while $\mathcal{V}^n_{k-1}\cap\mathcal{V}^n_k$, for $1\leq k\leq n$, is the span of vectors ${\left|\smash[t]{\vec{m}}\right>}$ satisfying: $$\label{eq:capconstr}
0\leq m_1\leq m_2\leq\ldots\leq m_k\;, \qquad\quad m_k>m_{k+1}>\ldots>m_n\geq 0 \;.$$

As a consequence $\pi^{(n)}_j(a)\pi^{(n)}_k(b)=0$ for all $a,b\in{\mathcal{O}}(L_q(p;{\underline}{\ell}))$, if $|j-k|>1$, and the maps $\pi_\pm^{(n)}:{\mathcal{O}}(L_q(p;{\underline}{\ell}))\to{\mathcal{B}}({\mathcal{H}}_n)$ defined by $$\label{irreps+-}
\pi_+^{(n)}(a):=\sum_{\substack{0\leq k\leq n \\[1pt] k\;{\mathrm}{even}}}\pi^{(n)}_k(a) \;,\qquad
\pi_-^{(n)}(a):=\sum_{\substack{0\leq k\leq n \\[1pt] k\;{\mathrm}{odd}}}\pi^{(n)}_k(a) \;,$$ are representations of the algebra ${\mathcal{O}}(L_q(p;{\underline}{\ell}))$. We then generalize [@DL10 Prop. 3].

\[prop\] For all $a\in{\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$, the operator $\pi_+^{(n)}(a)-\pi_-^{(n)}(a)$ is of trace class on ${\mathcal{H}}_n$; furthermore, the trace is given by a series which — as a function of $q$ — is absolutely convergent in the open interval $0<q<1$.

The space ${\mathcal{H}}_n$ is the orthogonal direct sum of ${\mathcal}{V}^n_{k-1}\cap{\mathcal}{V}^n_k$, for all $1\leq k\leq n$, plus the joint kernel of all the representations involved. From Lemma \[le:pl\], on ${\mathcal}{V}^n_{k-1}\cap{\mathcal}{V}^n_k$ only the representations $\pi^{(n)}_{k-1}$ and $\pi^{(n)}_k$ are different from zero (accordingly to the parity of $k$ one contributes to $\pi_+^{(n)}$ and the other to $\pi_-^{(n)}$). It then suffices to prove that $\pi^{(n)}_{k-1}(a)-\pi^{(n)}_k(a)$ is of trace class, and that the trace is absolutely convergent for any $0<q<1$. Moreover, it is enough to show this for $a=\xi_{i,j}$ a generator of ${\mathcal{O}}({\mathbb{P}}_q({\underline}{\ell}))$ as given in , with $0\leq i\leq j\leq n$. Remember that $\xi_{i,i} =z_i^*z_i$ 