# “Nice” basis for highest-weight irreducible module of a simple Lie algebra

Let $\mathfrak{g}$ be a simple complex Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $\Phi \subset \mathfrak{h}^*$ the associated root system, $\Sigma = \{\sigma_i : i\in I\}$ a basis of simple roots. Further, let $\lambda$ be a dominant weight (maybe even a fundamental weight if this simplifies anything), and let $V_\lambda$ be the irreducible $\mathfrak{g}$-module with heighest weight $\lambda$.

Vaguely stated, my question is:

Is there a "nice" basis for $V_\lambda$, which makes it algorithmically convenient to deal with?

Now I have to admit that I don't know exactly what I want of that basis, except at the very least that it has to be compatible with the weight decomposition (i.e., partitioned into sets of equal weight), that the action of $\mathfrak{g}$ on $V$ should be convenient to write (when $\mathfrak{g}$ itself is given a "nice" basis, see below), and that the symmetry of the Weyl group should be "as apparent as possible".

The standard construction of $V_\lambda$ as a quotient of the corresponding Verma module does not seem to give a nice basis (the Verma module itself is reasonably transparent, but when it comes to finding a basis for the quotient, while this is computable, I can't see how to do it in a nicely symmetric way).

However, for some particular $\lambda$, a nice basis is known:

If $\lambda$ is the highest root, corresponding to $V_\lambda$ being the adjoint representation, then this note by Meinolf Geck describes a basis for $V_\lambda = \mathfrak{g}$ which is due to Lusztig and earlier works of Chevalley and Tits (I don't fully understand how these contributions relate, but that doesn't really matter here). A Chevalley basis consists of elements $h_i$ for $i\in I$ (all having weight $0$) and $e_\alpha$ for $\alpha\in\Phi$ (having the corresponding weights). The adjoint action is given by $[h_i,h_j] = 0$ and $[h_i,e_\alpha] = (\alpha,\sigma_i^\vee)\, e_\alpha$ and $[e_{-\alpha},e_\alpha] = \sum_{i\in I} c_i h_i$ where $\alpha^\vee = \sum_{i\in I} c_i \sigma_i^\vee$ and lastly $[e_\alpha,e_\beta] = \pm (m+1) e_{\alpha+\beta}$ if $\alpha+\beta\in\Phi$ where $m = \max\{p\in\mathbb{N}: \beta-p\alpha\in\Phi\}$ — for some choice of signs (Geck explains how it can be made canonical). Further, the symmetry of the Weyl group is made apparent by the fact that for each $i\in I$ there is an automorphism taking $h_j$ to $h_j - (\sigma_i,\sigma_j^\vee)\, h_i$ and $e_\alpha$ to $\pm e_{s_i(\alpha)}$ (where $s_i(\alpha)$ is $\alpha-(\alpha,\sigma_i^\vee)\, \sigma_i$) — again for some choice of signs —, and these generate an extension of the Weyl group by an elementary abelian $2$-group.

In a related note, the same author provides a basis and construction for $V_\lambda$ when $\lambda$ is minuscule, which is very similar but even simpler.

So this suggests the question of what can be said for arbitrary $\lambda$. I suppose that nothing quite so canonical or explicit is known, but maybe if we lower the bar to "nice and algorithmically convenient", something can be said? (Certainly multiple weights are problematic; but after all, tensor powers of representations with a "nice" basis themselves have a "nice" basis, so not all hope is lost.)

I am not an expert, but it seems that Geck's constructions specialize the Lusztig-Kashiwara "canonical" or "crystal" bases of highest weight modules. These also have a Littelmann "path model", related to the earlier Hodge-Young "standard monomial" bases (for SL$_n$).