I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$. It is possible that they have been already computed; in this case I will appreciate a reference. Otherwise, How could I compute them?

These *particular* representations are as follows: let $\pi_{k,p}$ be the irreducible representation of $SO(2m)$ with highest weight
$$
\Lambda_{k,p}:=k \varepsilon_1+(\varepsilon_1+\varepsilon_2+...+\varepsilon_p) =
(k+1) \varepsilon_1+\varepsilon_2+...+\varepsilon_p,
$$
for $k\geq0$ and $1\leq p<m$.
Thus, for $\mu=\sum_{j=1}^m a_j\varepsilon_j$ with $a_j\in\mathbb Z$, I want a formula for $$\dim V_\pi(\mu).$$

**Extremal cases:**
If $k=0$, $\pi_{0,p}$ is the $p$-exterior representation $\bigwedge^p(\mathbb C^{2m})$ of the standard representation.
One can check that
$$
\dim V_\pi(\mu) = \binom{m-p+2r}{r}
$$
if $\mu=\sum a_j \varepsilon_j$ with $|a_j|\leq 1$ for all $j$ and $\|\mu\|_1:=\sum |a_j|= p-2r$ for some $r\in\mathbb Z_{\geq0}$, and $\dim V_\pi(\mu) = 0$ otherwise.

If $p=1$, then $\pi_{k,1}$ can be realized as the vector space of harmonic homogeneous polynomials of degree $k+1$ in $m$ variables. One can check that $$ \dim V_\pi(\mu) = \binom{r+m-2}{m-2} $$ if $\mu=\sum a_j \varepsilon_j$ with $\|\mu\|_1=k+1-2r$ for some $r\in\mathbb Z_{\geq0}$, and $\dim V_\pi(\mu) = 0$ otherwise.

**One idea:**
For $k\geq1$ and $2\leq p\leq m$, I have tried by using Steinberg's formula for the decomposition of $\pi_{k,0}\otimes\pi_{0,p}$ and then induction on $k$ and $p$. I failed even for $p=2$.