# highest weight the half-sum of positive roots

If I take the irreducible representation of $GL_n$ whose highest weight is the half-sum $\rho$ of positive roots, it has dimension $2^k$ where $k$ is the number of positive roots. Even better the weights are precisely (with multiplicity) $\sum_{\alpha} \pm \alpha$, where the sum is over all positive roots $\alpha$, and each sign can be either $+1$ or $-1$.

Is this true for other groups? Is there more known about these representations, e.g. a nice weight basis indexed by choices of sign for each positive root?

(At least for $GL_n$ there looks to be a similar story if you took the highest weight to be $2\rho, 3\rho$ etc.)

• Is it the observation attributed to Kostant in the first two paragraphs here? mathoverflow.net/questions/14770/… – David Treumann Jul 23 '15 at 22:46
• Oh yes, that's exactly it! Thanks. I found it by accident while checking the computations in a paper I'm refereeing. – tired_referee Jul 23 '15 at 23:09

Weyl's dimension formula (for semisimple groups) shows immediately that the irreducible representation with highest weight $\rho$ has the dimension you state: $2^k$ with $k$ the number of positive roots. Of course, you are starting with the reductive group $GL_n$ rather than its semisimple derived group $SL_n$; but these groups have the same irreducible representations up to multiplication by a power of det, which doesn't change the dimension. You are dealing with a particular weight which doesn't involve det.
Similarly, Weyl's formula makes it easy to find the dimensions for multiples of $\rho$. On the other hand, the subweights for the highest weight $\rho$ are fairly easy to work out in a similar way to what you've done in this special case, though I don't have a specific reference to quote.
ADDED: For other irreducible root systems, the subweights are certainly a little more complicated than in your special case: for example, the weight 0 (not a root) occurs for some types, and when there are two root lengths the computations get more complicated. The essential fact is that for a simple root $\alpha$, the reflection $s_\alpha$ sends $\rho$ to $\rho -\alpha$, and these reflections generate the entire Weyl group. But having two root lengths complicates the computations, as one sees already for type $B_2$ (odd orthogonal case) and especially the exceptional type $G_2$. Still, for the "very regular" highest weight $\rho$ there is a clear-cut way to compute everything case-by-case for simple root systems. I'm not sure whether all cases can be handled simultaneously.