Let $\mathbb{G}$ be a connected reductive group over $\mathbb{F}_q$. Let $R_{T}^{\theta}$ be a Deligne--Lusztig representation of $\mathbb{G}(\mathbb{F}_q)$. Assume that $R_{T}^{\theta}$ is cuspidal (i.e. at least one of its irreducible constituents does not appear in any parabolically induced representation).

Note that $R_{T}^{\theta}$ itself is not necessarily irreducible; e.g. some $\mathrm{dim}=(q-1)/2$ irreps of $\mathrm{SL}_2(\mathbb{F}_q)$ are the nontrivial subrepresentations of such a representation. This question is about the decomposition of such an $R_{T}^{\theta}$.

While in the parabolic induction case I learned that one can use Howlett--Lehrer's works to get many information about the decomposition (there is a whole chapter in Carter's book *Finite Groups of Lie Type* devoted to it), it seems there are less discussions in the cuspidal case.

**Question(s):** To what extent do we understand the irreducible constituents of the $R_{T}^{\theta}$'s (e.g. their multiplicies and dimensions)? Can we label them in some good way?

In the Coxeter torus case, if I understood it correctly, a solution was given in Lusztig's Coxeter orbit paper, where the labeling is by the eigenvalues of the Frobenius action.