such that $$\begin{gathered}
0\leq m_1\leq m_2\leq\ldots\leq m_n \, .\end{gathered}$$ An irreducible representation of the lens algebra ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$ is given on generators by $$\begin{aligned}
x_i{\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} &=
               q^{2\sum_{j=0}^{i-1}r_j}q^{2\sum_{j=1}^ip_{j-1}(m_j-m_{j-1})}
               \big(1-q^{2r_i}q^{2p_i(m_{i+1}-m_i)}\big){\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} \;, \\[2pt]
\zeta_i{\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} &=
               q^{p_i\sum_{j=0}^{i-1}r_j}q^{\sum_{j=1}^ip_{j-1}(m_j-m_{j-1})}
               \sqrt{ {\genfrac{ \{ }{ \} }{0pt}{}{ p_i(m_{i+1}-m_i+1) }{ p_i(m_{i+1}-m_i) }_{\!q}} }{\left|\smash[t]{\vec{m}+\vec{e}_{in};{\boldsymbol}{r}}\right>} \;, \\[-3pt]
\intertext{for all $i=0,\ldots,n-1$ and\vspace{-3pt}}
x_n{\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} &=
               q^{2\sum_{j=0}^{n-1}r_j}q^{2\sum_{j=1}^np_{j-1}(m_j-m_{j-1})}
               {\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} \;, \\[2pt]
\zeta_n{\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} &=
               q^{p_n\sum_{j=0}^{n-1}r_j}q^{\sum_{j=1}^np_{j-1}(m_j-m_{j-1})}
               {\left|\smash[t]{\vec{m};{\boldsymbol}{r}}\right>} \;,\end{aligned}$$ where $\vec{e}_{in}:=(\, \stackrel{i\;{\mathrm}{times}}{\overbrace{0,0,\ldots,0}}\,,\stackrel{n-i\;{\mathrm}{times}}{\overbrace{1,1,\ldots,1}}\, )$, for all $0\leq i<n$, with $m_0:=0$.

Fredholm modules {#sec:Khom}
================

Here we present some basic Fredholm modules for the algebra ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ constructed using faithful representations. Additional Fredholm modules can then be obtained by iterated pullbacks from ‘lower dimensions’. Indeed, the epimorphism ${\mathcal{O}}(S^{2n+1}_q)\twoheadrightarrow{\mathcal{O}}(S^{2n-1}_q)$ given by $z_n\mapsto 0$ induces an epimorphism ${\mathcal{O}}({\mathbb{P}}_q(\ell_0,\ldots,\ell_n))\twoheadrightarrow{\mathcal{O}}({\mathbb{P}}_q(\ell_0,\ldots,\ell_{n-1}))$.

The building block representations are the ones described in §\[sec:irreps\]. As done there, we assume that the hypothesis of Theorem \[thm:13\] are satisfied: that is ${\underline}{\ell}={\underline}{p}^{\,\sharp}$ with ${\underline}{p}$ pairwise coprime. The next definition is the analogue of [@DL10 Def. 1]. Through the whole section, we assume we fixed a sequence of integers ${\boldsymbol}{r}=(r_0,\ldots,r_{n-1})$ satisfying $$\label{eq:ri}
0\leq r_i<p_i \;,\qquad \textup{for} \quad i=0,\ldots,n-1 \, .$$

\[def:15\] Let ${\mathcal{H}}_n:=\ell^2({\mathbb{N}}^n)$, with orthonormal basis ${\left|\smash[t]{\vec{m}}\right>}$, $\vec{m}=(m_1,\ldots,m_n)\in{\mathbb{N}}^n$. For $0\leq k\leq n$ let ${\mathcal}{V}^n_k\subset{\mathcal{H}}_n$ be the linear span of basis vectors ${\left|\smash[t]{\vec{m}}\right>}$ satisfying 