S(\mathbb R^m)$.
E. The random vector $Z_{\pi}$ admits a density with respect to Lebesgue measure on $\mathbb R^m$, and its density belongs to $\mathcal S(\mathbb R^m)$ provided $m\ge 2$.


For $q\geq 0$, the poset     $\W_n^q$ is a lattice for any $n\geq 1$.

Let $q$ be a non-negative real number. A $q$-{\it decreasing word} of length $n\geq 0$ is a binary word of length $n$ satisfying the following constraint:

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows  that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have  $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
        $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
    with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors  of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

The notion of $q$-decreasing words has recently attracted significant attention in the literature. This family of words exhibits a striking combinatorial property whenever $q$ is a positive integer. Indeed, they are in one-to-one correspondence with binary strings that avoid the pattern $1^{q+1}$, i.e. binary strings without $q+1$ consecutive 1 (see \cite{bkv gray code}). So, this implies that $q$-decreasing words of length $n$ (when $q$ is a positive integer) are enumerated by the $(q+1)$-generalized Fibonacci numbers $F_{n+1}^{q+1}$ where $F_n^q$ is defined by

$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\sum_{n\geq 0}F_n^qx^n=\frac{1}{1-x-x^2-\cdots -x^q}.$$

Recently, Barcucci, Bernini, Bilotta and Pinzani \cite{Barc} extended this bijection to $q$-decreasing words for any positive rational number $q$, showing that $\mathcal{W}^q$ is in one-to-one correspondence with binary words avoiding some patterns.

These words have also been studied from a generative prospective. Baril et al. \cite{bkv gray code} provide efficient algorithms for the generation of all $q$-decreasing words whenever $q$ is a positive integer. In particular, they construct a $3$‑Gray code for general $q$, and notably a $1$‑Gray code for the case $q=1$, thus resolving a conjecture posed in the context of interconnection networks by \cite{egecioglu irsic}. More recently, Wong et al. \cite{Wong} present a two-stage algorithm for generating cyclic $2$-Gray codes for $q$-decreasing words.

More generally, for any $q>0$, the generating function $W_q(x)$ for the number of $q$-decreasing words with respect to the length $n$ is given by
\begin{equation}\label{eq:w_q general}
W_q(x)=\frac{1}{(1-x)\left(1-\sum_{i=0}^{+\infty}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}\right)},
\end{equation}
see \cite{sergey2,Serg}.
This expression can be simplified as follows when $q$ is rational, i.e. $q=c/d$ where $c$ and $d$ are positive integers:
\begin{equation}\label{eq:w_q rational}
W_q(x)=\frac{1-x^{c+d}}{(1-x)\left(1-x^{c+d}-\sum_{i=0}^{c-1}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}\right)}.
\end{equation}
Note that when $q$ is an integer, i.e. when we fix $d=1$ and $c=q$ in the previous formula,  we obtain $W_q(x)=\frac{F_{q+1}(x)-1}{x}$.  Using (\ref{eq:w_q general}) and (\ref{eq:w_q rational}), Dovgal and Kirgizov \cite{sergey2} proved that for all real $q>0$, $[x^n]W_q(x)\underset{n\to\infty}{\sim}C_q\cdot\Phi(q)^n$, for a positive constant $C_q$, and a function $\Phi(q)$ that interpolates the $q$-bonacci numbers. In particular, when $q=c/d$ is a rational number, $\Phi(q)^{-1}$ is the smallest root in modulus of the polynomial $x^{c+d}+\sum_{i=0}^{c-1}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}-1$. See \cite{sergey2} for additional properties of $\Phi(q)$.

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$,  then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
  meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
  An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
  $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An  element $w$ is said to {\it cover} an element $v$  if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write  $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

In this paper, we equip $\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\ldots v_n$ and  $w=w_1w_2\ldots w_n$ are two $q$-decreasing words in $\mathcal{W}_n^q$ then
$$v\leq w\Longleftrightarrow v_i\leq w_i \mbox{ for all } 1\leq i\leq n.$$

Let  $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation.  See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
 $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
 with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\sum_{n\geq 0}F_n^qx^n=\frac{1}{1-x-x^2-\cdots -x^q}.$$

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$, then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
 meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
 An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
 $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An element $w$ is said to {\it cover} an element $v$ if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

In this paper, we equip $\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\ldots v_n$ and $w=w_1w_2\ldots w_n$ are two $q$-decreasing words in $\mathcal{W}_n^q$ then
$$v\leq w\Longleftrightarrow v_i\leq w_i \mbox{ for all } 1\leq i\leq n.$$

Let $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation. See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
 $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
 with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$, then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
 meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
 An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
 $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An element $w$ is said to {\it cover} an element $v$ if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

Let $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation. See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\begin{figure}[h]
 \centering
\begin{tikzpicture}[scale=1.3]
 \node (11111) at (0,7.5) {$11111$};
 \node (11110) at (-1.5,6) {$11110$};
 \node (11001) at (0,4.5) {$11001$};
 \node (11100) at (-3,4.5) {$11100$};
 \node (00011) at (3,3) {$00011$};
 \node (10001) at (1,3) {$10001$};
 \node (10010) at (-1,3) {$10010$};
 \node (11000) at (-3,3) {$11000$};
 \node (00001) at (3,1.5) {$00001$};
 \node (00010) at (1,1.5) {$00010$};
 \node (10000) at (-3,1.5) {$10000$};
 \node (00100) at (-1,1.5) {$00100$};
 \node (00000) at (0,0) {$00000$};
 \draw (00000) -- (00100);\draw (00000) -- (10000);\draw (00000) -- (00001);\draw (00000) -- (00010);\draw (10000) -- (11000);\draw (10000) -- (10010);\draw (10000) -- (10001);\draw (00010) -- (10010);\draw (00010) -- (00011);\draw (00001) -- (10001);\draw (00001) -- (00011);\draw (00100) -- (11100);\draw (11000) -- (11001);\draw (11000) -- (11100);\draw (10010) -- (11110);\draw (10001) -- (11001);\draw (00011) -- (11111);\draw (11100) -- (11110);\draw (11001) -- (11111);\draw (11110) -- (11111);
\end{tikzpicture}
 \caption{The lattice $\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }
 \label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \( q > 0 \), we provide a closed-form expression for the generating function counting the number of coverings in \( \W^q_n \). When \( q \) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\Phi(q)$ defined in the introduction above. In Section~\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \( \W^q_n \) for any rational number \( q > 0 \). Finally, Section~\ref{sec:meet} presents the structure of meet-irreducible elements in $\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\W_n^q$ for any rational number $q>0$.

In this section, we provide enumerative results for classical parameters of a lattice, namely the join-irreducible elements, as well as covering relations. The enumeration of meet-irreducible elements is a bit more intricate, so we treat it in Section \ref{sec:meet}.
We first give the enumeration of join-irreducible elements for any $q>0$, and then we conclude by giving closed form for the generating functions of the covering for positive rational numbers $q$, and a method for computing arbitrarily many terms of the generating functions for positive irrational numbers $q$.
\subsection{Join-irreducible elements}
\begin{thm}\label{join irreducible}
 For $q>0$ and $n\geq1$, there are exactly $n$ join-irreducible elements in $\W_n^q$.
\end{thm}
\begin{proof}
 In a finite lattice, an element is join-irreducible if and only if it covers exactly one element. We therefore count the words in $\W_n^q$ that cover exactly one element. Note that a factor $0^a1^b$, with $qa>b\geq 1$ covers only one element if and only if $b=1$ (otherwise it covers $0^a1^{b-1}0$ and $0^{a+1}1^{b-1}$). Now we investigate the elements covered by $1^m$. Given $i\geq0$, there is at most one word covered by $1^m$ with suffix $01^i$. So the words covered by $1^m$ are exactly

\begin{proof} For any words $v$ and $w$ in $\mathcal{W}_n^q$, we consider the binary word $a=a_1\ldots a_n$  where $a_i=1$ if and only if $v_i=w_i=1$. It is straightforward to see that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$,  which implies that $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.
\end{proof}

\begin{figure}[h]
    \centering
\begin{tikzpicture}[scale=1.3]
    \node (11111) at (0,7.5) {$11111$};
    \node (11110) at (-1.5,6) {$11110$};
    \node (11001) at (0,4.5) {$11001$};
    \node (11100) at (-3,4.5) {$11100$};
    \node (00011) at (3,3) {$00011$};
    \node (10001) at (1,3) {$10001$};
    \node (10010) at (-1,3) {$10010$};
    \node (11000) at (-3,3) {$11000$};
    \node (00001) at (3,1.5) {$00001$};
    \node (00010) at (1,1.5) {$00010$};
    \node (10000) at (-3,1.5) {$10000$};
    \node (00100) at (-1,1.5) {$00100$};
    \node (00000) at (0,0) {$00000$};
    \draw (00000) -- (00100);\draw (00000) -- (10000);\draw (00000) -- (00001);\draw (00000) -- (00010);\draw (10000) -- (11000);\draw (10000) -- (10010);\draw (10000) -- (10001);\draw (00010) -- (10010);\draw (00010) -- (00011);\draw (00001) -- (10001);\draw (00001) -- (00011);\draw (00100) -- (11100);\draw (11000) -- (11001);\draw (11000) -- (11100);\draw (10010) -- (11110);\draw (10001) -- (11001);\draw (00011) -- (11111);\draw (11100) -- (11110);\draw (11001) -- (11111);\draw (11110) -- (11111);
\end{tikzpicture}
    \caption{The lattice $\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }
    \label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \( q > 0 \), we provide a closed-form expression for the generating function counting the number of coverings in \( \mathbb{W}^q_n \). When \( q \) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\Phi(q)$ defined in the introduction above. In Section~\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \( \mathbb{W}^q_n \) for any rational number \( q > 0 \). Finally, Section~\ref{sec:meet} presents the structure of meet-irreducible elements in $\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet  of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\W_n^q$ for any rational number $q>0$.

For any words $v$ and $w$ in $\mathcal{W}_n^q$, consider the binary word $a=a_1\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \"straightforward to see\" that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.

For any words $v$ and $w$ in $\mathcal{W}_n^q$, consider the binary word $a=a_1\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is "straightforward to see" that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.,

Question: Let \(q\ge 0\) and \(n\ge 1\). A binary word of length \(n\) is called \(q\)-decreasing if every maximal factor of the form \(0^a1^b\) satisfies either \(a=0\) or \(q\,a>b\). Let \(\mathcal{W}_n^q\) be the set of all such words, and equip it with the componentwise order: for \(v=v_1\cdots v_n\) and \(w=w_1\cdots w_n\), define \(v\le w\) iff \(v_i\le w_i\) for all \(1\le i\le n\). Write \(\W_n^q=(\mathcal{W}_n^q,\le)\). Which statement holds for every choice of \(q\ge 0\) and \(n\ge 1\)?

The poset \(\W_n^q\) is a lattice; equivalently, every pair of elements of \(\mathcal{W}_n^q\) has both a meet (greatest lower bound) and a join (least upper bound) in the componentwise order.

Choices:
B. The poset \(\W_n^q\) is a meet-semilattice but, for some choices of \(q\ge 0\) and \(n\ge 1\), it is not a lattice; in general, componentwise meets of elements of \(\mathcal{W}_n^q\) always exist, whereas componentwise joins need not belong to \(\mathcal{W}_n^q\).
C. For every \(q\ge 0\) and \(n\ge 1\), every pair of elements of \(\mathcal{W}_n^q\) has a meet (greatest lower bound) in the componentwise order; equivalently, \(\W_n^q\) is a meet-semilattice.
D. The poset \(\W_n^q\) is a lattice for every irrational \(q>0\), but this can fail for rational values of \(q\) because the distinction between the strict condition \(q\,a>b\) and the weak condition \(q\,a\ge b\) prevents componentwise joins from being well defined in general.
E. The poset \(\W_n^q\) is a lattice precisely for \(q>0\); when \(q=0\), the condition on maximal factors forces a boundary case in which \(\mathcal{W}_n^q\) is no longer closed under the meet and join operations in the componentwise order.


Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\mathbb R[x_1,\dots,x_n],$ contains more simple modules than $\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\mathbb C$-algebra $A_{\mathbb R}$ such that $\simp(A_{\mathbb R})\cong\mathbb R^n.$ Thus the points in $\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\mathbb R}$-modules for which $\aspec(A_{\mathbb R})$ is a fine moduli, see the book \cite{S23} or the preprint \cite{S241}. For any real manifold $M$ we define an associative variety $(\mathcal M,\mathcal O^A_L)$ over $\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\mathcal M,$ and such that the charts $U$ in $\mathcal M$ corresponds to $\mathcal O^A_L(U)=A_{\mathbb R}.$

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

Because of the discovery of localization in associative rings, \cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.
Let $k$ be a field and consider the polynomial algebra in $n\geq 1$ variables $$A=k[x_1,\dots,x_n]=k[\underline x].$$ For a point $P=(p_1,\dots,p_n)\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\dots,x_n-p_n).$$

\begin{abstract}
In the preprint \cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$
Then we define the set of aprime right $A$-modules $\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\aspec A=\spec A,$ and we define a topology on $\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on $\aspec A,$ agreeing with the usual sheaf of rings on $\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq\aspec A.$ In particular, this
proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.
\end{abstract}

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

\begin{proof} We have a homomorphism $f:A\rightarrow A/\mathfrak m$ such for every $s\in A\setminus\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\phi:A_\mathfrak m\rightarrow A/\mathfrak m$ such that $\phi(\frac{a}{s})=\iota(a)\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\mathfrak m A_\mathfrak m$ giving the wanted isomorphism.
\end{proof}

\begin{proof} We send the prime ideal $\mathfrak p\subset A$ to $A_\mathfrak p/\mathfrak p A_\mathfrak p$ which is $A$-prime by definition. On the other hand, let
$M$ be an aprime $A$-module, defined by $\iota_M: A\rightarrow B$ such that $M$ is a simple $B$-module. Let $\mathfrak m\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\iota_M^{-1}(\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\mathfrak m\simeq B_{\mathfrak m}/\mathfrak m B_\mathfrak m$ as proven i Lemma \ref{anotherloclemma}.
\end{proof}

\begin{definition} Define the subset of $k$-points in $\mathbb X$ by $$\tilde{\mathbb X}(k)=\{x\in X\subseteq\mathbb X|x\text{ is simple}\}\subseteq\mathbb X.$$ Then the induced associative subscheme $\mathbb X(k)$ is called the associative subscheme of $k$-points.
\end{definition}

\begin{proof} Because $k\subseteq\Bbbk$ is a sub-algebra, it follows that if $\phi\otimes\id$ is an isomorphism, then $\dim_k V_1=\dim_k V_2,$ and that choosing corresponding bases, $$0\neq\det(\phi\otimes\id)=\det\phi.$$
\end{proof}

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P=Q$ we can assume $M_1=M_2=k[\underline x]/(\underline x).$ Then any inner derivation is on the form $\ad_\phi$ for which $\ad_\phi(x_i)=[\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\ext^1_A(M_1,M_2)\simeq\der_k(A,k)$ where
$\operatorname d_i(x_i)=1,\ i=1,\dots,n$ gives a basis for the derivations.

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

It follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \cite{S23} for a lot of examples.

From \cite{S23}: \(\ext^1_A(M_1,M_2)\simeq \der_k(A,\hmm_k(M_1,M_2))/\inner\), with \(\hmm_k(M_1,M_2)\) an \(A\)-\(A\) bimodule via \(a\phi(m)=\phi(am)\), \(\phi a(m)=\phi(m)a\).

- Case \(P=Q\): assume \(M_1=M_2=k[\underline x]/(\underline x)\). Any inner derivation is \(\ad_\phi\) and \(\ad_\phi(x_i)=[\phi,x_i]=0\), so \(\dim_k\inner=0\). Hence \(\ext^1_A(M_1,M_2)\simeq \der_k(A,k)\), and the derivations \(\operatorname d_i\) with \(\operatorname d_i(x_i)=1\) for \(i=1,\dots,n\) give a basis, so the dimension is \(n\).

- Case \(P\neq Q\): consider \(M_{\underline 0}, M_P\) with \(p_1\neq 0\). For \(\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P))\), comparing \(\delta(x_ix_1)=\delta(x_1x_i)\) yields
\[\delta(x_i)x_1+x_i\delta(x_1)=\delta(x_1)x_i+x_1\delta(x_i)\Rightarrow \delta(x_i)p_1=\delta(x_1)p_i\Rightarrow \delta(x_i)=\delta(x_1)\frac{p_i}{p_1}.\]
This relation also holds for inner derivations, and it is concluded that \(\dim_k\inner=1\), so \(\ext^1_A(M_P,M_Q)=0\) when \(P\neq Q\).

For two points $P,Q\in k^n$ we have that $$\dim_k\ext^1_A(M_P,M_Q)=\begin{cases}n,\ P=Q\\0,\ P\neq Q.\end{cases}$$,

Question:
Let $k$ be a field, let $A=k[x_1,\dots,x_n]$ be the polynomial algebra in $n\ge 1$ variables over $k$, and for each point $P=(p_1,\dots,p_n)\in k^n$ define the simple $A$-module
$$M_P:=A/(x_1-p_1,\dots,x_n-p_n).$$
Which statement holds for every pair of points $P,Q\in k^n$ about the $k$-dimension of $\operatorname{Ext}^1_A(M_P,M_Q)$?

$$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\ne Q.\end{cases}$$

Choices:
B. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\1,& P\ne Q.\end{cases}$$
C. $$\dim_k\operatorname{Ext}^1_A(M_P,M_P)=n\qquad\text{for every }P\in k^n.$$
D. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\text{ and }Q\text{ differ in every coordinate}.\end{cases}$$
E. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n-1,& P=Q,\\0,& P\ne Q.\end{cases}$$

quotient by inner derivations in the off-diagonal case

use of a coordinate with nonzero difference after reducing to one point at the origin

dimension of derivations in the diagonal case


\label{thm:main}
Let $E_0\in \mathbb R$ be a fixed energy for which $L_d(\omega,E_0)>0$.
For any $\omega\in \mathrm{DC}$, the integrated density of states of $H_{\omega,\theta}$ satisfies
    \begin{align}
        |\mathcal{N}(\omega,E)-\mathcal{N}(\omega,E')|\leq |E-E'|^{\beta},
    \end{align}
    for any $E,E'\in I_{E_0}$, a neighborhood of $E_0$, and any H\"older exponent $0<\beta<1/(2\kappa^d(\omega,E_0))$.

\begin{theorem}\label{thm:main}
Let $E_0\in \R$ be a fixed energy for which $L_d(\omega,E_0)>0$.
For any $\omega\in \mathrm{DC}$, the integrated density of states of $H_{\omega,\theta}$ satisfies
    \begin{align}
        |\mathcal{N}(\omega,E)-\mathcal{N}(\omega,E')|\leq |E-E'|^{\beta},
    \end{align}
    for any $E,E'\in I_{E_0}$, a neighborhood of $E_0$, and any H\"older exponent $0<\beta<1/(2\kappa^d(\omega,E_0))$.
\end{theorem}

\begin{theorem}\label{thm:acc=zeros}
Assume that for some $E\in \R$, $L^1(\omega,E)\geq \nu>0$.
Then for some $\gamma\in (0,1)$ and for $n$ large enough,
\begin{align}
\left| \frac{1}{2n} N_n (E,\delta/2)-\kappa^1(\omega,E)\right|\leq \delta^{-1} n^{-\gamma}.
\end{align}
\end{theorem}

\begin{aligned}\label{def:DC}
    \omega\in \mathrm{DC}:=&\bigcup_{a>0, A>1} \mathrm{DC}_{a,A}, \text{ where}\\
    \mathrm{DC}_{a,A}=&\left\{\omega\in \T:\, \|k\omega\|_{\T}\geq \frac{a}{|k|^A}\, \text{ for all } k\in \Z\setminus \{0\}\right\}.
\end{aligned}

\begin{lemma}\label{lem:numerator_diag}
    Let $\omega\in \mathrm{DC}$ and $x=y\in [2d,n-2d-1]$,
    Then for $\gamma>0$ as in Lemma \ref{lem:upperbd}, $n$ large, and $E'\in \C$ such that $|E'-E|\leq n^{-2}e^{-n^{\gamma}}$, uniformly in $\theta\in \T$:
    \begin{align}
        |\mu^{E'}_{[0,nd-1],x,x}(\theta)|\leq  e^{n(L^d(E)+\langle \log |\det B|\rangle+5n^{-\gamma})}.
    \end{align}
\end{lemma}

\begin{align}\label{eq:SDC}
 \mathrm{DC}_{\mathrm{strong}}:= \bigcup_{a>1, c>0} \left\{\omega\in \T:\, \|k\omega\|_{\T}\geq \frac{c}{|k|(\log |k|)^a}\, \text{ for all } k\in \Z\setminus \{0,\pm 1\}\right\}.
\end{align}

\begin{lemma}\label{lem:local_zero*}
Let $n$ be $\kappa_0$-admissible and large.
For each ball $B(z_0,r_n)$, with $z_0\in \kreis^1$ and $r\simeq r_n:=e^{-(\log n)^{C_0}}$ with $C_0>1$, there exists $|k|<(1-\epsilon)n/2$ such that $f_n(e^{2\pi ik\omega} z,E)$ has at most $2\kappa^d$ zeros in $B(z_0,r)$.
\end{lemma}

In this paper we study Schr\"odinger cocycles on a strip with Hamiltonian
\begin{equation}\label{eq:BVsys}
 (H_{\omega,\theta} \Phi)_n = B_{n+1}(\theta)\Phi_{n+1}+ B_n^{(*)}(\theta) \Phi_{n-1}+ V_n (\theta)\Phi_n
\end{equation}
where $F_n(\theta):=F(\theta+n\omega)$ for any $d\times d$-matrix valued function. We set $B^{(*)}(\theta)=(B(\theta))^*$ for $\theta\in \mathbb T$, and require it to be the analytic extension of $(B(\theta))^*$ off of the real torus. Here $\omega\in\mathbb T$ and we assume that $\omega$ is Diophantine, i.e.,
\begin{equation}\begin{aligned}\label{def:DC}
 \omega\in \mathrm{DC}:=&\bigcup_{a>0, A>1} \mathrm{DC}_{a,A}, \text{ where}\\
 \mathrm{DC}_{a,A}=&\left\{\omega\in \mathbb T:\, \|k\omega\|_{\mathbb T}\geq \frac{a}{|k|^A}\, \text{ for all } k\in \mathbb Z\setminus \{0\}\right\}.
\end{aligned}
\end{equation}
We further assume that $B,V\in C^{\omega}(\T_{\delta}, \mathrm{Mat}(d,\mathbb C))$ are analytic, where $$\T_{\delta}:=\{\theta+i\varepsilon:\, \theta\in \mathbb T,\, \varepsilon\in \mathbb R, \text{ and } |\varepsilon|\leq \delta\}$$with some positive $\delta>0$.
We assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\det B(\theta)\neq 0$ for any $\theta\in \T_{\delta}$).
The difference equation $H_{\omega,\theta} \Phi=E\Phi$ is equivalent to the cocycle
\begin{align}\label{def:M_E}
 \mathcal{C} &\:: (\theta,\Psi)\in \mathbb T\times \mathbb C^{2d}\mapsto (\theta+\omega, M_E(\theta)\Psi), \notag \\
 \quad M_{E}(\theta) &= \left [ \begin{matrix}
 ( E-V(\theta))B(\theta)^{-1} & -B^{(*)}(\theta) \\
 B(\theta)^{-1} & 0
\end{matrix}\right]
\end{align}
in the sense that for $n\ge1$,
\begin{equation}\label{eq:MnE}
\mathcal{C}^n (\theta,\Psi)=(\theta+n\omega, M_{n,E}(\theta)\Psi), \quad M_{n,E}(\theta)=\prod_{j=n-1}^0 M_E(\theta+j\omega), \quad
\Psi_n:=\binom{B_{n}\Phi_n}{\Phi_{n-1}}
\end{equation}
satisfies $\Psi_n = M_{n,E} (\theta) \Psi_0$. The model~\eqref{eq:BVsys} was studied systematically by S.~Klein~\cite{SK_block}, and Duarte and Klein~\cites{DK1, DK2} within a very general framework, including the identically singular case when $\det B(\theta)\equiv 0$ on $\mathbb T^k$, $k\geq 1$. These models generalize the quasi-periodic Schr\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\cite{BJ}, who proved localization perturbatively for large disorder.

Since $M_E(\theta)$ is (complex) symplectic, see \eqref{def:symp}, for $\theta\in \mathbb T$, the Lyapunov exponents $\{L_j(\omega,M_E)\}_{j=1}^{2d}$, see definition in \eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\le j\le d$.
Let $L^d(\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\kappa^d(\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\omega,M_E(\cdot+i\varepsilon))$ in the imaginary direction $\varepsilon$, see \eqref{eq:acc}.

Throughout the paper we shall fix an energy $E_0\in \mathbb R$ such that $L_d(\omega,E_0)\geq \tau>0$. It is known, see \cite{AJS}, that $L^d(\omega,M_E(\cdot+i\varepsilon))$ and $L_d(\omega,M_E)$ are continuous in $E,\varepsilon$ and $\kappa^d(\omega,M_E)$ is upper semi-continuous in $E,\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\in I_{E_0}$ and $|\varepsilon|\leq \delta$, $\varepsilon\in \mathbb R$, for some $\delta=\delta(E_0)>0$:
\begin{align}\label{def:IE0}
\begin{cases}
L_d(\omega,M_E(\cdot+i\varepsilon))\geq \tau/2,\\
\kappa^d(\omega,M_E(\cdot+i\varepsilon))\leq \kappa^d(\omega,M_{E_0})\\
L^d(\omega,M_E(\cdot+i\varepsilon))\leq L^d(\omega,M_E)+2\pi \kappa^d(\omega,M_{E_0})\varepsilon.
\end{cases}
\end{align}
Note the third inequality is a corollary of the second one.

The density of states $\mathcal{N}(\omega,\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues.
Let $H_{\omega,\theta}|_{\Lambda}$ be the restriction of $H_{\omega,\theta}$ to the interval $\Lambda=[a,b]\subset \mathbb Z$.
Let $\{E_{\Lambda,j}(\omega,\theta)\}_{j=1}^{|\Lambda|}$ be the eigenvalues of $H_{\omega,\theta}|_{\Lambda}$.
Consider
\begin{align}
 N_{\Lambda}(\omega,E,\theta):=\frac{1}{|\Lambda|}\sum_{j=1}^{|\Lambda|}\chi_{(-\infty,E)}(E_{\Lambda,j}(\omega,\theta)).
\end{align}
It is well-known that the weak-limit
\begin{align}
 \lim_{a\to-\infty, b\to \infty}\mathrm{d} N_{\Lambda}(\omega,E,\theta)=:\mathrm{d} \mathcal{N}(\omega,E).
\end{align}
exists and is independent of $\theta$. In this paper we prove the following result.

\begin{align}\label{def:LE}
L_j(\omega, A,n):=\frac{1}{n}\int_{\T} \log \sigma_j(A_n(\omega,\theta))\, \mathrm{d}\theta, \text{ for } 1\leq j\leq k,
\end{align}

\begin{align}\label{def:symp}
    M^*\Omega M=\Omega,
\end{align}

\begin{equation}\label{eq:BVsys}
   (H_{\omega,\theta} \Phi)_n   = B_{n+1}(\theta)\Phi_{n+1}+ B_n^{(*)}(\theta) \Phi_{n-1}+ V_n (\theta)\Phi_n
\end{equation}

\begin{align}\label{eq:acc}
\kappa^1_{\varepsilon}(\omega, A):=\lim_{\varepsilon'\to 0^+} \frac{L^1_{\varepsilon+\varepsilon'}(\omega, A)-L^1_{\varepsilon}(\omega, A)}{2\pi \varepsilon'}.
\end{align}

In \cite{HS3}*{Lemma 2.8} we established the following large deviation theorem for $f_n$.
\begin{lemma}\label{lem:deno}
Let $\omega\in \mathrm{DC}$, and $\gamma>0$ be as in Lemma \ref{lem:LDTsig}.
Assume $L_d(\omega,M_E)\geq \nu>0$.
There exist $\gamma>0$, $N_0>1$ large and $0<\kappa_0\ll 1$ so that the {\it $\kappa_0$-admissible} sequence
\begin{align}\label{def:admissible}
\mathcal{N}:=\{n\geq N_0: \|n\omega\|_{\T}\leq \kappa_0\}
\end{align}
has the following property:
for any $|\varepsilon|\leq \delta/2$, and all large $\kappa_0$-admissible $n$, the following large deviation set
\begin{align}\label{def:B_fEn}
\mathcal{B}_{f,E,n,\varepsilon}:=
\big\{\theta\in \T: n^{-1}\log |f_{n}(\theta+i\varepsilon,E)|<\langle \log |\det B(\cdot+i\varepsilon)|\rangle +{L}^d_{\varepsilon}(\omega,M_E)- n^{-\gamma}\big\}
\end{align}
satisfies $\mathrm{mes}(\mathcal{B}_{f,E,n,\varepsilon})<e^{-n^{\gamma}}$.
\end{lemma}
\begin{remark}\label{rem:admissible}
 For every large integer $n>0$ there exists an admissible $\tilde n>0$ with $|n-\tilde n|\le C_*$ for some constant $C_*$.
\end{remark}

Recall that $E\in I_{E_0}$.
Our goal is to analyze the following for an arbitrary fixed $\theta$ as $N\to\infty$:
\begin{align}\label{eq:goal}
 &d_{\omega,N}(\theta,E-\eta,E+\eta)=\frac{1}{N}\mathrm{tr}(P_{[E-\eta, E+\eta)}(H_{\omega,\theta}|_{[0,N-1]})),
\end{align}
in which $H_{\omega,\theta}|_{[0,N-1]}$ is $H_{\omega,\theta}$ restricted to the interval $[0,N-1]$ with Dirichlet boundary condition and $P_{[E-\eta,E+\eta)}$ stands for the spectral projection.
Bounding \eqref{eq:goal} through the trace of the resolvent yields
\begin{align}\label{eq:dN<G}
 d_{\omega,N}(\theta,E-\eta,E+\eta)
 &\leq \frac{2\eta}{N} \mathrm{Im}(\mathrm{tr}(G_{[0,N-1]}^{E+i\eta}(\theta)))\\
 &=\frac{2\eta}{N}\sum_{k=0}^{N-1} \mathrm{Im} (G_{[0,N-1]}^{E+i\eta}(\theta;k,k)),
\end{align}
in which $G^{E+i\eta}_{[0,N-1]}(\theta)=(H_{\omega,\theta}|_{[0,N-1]}-(E+i\eta))^{-1}$ is the Green's function, i.e., the kernel of the resolvent.
Let an auxiliary parameter $\tau$ be defined as
\begin{align}\label{eq:choose_tau}
 \tau=\eta^{\frac{1}{2\kappa}}.
\end{align}
For any $n\in \N$, let
\begin{align}
 \mathcal{B}_{n,h}^{\eta}:=\{\theta\in \T: u_n(e^{2\pi i\theta},E+i\eta)<L(E,n)-h\cdot n^{-\gamma}\}.
\end{align}
be the large deviation set for $u_n(z,E+i\eta)=n^{-1}\log |D_n(z,E+i\eta)|$.
We choose $m\in \N$ such that
$e^{-(2m)^{\gamma_1}}\simeq \eta$.
The set $\mathcal{B}_{2m,2}^{\eta}$ satisfies a large deviation estimate
\begin{align}\label{eq:mes_B2m_2}
 \mes\;\mathcal{B}_{2m,2}^{\eta}\leq e^{-(2m)^{\gamma}}.
\end{align}
In fact, the large deviation estimate of Lemma \ref{lem:un_LDT} applies directly to $\mathcal{B}_{2m,1}^0$, implying that
\begin{align}\label{eq:mes_B2m_1}
 \mathrm{mes}\; \mathcal{B}_{2m,1}^0\leq e^{-(2m)^{\gamma}}.
\end{align}
If $\theta\in (\mathcal{B}_{2m,1}^0)^c$, $|D_{2m}(e^{2\pi i\theta},E)|>\exp(2m(L(E,2m)-(2m)^{-\gamma}))$. A telescoping argument, as in the proof of Lemma \ref{lem:un_lower_zeros_eta}, using that $\gamma_1>1-\gamma/2$, yields
\begin{align}
 |D_{2m}(e^{2\pi i\theta},E+i\eta)|>\frac{1}{2}|D_{2m}(e^{2\pi i\theta},E)|>\frac{1}{2}\exp(2m(L(E,2m)-(2m)^{-\gamma})).
\end{align}
This implies $(\mathcal{B}_{2m,1}^0)^c\subset (\mathcal{B}_{2m,2}^{\eta})^c$,
and hence \eqref{eq:mes_B2m_2}, when combining with \eqref{eq:mes_B2m_1}.

The proof is analogous to the scalar-valued case.
Our goal is to estimate the following expression for arbitrary fixed $E\in I_{E_0}$ and $\theta\in \T$ as $N\to\infty$:
\begin{align}\label{eq:goal*}
 &d_{\omega,dN}(\theta,E-\eta,E+\eta)=\frac{1}{dN}\mathrm{tr}(P_{[E-\eta, E+\eta)}(H_{\omega,\theta}|_{[0,dN-1]})),
\end{align}
cf.\ \eqref{eq:goal}. As before, we
bound~\eqref{eq:goal*} through the trace of the Green's function:
\begin{align}\label{eq:dN<G*}
 d_{\omega,dN}(\theta,E-\eta,E+\eta)
 &\leq \frac{2\eta}{dN} \mathrm{Im}(\mathrm{tr}(G_{[0,dN-1]}^{E+i\eta}(\theta)))\\
 &=\frac{2\eta}{dN}\sum_{k=0}^{dN-1} \mathrm{Im} (G_{[0,dN-1]}^{E+i\eta}(\theta;k,k)),
\end{align}
in which $G^{E+i\eta}_{[0,dN-1]}(\theta)=(H_{\omega,\theta}|_{[0,dN-1]}-(E+i\eta))^{-1}$ is the kernel of the resolvent.
Let the auxiliary parameter $\tau$ be defined as
\begin{align}\label{def:tau_d}
 \tau=\eta^{\frac{1}{2\kappa^d}},
\end{align}
in this block-valued case.
For any $\kappa_0$-admissible $n\in \N$, let
\begin{align}
 \mathcal{B}_{n,h}^{\eta}:=\{\theta\in \T: u_n(e^{2\pi i\theta},E+i\eta)<L^d(E,n)+\langle \log |\det B(\cdot)|\rangle-h\cdot n^{-\gamma}\}
\end{align}
be the large deviation set for $u_n(z,E+i\eta)=n^{-1}\log |f_n(z,E+i\eta)|$.
We choose $m\in \N$ such that $2m$ is $\kappa_0$-admissible and $e^{-(2m)^{\gamma_1}}\simeq \eta$.
The set $\mathcal{B}_{2m,2}^{\eta}$ satisfies a large deviation estimate
\begin{align}\label{eq:B2m_eta_block}
 \mes\,\mathcal{B}_{2m,2}^{\eta}\leq e^{-(2m)^{\gamma}}.
\end{align}
In fact, note $2m$ is $\kappa_0$ admissible, the large deviation estimate of Lemma \ref{lem:deno} applies to $\mathcal{B}_{2m,1}^0$ implying
\begin{align}\label{eq:B2m_0_block}
 \mathrm{mes}\,\mathcal{B}_{2m,1}^0\leq e^{-(2m)^{\gamma}}.
\end{align}
Suppose $\theta\in (\mathcal{B}_{2m,1}^0)^c$, then
\begin{align}
 |f_{2m}(e^{2\pi i\theta},E)|\geq \exp(2m(L^d(E,2m)+\langle \log |\det B(\cdot) |\rangle-(2m)^{-\gamma})).
\end{align}
By the same telescoping argument in the proof of Lemma \ref{lem:un_lower_zeros_eta*}, using, $\gamma_1>1-\gamma/2$, we have
\begin{align}
 |f_{2m}(e^{2\pi i\theta},E+i\eta)|\geq \frac{1}{2} |f_{2m}(e^{2\pi i\theta}, E)|\geq \exp(2m(L^d(E,2m)+\langle \log |\det B(\cdot) |\rangle-2(2m)^{-\gamma})).
\end{align}
This implies $(\mathcal{B}_{2m,1}^0)^c\subset (\mathcal{B}_{2m,2}^{\eta})^c$, hence the estimate \eqref{eq:B2m_eta_block} follows from \eqref{eq:B2m_0_block}.
Let $\{\xi_j\, e^{2\pi i\theta_j}\}_{j=1}^{J_0}$, $\xi_j>0$ and $\theta_j\in \T$, be the zeros of $f_{2m}(z,E+i\eta)$.
Cartan's estimate implies that
\begin{align}\label{eq:B2m_theta_pm_rm*}
\mathcal{B}_{2m,m^{\gamma/2}}^{\eta}\subset \bigcup_{j=1}^{J_0}(\theta_j-\tilde{r}_m, \theta_j+\tilde{r}_m),
\end{align}
with $\tilde{r}_m:=e^{-cm^{\gamma/2}}$ for some constant $c>0$, and $J_0\lesssim m$.
Applying Lemma~\ref{lem:un_lower_zeros_eta*} to $n=2m$ and $z_0=e^{2\pi i\theta_j}$, for each $1\leq j\leq J_0$, yields the existence of $|k_j|<(1-\varepsilon)m$, $C_j\in [1,2\kappa^d+1]$, and $\ell_j\in [1,2\kappa^d]$, such that $w_j(z):=f_{2m}(z e^{2\pi ik_j\omega},E+i\eta)$
has $\ell_j$ zeros in $B(e^{2\pi i\theta_j}, (4C_j+1)r_m)$, $r_m\simeq \exp(-(\log m)^{C_0})\gg \tilde{r}_m$.
Denoting the zeros of $f_{2m}(z e^{2\pi ik_j\omega},E+i\eta)$ by $\{\xi_{j,\ell}\, e^{2\pi i\theta_{j,\ell}}\}_{\ell=1}^{\ell_j}$ by $\xi_{j,\ell}>0$, $\theta_{j,\ell}\in \T$, one has
\begin{align}\label{eq:wj_lower_zeros*}
 \log |w_j(z)|\geq 2m(L^d(E,2m)+\langle \log |\det B(\cdot)|\rangle-m^{-\gamma/2})+\sum_{\ell=1}^{\ell_j}\log |z-\xi_{j,\ell}\, e^{2\pi i\theta_{j,\ell}}|,
\end{align}
for all $z\in B(e^{2\pi i\theta_j},(4C_j+2)r_m)$.
For each $1\leq j\leq J_0$ and $1\leq \ell\leq \ell_j$, let
 $ \tilde{I}_j,
 I_{j,\ell}$ be defined as in~\eqref{eq:Ijdef}.
With $\tau$ as in \eqref{eq:choose_tau} and $\tau\ll r_m$, we have for each $j,\ell$ that $I_{j,\ell}\subset \tilde{I}_j$.
As before, we let $\tilde{\mathcal{B}}_{2m}^{\eta}:=\bigcup_{j=1}^{J_0} \tilde{I}_j$.
Clearly, by \eqref{eq:B2m_theta_pm_rm*}, $\mathcal{B}_{2m,m^{\gamma/2}}^{\eta}\subset \tilde{\mathcal{B}}_{2m}^{\eta}$.
We again divide the analysis of $G^{E+i\eta}_{[0,dN-1]}(\theta; k,k)$ into three cases.

\begin{remark}
    The exponent $2\kappa^d(\omega,E)$ comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count. When $d\geq 2$, this is the zero count of $f_n(\theta,E)$, see \eqref{def:fn}. If the blocks $B,V$ satisfy additional symmetries, then it is possible to deduce  smaller local zero counts, and hence improved H\"older exponents. We refer interested readers to \cite[Theorem 1.4 \& Theorem 1.6]{HS3} where the effects of additional symmetries are studied.
\end{remark}

As for the history of this type of regularity result: $\log$--H\"older continuity was established by Craig and Simon in \cite{CS} for general ergodic potentials. The stronger H\"older continuity of the integrated density of states associated with~\eqref{eq:BVsys} was first proved by Goldstein and the second author in \cite{GS1} for the scalar case $d=1$.
Later, \cite{GS2}*{Theorem 1.1} established that for a trigonometric polynomial $f$ of degree $k_0\ge1$ and assuming $\omega\in \mathrm{DC}_{\mathrm{strong}}$, see \eqref{eq:SDC}, and
$L(\omega,E)>0$  the IDS is $\beta$-H\"older continuous for any $\beta<1/(2k_0)$.
This result, for the same set of frequencies, was improved to $\beta<1/(2\kappa^1(\omega,E))$ in \cite{HS1}. Note for trigonometric $f$ one has $\kappa^1(\omega,E)\leq k_0$.
Theorem~\ref{thm:main}, when $d=1$, is an improvement of \cite{HS1}*{Theorem 1.3} in the sense that it covers more general Diophantine frequencies: $\mathrm{DC}_{\mathrm{strong}}\subset \mathrm{DC}$. When $d\geq 2$, Theorem~\ref{thm:main} is completely new.

For the almost Mathieu operator and $\alpha\in \mathrm{DC}$, the IDS was proved to be $1/2$-H\"older if $|\lambda|\neq 0,1$, see Avila, Jitomirskaya~\cite{AJ}.
The $1/2$-H\"older exponent was also proved for Diophantine $\alpha$ and $f=\lambda\, g$ with $g$ being analytic and the coupling constant $|\lambda|$ being small, first in the perturbative regime (smallness depends on $\alpha, g$) by Amor~\cite{Amor}, and then in the non-perturbative regime \cite{AJ} (with dependence on $\alpha$ removed).
For quasi-periodic long-range operators with large trigonometric polynomial potentials and Diophantine frequencies, the H\"older exponent in \cite{GS2} was improved recently in the perturbative regime~\cite{GYZ}.
These results are proved using the reducibility method.

For the proof of Theorem~\ref{thm:main} we first single out the Schr\"odinger operator case, where $d=1$, $B\equiv 1$ and $V=v$ is a non-constant analytic function. Sections~\ref{sec:sch1}, \ref{sec:sch2} present this case in details. The argument is variant of the one in~\cite{GS2}, but is simpler and more robust.
The difference lies in how we derive a favorable local zero count for a finite volume determinant relative to some interval $\Lambda\subset\mathbb Z$. In~\cite{GS2}, this is accomplished by slightly varying the size of~$\Lambda$, but not the position. Here we keep the size fixed but change the position. The strip case ($d>1$ in~\eqref{eq:BVsys}) follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case. Thankfully,  these were developed in~\cite{HS3}, so we can treat them as a black box, cf.~Theorem~\ref{thm:acc=zeros}.

We also point out that the Aubry dual of a subcritical Schr\"odinger operator $H$, with trigonometric potential with degree $d$, is a special family of $d\times d$ Jacobi block-valued matrix $\hat{H}$ with uniformly positive $L_d(\omega,E)>0$, see e.g.~\cite[Corollary 1.12]{HS2}. An easy computation shows that $\kappa^d(\omega,E)\leq d$, but one can show the local zero count is at most $2$, instead of $2d$. This is a small modification of Lemma \ref{lem:local_zero*} with considering not only $f_n(e^{2\pi ik\omega}z,E)$ with $|k|<(1-\varepsilon)n/2$, but also $f_n(e^{2\pi ik\omega/d}z,E)$ with $|k|<(1-\varepsilon)nd/2$. This is a feature due to the special form of $\hat{H}$, and has been explored in the proof of arithmetic Anderson localization of $\hat{H}$ in \cite[Theorem 1.7]{HS2} and \cite[Corollary 1.3]{HS3}. The local zero count being at most $2$ then implies almost $1/2$ H\"older continuity of the integrated density of states of the original subcritical Schr\"odinger with arbitrary trigonometric potentials with Diophantine frequencies.

Finally, we comment that the (non-)H\"older continuity of the integrated density of states for Liouville frequencies has also been studied \cite{YZ,HZ,ALSZ,HS4} when $d=1$,
and it would be natural to combine those considerations with the techniques developed here for $d\geq 2$.

Our proofs are essentially self-contained. The only ingredients we require are the large deviation estimates, developed in \cites{BG,GS1,HS3} and global zero counts of finite-volume determinants in \cites{HS1,HS3}.
As we mentioned, the Diophantine condition \eqref{def:DC} is weaker than that in \cites{GS2,HS1} where it was defined to be
\begin{align}\label{eq:SDC}
 \mathrm{DC}_{\mathrm{strong}}:= \bigcup_{a>1, c>0} \left\{\omega\in \mathbb T:\, \|k\omega\|_{\mathbb T}\geq \frac{c}{|k|(\log |k|)^a}\, \text{ for all } k\in \mathbb Z\setminus \{0,\pm 1\}\right\}.
\end{align}
The Anderson localization result of \cite{HS1} holds for the weaker Diophantine condition \eqref{def:DC}, see \cite{HS1}*{Remark 1.9}. However,  the H\"older continuity result of \cite{HS1} holds under \eqref{eq:SDC} due to the need for a sharp large deviation estimate as in \cite{GS2}.

We organize the paper as follows: Section~\ref{sec:Pre} contains the preliminary results. Sections~\ref{sec:sch1} and \ref{sec:sch2} prove Theorem~\ref{thm:main} in the Schr\"odinger case, more specifically, Sec.~\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\ref{thm:main} in Sec.~\ref{sec:sch2}. The Jacobi block-valued case is proved in Sections~\ref{sec:block1} and \ref{sec:block2}. Finally, the proof of Lemma~\ref{lem:numerator_diag} is contained in Sec.~\ref{sec:numerator}.

For the proof of Theorem~\ref{thm:main} the authors “first single out the Schr\"odinger operator case, where $d=1$, $B\equiv 1$ and $V=v$ is a non-constant analytic function.” Sections~\ref{sec:sch1}, \ref{sec:sch2} treat this case; “the argument is variant of the one in~\cite{GS2}, but is simpler and more robust.” The key difference is “how we derive a favorable local zero count for a finite volume determinant relative to some interval $\Lambda\subset\mathbb Z$”: in~\cite{GS2} this is done “by slightly varying the size of~$\Lambda$, but not the position,” whereas here “we keep the size fixed but change the position.”

The strip/block case (“$d>1$ in~\eqref{eq:BVsys}”) “follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case”; these zero counting tools were developed in~\cite{HS3} and are used “as a black box, cf.~Theorem~\ref{thm:acc=zeros}.”

Structurally, “Sec.~\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\ref{thm:main} in Sec.~\ref{sec:sch2},” and “the Jacobi block-valued case is proved in Sections~\ref{sec:block1} and \ref{sec:block2}.” The underlying inputs are stated to be “large deviation estimates” (from \cites{BG,GS1,HS3}) and “global zero counts of finite-volume determinants” (from \cites{HS1,HS3}); the remark also explains that the exponent $2\kappa^d(\omega,E)$ “comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count.”

Question:
Consider the analytic quasi-periodic Schr\"odinger operator on a strip acting on sequences \(\Phi_n\in\mathbb C^d\) by
\[
(H_{\omega,\theta}\Phi)_n=B(\theta+(n+1)\omega)\Phi_{n+1}+B(\theta+n\omega)^*\Phi_{n-1}+V(\theta+n\omega)\Phi_n,
\]
where \(B,V\in C^{\omega}(\mathbb T_{\delta},\mathrm{Mat}(d,\mathbb C))\), \(V\) is Hermitian, and \(\det B(\theta)\neq 0\) on \(\mathbb T_{\delta}\). Assume the frequency \(\omega\) is Diophantine, i.e.
\[
\omega\in \mathrm{DC}:=\bigcup_{a>0,\,A>1}\left\{\omega\in\mathbb T:\ \|k\omega\|_{\mathbb T}\ge \frac{a}{|k|^A}\ \text{for all }k\in\mathbb Z\setminus\{0\}\right\}.
\]
For each energy \(E\in\mathbb R\), let the associated cocycle be
\[
M_E(\theta)=\begin{pmatrix}
(E-V(\theta))B(\theta)^{-1} & -B(\theta)^*\\
B(\theta)^{-1} & 0
\end{pmatrix}.
\]
Let \(L_d(\omega,E)\) denote the \(d\)-th Lyapunov exponent of this cocycle, let \(L^d\) be the sum of its top \(d\) Lyapunov exponents, and let \(\kappa^d(\omega,E)\) be the right derivative at \(\varepsilon=0\) of \(L^d(\omega,M_E(\cdot+i\varepsilon))\) with respect to the imaginary shift \(\varepsilon\). Let \(\mathcal N(\omega,E)\) be the integrated density of states, i.e. the phase-independent limiting cumulative distribution function of the finite-volume eigenvalues. Fix \(E_0\in\mathbb R\) such that \(L_d(\omega,E_0)>0\). Which statement holds for every such Diophantine \(\omega\)?

There exists a neighborhood \(I_{E_0}\) of \(E_0\) such that for every \(E,E'\in I_{E_0}\) and every H\"older exponent \(0<\beta<1/(2\kappa^d(\omega,E_0))\), one has
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}.
\]

Choices:
B.
There exists a neighborhood \(I_{E_0}\) of \(E_0\) such that for every \(E,E'\in I_{E_0}\) one has
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}
\]
for every H\"older exponent \(0<\beta<1/\kappa^d(\omega,E_0)\).
C.
There exists a neighborhood \(I_{E_0}\) of \(E_0\) and some H\"older exponent \(\beta>0\) such that for every \(E,E'\in I_{E_0}\),
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}.
\]
D.
For every H\"older exponent \(0<\beta<1/(2\kappa^d(\omega,E_0))\), there exists a neighborhood \(I_{E_0,\beta}\) of \(E_0\) such that for every \(E,E'\in I_{E_0,\beta}\),
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta},
\]
and one may choose \(I_{E_0,\beta}\) independently of the Diophantine frequency \(\omega\in\mathrm{DC}\).
E.
There exists a neighborhood \(I_{E_0}\) of \(E_0\) such that for every \(E,E'\in I_{E_0}\) and every H\"older exponent \(0<\beta<1/(2\kappa^d(\omega,E))\), one has
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}.
\]

uniformity over all \(0<\beta<1/(2\kappa^d(\omega,E_0))\) replaced by existence of some positive \(\beta\)

dependence of the neighborhood on the fixed frequency \(\omega\)

acceleration evaluated at \(E_0\) replaced by acceleration at the variable energy \(E\)


Let $X,\, Y$ be closed oriented surfaces and $f \,:\, Y\,\longrightarrow\, X$ a degree $d$ branched
covering. Given a flat vector
bundle $(V,\, \nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle
$(f_*V,\, f_*\nabla)$ of rank $dr$ on $X\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.

Denote $G_m\,=\,\mathrm{GL}(m,\mathbb{C})$. The isomorphism class of the bundle $(V, \,\nabla)$ is determined by the
$G_r$--conjugation orbit of its holonomy $\rho\,:\,\pi_1(Y)\,\longrightarrow\, G_r$. Denote this orbit
(conjugacy class) by $[\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)
Geometric Invariant Theory (GIT) quotient $\mathcal{M}_r(Y)\,:=\,\mathrm{Hom}(\pi_1(Y),\,G_r)/\!\!/G_r$.
We note that the {\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of
$\mathrm{Inn}(G_r)$-invariant polynomials on $\mathrm{Hom}(\pi_1(Y),\,G_r)$. The above GIT quotient
$\mathrm{Hom}(\pi_1(Y),\,G_r)/\!\!/G_r$ is also known as a {\it character variety} or more precisely
the $G_r$--character variety of $\pi_1(Y)$. We likewise consider the character variety
$\mathcal{M}_{dr}(X-B)\,:=\,\mathrm{Hom}(\pi_1(X-B),\,G_{dr})/\!\!/G_{dr}$.

By \cite{Go, La, BJ}, we know $\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\mathcal{M}_{dr}(X-B)$ is
(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\, \nabla)$ to the holonomy of
$(f_*V,\, f_*\nabla)$, we get a natural map $\mathsf{Frob} \,:\, \mathcal{M}_r(Y)
\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ which we call the {\it Frobenius reciprocity map}.

\begin{abstract}
We show that a Frobenius reciprocity map on character varieties of surfaces is a Poisson embedding.
\end{abstract}

By \cite{Go, La, BJ}, we know $\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\mathcal{M}_{dr}(X-B)$ is
(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\, \nabla)$ to the holonomy of
$(f_*V,\, f_*\nabla)$, we get a natural map $\mathsf{Frob} \,:\, \mathcal{M}_r(Y)
\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ which we call the {\it Frobenius reciprocity map}.

Let $Y_0\,=\,Y-f^{-1}(B)$ and $\mathcal{M}_r(Y_0)\,:=\,\mathrm{Hom}(\pi_1(Y_0),\,G_r)/\!\!/G_r$. Again,
$\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\,\subset\, Y$ there is a
natural map $\iota\,:\,\pi_1(Y_0)\,\longrightarrow\, \pi_1(Y)$. The map $\iota$ is surjective (and hence has a
right inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\dim f^{-1}(B)\,=\,0$.

Let $\mathsf{Res}\,:\,\mathcal{M}_r(Y)\,\longrightarrow\, \mathcal{M}_r(Y_0)$ be given by $\mathsf{Res}([h])
\,=\,[h\circ \iota]$, where $h\,:\,\pi_1(Y)\,\longrightarrow\, G_r$. If
$\mathsf{Res}([h_1])\,=\,\mathsf{Res}([h_1])$, then
$[h_1\circ\iota]\,=\,[h_2\circ\iota]$ and so $[h_1]=[h_2]$ since $\iota$ has a right inverse. Therefore,
$\mathsf{Res}$ is an embedding. The inclusion map $Y_0\,\subset\, Y$ preserves transversality of based loops
and thus by \cite[Theorem A]{BHJL} the ``restriction" map $\mathsf{Res}$ is Poisson.

Since $f\,:\,Y_0\,\longrightarrow\, X-B$ is an unramified covering map, $\pi_1(Y_0)$ is isomorphic to a
subgroup of $\pi_1(X-B)$. Therefore, there
is a map $\mathsf{Ind}\,:\,\mathcal{M}_r(Y_0)\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ given by
$\mathsf{Ind}([h])\,=\,[\mathsf{ind}(h)]$, where $\mathsf{ind}(h)\,:\,\pi_1(X-B)\,\longrightarrow\, G_{dr}$
is the representation induced by $h\,:\,\pi_1(Y_0)\,\longrightarrow\, G_r$. Since $f\,:\,Y_0\,\longrightarrow\,
X-B$ is a $d$--sheeted cover, the index $[\pi_1(X-B):\,\pi_1(Y_0)]\,=\,d$ is finite and
$\pi_1(X-B)/\pi_1(Y_0)\,=\, \{\gamma_1\pi_1(Y_0),\, \cdots,\,\gamma_d\pi_1(Y_0)\}$. And
so $\mathsf{ind}(h)$ is defined
by the natural action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\C^r\,\cong\,\C^{dr}$ by $\gamma\cdot
\sum_{i=1}^d\gamma_iv_i\,:=\,\sum_{i=1}^d\gamma_{j_i}\pi(\delta_i)v_i$, where
$\gamma\gamma_i\,=\,\gamma_{j_i}\delta_i$ with $\delta_i\,\in\,\pi_1(Y_0)$. Thus, $\mathsf{ind}(h)$ takes
values in $\mathsf{Sym}_d\times G_r$ inside $G_{dr}$ where $\mathsf{Sym}_d$ is the symmetric group on $d$ symbols
(realized by permutation matrices).

By construction of the direct image sheaf, we have $\mathsf{Frob}\,=\,\mathsf{Ind}\circ \mathsf{Res}$ (see
Section \ref{sec:directimage} for details). Consequently, $\mathsf{Frob}$ is algebraic. By the Mackey
Formula \cite{Mackey} applied to this context, we see that $\mathsf{Ind}$ is injective. Therefore,
$\mathsf{Frob}$ is an embedding. In Section \ref{sec3}, we show that $\mathsf{Ind}$ is Poisson. Having
both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem
\ref{thm:main}.

Let $\text{Hom}(\Delta,\, \text{GL}(r,{\mathbb C}))^{\rm ir}\, \subset\, \text{Hom}(\Delta,\, \text{GL}
(r,{\mathbb C}))$ be the space of irreducible representations of $\Delta$ in $\text{GL}(r,{\mathbb C})$.
The conjugation action of $\text{GL}(r,{\mathbb C})$ on itself produces an action of $\text{GL}
(r,{\mathbb C})$ on $\text{Hom}(\Delta,\, \text{GL}(r,{\mathbb C}))^{\rm ir}$. The corresponding quotient
\begin{equation}\label{e12}
{\mathcal C}_r(Y_0)\ :=\ \text{Hom}(\Delta,\, \text{GL}(r,{\mathbb C}))^{\rm ir}/\text{GL}(r,{\mathbb C})
\end{equation}
is a complex manifold equipped with a holomorphic Poisson structure \cite{La}, \cite{BJ}, \cite{Go}.
Similarly, $\text{Hom}(\Gamma,\, \text{GL}(rd,{\mathbb C}))^{\rm ir}\, \subset\, \text{Hom}(\Gamma,\, \text{GL}
(rd,{\mathbb C}))$ is the space of irreducible representations of $\Gamma$ in $\text{GL}(rd,{\mathbb C})$.
As before,
\begin{equation}\label{e13}
{\mathcal C}_{rd}(X_0)\ :=\ \text{Hom}(\Gamma,\, \text{GL}(rd,{\mathbb C}))^{\rm ir}/\text{GL}(rd,{\mathbb C})
\end{equation}
is the quotient complex manifold equipped with a holomorphic Poisson structure. Sending
any $(V,\, \nabla)\, \in\, {\mathcal C}_r(Y_0)$ to its direct image $(\phi_* V,\, \phi_*\nabla)\, \in\,
{\mathcal C}_{rd}(X_0)$ (see \eqref{e12} and \eqref{e13}) we obtain a map
\begin{equation}\label{e14}
\Psi\ :\ {\mathcal C}_r(Y_0)\ \longrightarrow\ {\mathcal C}_{rd}(X_0).
\end{equation}

\begin{proof}
The holomorphic tangent (respectively, cotangent) bundle of ${\mathcal C}_r(Y_0)$ will be
denoted by $T{\mathcal C}_r(Y_0)$ (respectively, $T^*{\mathcal C}_r(Y_0)$). The Poisson structure
on ${\mathcal C}_r(Y_0)$ is given by a holomorphic homomorphism
\begin{equation}\label{e15}
P\ :\ T^*{\mathcal C}_r(Y_0) \ \longrightarrow\ T{\mathcal C}_r(Y_0)
\end{equation}
which is skew-symmetric and the Poisson bracket defined by $P$ satisfies the Jacobi identity. Similarly,
\begin{equation}\label{e16}
Q\ :\ T^*{\mathcal C}_{rd}(X_0) \ \longrightarrow\ T{\mathcal C}_{rd}(X_0)
\end{equation}
is the Poisson structure on ${\mathcal C}_{rd}(X_0)$. The map $\Psi$ in \eqref{e14} is said to be
compatible with the Poisson structures on ${\mathcal C}_r(Y_0)$ and ${\mathcal C}_{rd}(X_0)$ if the
following diagram of homomorphisms of vector bundles over ${\mathcal C}_r(Y_0)$ is commutative:
\begin{equation}\label{e17}
\begin{matrix}
T^*{\mathcal C}_r(Y_0) & \xrightarrow{\,\,\, P\,\,\,} & T{\mathcal C}_r(Y_0)\\
\, \,\,\,\,\, \Big\uparrow (d\Psi)^* && \,\,\,\, \Big\downarrow d\Psi\\
\Psi^*T^*{\mathcal C}_{rd}(X_0) & \xrightarrow{\,\,\, \Psi^*Q\,\,\,} & \Psi^* T{\mathcal C}_{rd}(X_0)
\end{matrix}
\end{equation}
where $P$ (respectively, $Q$) is constructed in \eqref{e15} (respectively, \eqref{e16})
and $d\Psi$ is the differential of the map $\Psi$ in \eqref{e14} while $(d\Psi)^*$ is its dual.

\begin{theorem}\label{thm:main}
 $\mathsf{Frob}$ is a Poisson embedding.
\end{theorem}

Let $Y_0\,=\,Y-f^{-1}(B)$ and $\mathcal{M}_r(Y_0)\,:=\,\mathrm{Hom}(\pi_1(Y_0),\,G_r)/\!\!/G_r$. Again,
$\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\,\subset\, Y$ there is a
natural map $\iota\,:\,\pi_1(Y_0)\,\longrightarrow\, \pi_1(Y)$. The map $\iota$ is surjective (and hence has a
right inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\dim f^{-1}(B)\,=\,0$.

Let $\mathsf{Res}\,:\,\mathcal{M}_r(Y)\,\longrightarrow\, \mathcal{M}_r(Y_0)$ be given by $\mathsf{Res}([h])
\,=\,[h\circ \iota]$, where $h\,:\,\pi_1(Y)\,\longrightarrow\, G_r$. If
$\mathsf{Res}([h_1])\,=\,\mathsf{Res}([h_1])$, then
$[h_1\circ\iota]\,=\,[h_2\circ\iota]$ and so $[h_1]=[h_2]$ since $\iota$ has a right inverse. Therefore,
$\mathsf{Res}$ is an embedding. The inclusion map $Y_0\,\subset\, Y$ preserves transversality of based loops
and thus by \cite[Theorem A]{BHJL} the ``restriction" map $\mathsf{Res}$ is Poisson.

Since $f\,:\,Y_0\,\longrightarrow\, X-B$ is an unramified covering map, $\pi_1(Y_0)$ is isomorphic to a
subgroup of $\pi_1(X-B)$. Therefore, there
is a map $\mathsf{Ind}\,:\,\mathcal{M}_r(Y_0)\,\longrightarrow\, \mathcal{M}_{dr}(X-B)$ given by
$\mathsf{Ind}([h])\,=\,[\mathsf{ind}(h)]$, where $\mathsf{ind}(h)\,:\,\pi_1(X-B)\,\longrightarrow\, G_{dr}$
is the representation induced by $h\,:\,\pi_1(Y_0)\,\longrightarrow\, G_r$. Since $f\,:\,Y_0\,\longrightarrow\,
X-B$ is a $d$--sheeted cover, the index $[\pi_1(X-B):\,\pi_1(Y_0)]\,=\,d$ is finite and
$\pi_1(X-B)/\pi_1(Y_0)\,=\, \{\gamma_1\pi_1(Y_0),\, \cdots,\,\gamma_d\pi_1(Y_0)\}$. And
so $\mathsf{ind}(h)$ is defined
by the natural action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\mathbb{C}^r\,\cong\,\mathbb{C}^{dr}$ by $\gamma\cdot
\sum_{i=1}^d\gamma_iv_i\,:=\,\sum_{i=1}^d\gamma_{j_i}\pi(\delta_i)v_i$, where
$\gamma\gamma_i\,=\,\gamma_{j_i}\delta_i$ with $\delta_i\,\in\,\pi_1(Y_0)$. Thus, $\mathsf{ind}(h)$ takes
values in $\mathsf{Sym}_d\times G_r$ inside $G_{dr}$ where $\mathsf{Sym}_d$ is the symmetric group on $d$ symbols
(realized by permutation matrices).

By construction of the direct image sheaf, we have $\mathsf{Frob}\,=\,\mathsf{Ind}\circ \mathsf{Res}$ (see
Section \ref{sec:directimage} for details). Consequently, $\mathsf{Frob}$ is algebraic. By the Mackey
Formula \cite{Mackey} applied to this context, we see that $\mathsf{Ind}$ is injective. Therefore,
$\mathsf{Frob}$ is an embedding. In Section \ref{sec3}, we show that $\mathsf{Ind}$ is Poisson. Having
both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem
\ref{thm:main}.

Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\mathcal{M}_r(Y_0)=\mathrm{Hom}(\pi_1(Y_0),G_r)/\!\!/G_r$. Since $Y_0\subset Y$ there is a natural surjection $\iota:\pi_1(Y_0)\to\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\dim f^{-1}(B)=0$), hence $\iota$ has a right inverse.

Define the “restriction” map $\mathsf{Res}:\mathcal{M}_r(Y)\to\mathcal{M}_r(Y_0)$ by $\mathsf{Res}([h])=[h\circ\iota]$. If $\mathsf{Res}([h_1])=\mathsf{Res}([h_2])$ then $[h_1\circ\iota]=[h_2\circ\iota]$, and since $\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\mathsf{Res}$ is an embedding. The inclusion $Y_0\subset Y$ “preserves transversality of based loops” and hence by \cite[Theorem A]{BHJL} $\mathsf{Res}$ is Poisson.

Because $f:Y_0\to X-B$ is an unramified $d$--sheeted cover, $\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\pi_1(X-B)$. Define an induction map $\mathsf{Ind}:\mathcal{M}_r(Y_0)\to\mathcal{M}_{dr}(X-B)$ by $\mathsf{Ind}([h])=[\mathsf{ind}(h)]$, where $\mathsf{ind}(h):\pi_1(X-B)\to G_{dr}$ is the induced representation via the action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\mathbb{C}^r\cong\mathbb{C}^{dr}$; thus $\mathsf{ind}(h)$ lands in $\mathsf{Sym}_d\times G_r\subset G_{dr}$. By construction of the direct image sheaf, $\mathsf{Frob}=\mathsf{Ind}\circ\mathsf{Res}$.

Consequently $\mathsf{Frob}$ is algebraic. By the Mackey Formula \cite{Mackey}, $\mathsf{Ind}$ is injective, hence $\mathsf{Frob}$ is an embedding. In Section \ref{sec3} it is shown that $\mathsf{Ind}$ is Poisson. With both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, this “outlines the proof of Theorem~\ref{thm:main},” i.e. that $\mathsf{Frob}$ is a Poisson embedding.

Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\mathcal{M}_r(Y_0)=\mathrm{Hom}(\pi_1(Y_0),G_r)/\!\!/G_r$. Since $Y_0\subset Y$ there is a natural surjection $\iota:\pi_1(Y_0)\to\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\dim f^{-1}(B)=0$), hence $\iota$ has a right inverse.

Define the “restriction” map $\mathsf{Res}:\mathcal{M}_r(Y)\to\mathcal{M}_r(Y_0)$ by $\mathsf{Res}([h])=[h\circ\iota]$. If $\mathsf{Res}([h_1])=\mathsf{Res}([h_2])$ then $[h_1\circ\iota]=[h_2\circ\iota]$, and since $\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\mathsf{Res}$ is an embedding. The inclusion $Y_0\subset Y$ “preserves transversality of based loops” and hence by \cite[Theorem A]{BHJL} $\mathsf{Res}$ is Poisson.

Because $f:Y_0\to X-B$ is an unramified $d$--sheeted cover, $\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\pi_1(X-B)$. Define an induction map $\mathsf{Ind}:\mathcal{M}_r(Y_0)\to\mathcal{M}_{dr}(X-B)$ by $\mathsf{Ind}([h])=[\mathsf{ind}(h)]$, where $\mathsf{ind}(h):\pi_1(X-B)\to G_{dr}$ is the induced representation via the action of $\pi_1(X-B)$ on $\oplus_{i=1}^d\gamma_i\mathbb{C}^r\cong\mathbb{C}^{dr}$; thus $\mathsf{ind}(h)$ lands in $\mathsf{Sym}_d\times G_r\subset G_{dr}$. By construction of the direct image sheaf, $\mathsf{Frob}=\mathsf{Ind}\circ\mathsf{Res}$.

Consequently $\mathsf{Frob}$ is algebraic. By the Mackey Formula \cite{Mackey}, $\mathsf{Ind}$ is injective, hence $\mathsf{Frob}$ is an embedding. In Section \ref{sec3} it is shown that $\mathsf{Ind}$ is Poisson. With both $\mathsf{Res}$ and $\mathsf{Ind}$ injective and Poisson, this outlines the proof of the main theorem, i.e. that $\mathsf{Frob}$ is a Poisson embedding.

\label{thm:main}
 $\mathsf{Frob}$ is a Poisson embedding.,

Question:
Let $X$ and $Y$ be closed oriented surfaces, and let $f:Y\to X$ be a degree $d$ branched covering with branch locus $B\subset X$. For $m\ge 1$, write $G_m=\mathrm{GL}(m,\mathbb C)$ and define the character varieties
\[\mathcal M_r(Y):=\mathrm{Hom}(\pi_1(Y),G_r)/\!/G_r,\qquad \mathcal M_{dr}(X\setminus B):=\mathrm{Hom}(\pi_1(X\setminus B),G_{dr})/\!/G_{dr}.\]
If $(V,\nabla)$ is a rank-$r$ flat vector bundle on $Y$, its holonomy class gives a point of $\mathcal M_r(Y)$, and the direct image flat bundle $(f_*V,f_*\nabla)$ on $X\setminus B$ gives a point of $\mathcal M_{dr}(X\setminus B)$. Let
\[\mathsf{Frob}:\mathcal M_r(Y)\longrightarrow \mathcal M_{dr}(X\setminus B)\]
be the resulting map on character varieties, called the Frobenius reciprocity map. Which statement holds for this map, with respect to the natural holomorphic Poisson structures on these character varieties?

The map $\mathsf{Frob}$ is a Poisson embedding; that is, it is an embedding of $\mathcal M_r(Y)$ into $\mathcal M_{dr}(X\setminus B)$ and it is compatible with the natural holomorphic Poisson structures.

Choices:
B. The map $\mathsf{Frob}$ is a Poisson immersion on the smooth locus of $\mathcal M_r(Y)$, but it need not be injective as a map into $\mathcal M_{dr}(X\setminus B)$.
C. The map $\mathsf{Frob}$ is Poisson; that is, it is compatible with the natural holomorphic Poisson structures on $\mathcal M_r(Y)$ and $\mathcal M_{dr}(X\setminus B)$.
D. The map $\mathsf{Frob}$ is an embedding of $\mathcal M_r(Y)$ into $\mathcal M_{dr}(X\setminus B)$, but in general it is only algebraic and does not preserve the natural holomorphic Poisson structures.
E. The map $\mathsf{Frob}$ is a holomorphic symplectic embedding onto its image; equivalently, its image is a symplectic submanifold of $\mathcal M_{dr}(X\setminus B)$ and $\mathsf{Frob}$ preserves the symplectic forms.

Poisson target is not symplectic in general