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 $p=2$. 



Weight multiplicities for some particular representations of SO(2m).
For $\mu$ a weight, let $||\mu||\_1$ denote the one-norm of $\mu$ (the sum of the absolute values of its entries) and let $Z(\mu)$ be the number of zero coordinates of $\mu$. 
Let $k\geq0$ and $1\leq p\leq n$.
Write $r(\mu)=(k+p-||\mu||\_1)/2$.
If $r(\mu)$ is a non-negative integer, then
\begin{align\*}
m\_{\pi\_{\Lambda\_{k,p}}}(\mu)
&= \sum\_{j=1}^{p} (-1)^{j-1} \sum\_{t=0}^{\lfloor\frac{p-j}{2}\rfloor} \binom{n-p+j+2t}{t} \sum\_{\beta=0}^{p-j-2t} 2^{p-j-2t-\beta} \binom{n-Z(\mu)}{\beta} \binom{Z(\mu)}{p-j-2t-\beta} \\
&\quad \sum\_{\alpha=0}^\beta \binom{\beta}{\alpha} \sum\_{i=0}^{j-1} \binom{r(\mu)-i-p+\alpha+t+j+n-2}{n-2},
\end{align\*}
and $m\_{\pi\_{k,p}}(\mu)=0$ otherwise.


This is Theorem 4.1 in [this article](https://arxiv.org/abs/1706.07839v1), where there are also analogous weight multiplicity formulas for the rest of the classical complex Lie algebras. 


