Weight multiplicities for some particular representations of SO(2m).

Question

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 $p=2$.

Answer 1

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, where there are also analogous weight multiplicity formulas for the rest of the classical complex Lie algebras.

Answer 2

I suspect that the answer you seek can be deduced from the Kostant Multiplicity Formula. It gives the multiplicity of a weight in an irreducible complex representation of a compact Lie group $G$. One reference would be: A Formula for the Multiplicity of a Weight, by Kostant.