that $m\geq \xi_1$ whenever we talk about level $m$–Demazure flags of $V(\xi)$.

 {#section-2}

The number of times a particular level $m$–Demazure module appears as a quotient in a level $m$–flag is independent of the choice of the flag. We encode these multiplicities in a polynomial $$\label{poly}
\mathcal{V}_{n}^{\xi\rightarrow m}(q):=\sum_{p\ge 0}\ [V(\xi):\tau_p^* D(m,n)] \hspace{0,04cm} q^p,$$ where $[V(\xi):\tau_p^* D(m,n)]=\sharp\{1\leq i\leq k : V_i/V_{i-1}\cong \tau_p^* D(m,n)\}$. It is known that $$\label{initial12}
\mathcal{V}_{s}^{\xi(m',s)\rightarrow m}(q)=1,\quad \mathcal{V}_{n}^{\xi(m,s)\rightarrow m}(q)=\delta_{s,n},\ \ \mathcal{V}_{n}^{\xi\rightarrow m}(q)=0,\ \ \text{if }\  |\xi|-n\notin2{\mathbb Z}_+.$$ Moreover, for $m\geq m'\geq \xi_1$ we have $$\label{mat}\mathcal{V}_{n}^{\xi\rightarrow m}(q)=\sum_{p\geq 0} \mathcal{V}_{p}^{\xi\rightarrow m'}(q)\ \mathcal{V}_{n}^{\xi(m',p)\rightarrow m}(q).$$ For convinience, set $$\label{initial1}
\mathcal{V}_{n}^{\xi\rightarrow m}(q)=0,\  \ {\rm{if}} \ \ |\xi|<0\ \   {\rm{or}}\  n<0.$$ The following lemma will be needed later; for a proof see [@CSSW14 Lemma 3.8].

\[ausc\]Let $\xi=(\xi_1\geq \xi_2\geq \cdots \geq \xi_{\ell})$ a partition and set $\xi'=(\xi_2\geq \xi_3\geq \cdots \geq \xi_{\ell})$. We have $$\mathcal{V}_{n}^{\xi\rightarrow \xi_1}(q)=q^{(|\xi|-n)/2}\hspace{0,04cm} \mathcal{V}_{n-\xi_1}^{\xi'\rightarrow \xi_1}(q).$$

We will freuquently deal with the polynomials when $\xi=\xi(m',s)$, i.e. $V(\xi)$ itself is a level Demazure module of level $m'$. So we abbreviate $$\mathcal{V}_{s,n}^{m'\rightarrow m}(q):=\mathcal{V}_{n}^{\xi(m',s)\rightarrow m}(q)$$ and define the associated generating series by $$\mathcal{A}^{m'\rightarrow m}_n(x,q)=\sum_{s\geq 0}\mathcal{V}_{s,n}^{m'\rightarrow m}(q)\,x^{(s-n)/2},\ \ \ n\ge 0.$$

\[rem21\] The proof of Proposition \[existflag\] uses the short exact sequence of [@CV13 Theorem 5]. It has 