$$0=V_0\subset V_1\subset \cdots V_{k-1}\subset V_k=V(\xi)$$ such that $$V_i/V_{i-1}\cong \tau_{p_i}^*D(m, n_i),\ \ (p_i,n_i)\in {\mathbb Z}_+\times {\mathbb Z}_+,\ \ 1\leq i\leq k$$ if and only if $m\geq \xi_1$.

These types of filtrations are called excellent filtrations [@Jo03; @J06] but are also known in the literature as level $m$–Demazure flags [@BCSV15; @CSSW14]. The aim of the present paper is to give a combinatorial formula for the graded multiplicities in excellent filtrations.

\[rem23\] We discuss two further specializations of the partition $\xi$.

1.  The specialization $\xi_i=1$, $1\leq i\leq \ell$ leads to a representation which is isomorphic to the local Weyl module $W_{\text{loc}}(|\xi|)$ which is the module generated by an element $w_{|\xi|}$ subject to the relations . The interest in local Weyl modules is its connections with quantum affine algebras [@CP01], the theory of Macdonald polynomials [@I03], $q$–Whittaker functions [@BF14a], $Q$–systems [@CV13; @KV14] and more recently with hypergeometric series [@BCK16].

2.  Let $N\in {\mathbb N}$. The truncated Weyl module $W_{\text{loc}}(N,n)$ is a quotient of $W_{\text{loc}}(n)$ by the additional relation $(\mathfrak{sl}_2\otimes t^N{\mathbb C}[t])w_{n}=0$. The special choice $\xi=((d+1)^j,d^{N-j})$ yields an isomorphism $V(\xi)\cong W_{\text{loc}}(N,n)$, where $d\in {\mathbb Z}_+$ and $j<N$ are such that $n=dN+j$ (see [@KL15 Theorem 4.3]).

We will secretly assume in the rest of the paper 