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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.

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    $\begingroup$ Since Lusztig's treatment of the special "Coxeter orbit" case, he and others have filled in all essential details for the general case. But it isn't easy to summarize, and so far there seems to be no comprehensive treatment. It may be helpful to look at Geck's survey front.math.ucdavis.edu/1705.05083 and his references. $\endgroup$ – Jim Humphreys Aug 16 '17 at 17:18

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