Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Question

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.

I'll start with a somewhat vague question and make my question more specific further down:

How do cuspidal representations fit into Deligne-Lusztig characters $R_{T,\theta}$?

A little about what is known classically: if $\rho$ is a cuspidal representation of $G^F$, then $\rho$ is a subrepresentation of $\pm R_{T,\theta}$ for some pair $(T,\theta)$, where $T$ is a minisotropic torus (so it is not a subtorus of any proper Levi subgroup of $G$). Moreover, for such a $T$, if $\theta: T^F \to \mathbb{C}^\times$ is in general position (so not fixed by any nontrivial element of the Weyl group), then $\pm R_{T,\theta}$ is irreducible cuspidal.

In the case of $G = GL_n$, it turns out that these are all of the irreducible cuspidal representations; i.e. every irreducible cuspidal representation is isomorphic to some $\pm R_{T,\theta}$, where $T$ is ministropic, and $R_{T,\theta} \cong R_{T',\theta'}$ if and only if $(T,\theta)$ is $G^F$-conjugate to $(T',\theta')$.

Of course, it's too much to hope that this should happen in every case. Let $T$ be an anisotropic torus in $SL_2$. Then there is a character $\theta: T^F \to \mathbb{C}^\times$ that is not in general position, where $\pm R_{T,\theta}$ splits as a sum of two irreducible cuspidal representations (perhaps you need the base field to have odd characteristic).

Now to specify my question. Throughout, $T$ is a minisotropic torus of $G$, and $\theta:T^F \to \mathbb{C}^\times$.

1). Are there groups $G$ with pairs $(T,\theta)$ and cuspidal representations $\rho,\, \rho'$ such that $\rho$ occurs in $R_{T,\theta}$ with positive multiplicity, and $\rho'$ occurs with negative multiplicity?

2). Are there groups $G$ with pairs $(T,\theta)$ such that $R_{T,\theta}$ such that $R_{T,\theta}$ contains both cuspidal and non-cuspidal representations with nonzero multiplicity?

Answer 1

EDIT: It's easy to answer Question 2 affirmatively by pointing to the groups $G=\mathrm{Sp}(4,q)$ of Lie type $B_2= C_2$ in odd characteristic. Bhama Srinivasan first worked out the ordinary irreducible characters of $G$ using ad hoc methods in her thesis work at Manchester: the resulting paper is here and can be downloaded freely from the AMS site. She was advised by J.A. Green, whose 1955 paper on characters of finite general linear groups was the inspiration for this approach. After the landmark Annals of Mathematics (1976) paper by Deligne and Lusztig, their generalized characters $R_T^\theta$ could be compared with her work (and then reduced modulo $p$ by Jantzen in his 1987 exposition for the Arcata Conference on Finite Groups). Some of this comparison is not formally published but was written up in her own notes.

A striking feature of the family of groups of Lie type is the existence of just one cuspidal "unipotent" character (in the later terminology): this was called $\theta_{10}$ by Srinivasan. She later computed (unpublished) the explicit decomposition of various unipotent characters $R_T^\theta$ (where $\theta$ is the trivial character but $T$ ranges over all types of maximal tori). This shows that $\theta_{10}$ occurs with coefficient $\pm 1, \pm 2$ in two cases where $T$ is anisotropic, along with other non-cuspidal irreducible unipotent characters.

The answer to Question 1 is also yes, but it's most conveniently seen in type $G_2$ (again using unipotent characters). Following Srinivasan's work, but still prior to 1976, B. Chang and R. Ree (1974) worked out the ordinary irreducible characters of these groups (again assuming odd characteristic) in the spirit of Macdonald's conjectures which were later proved by Deligne-Lusztig. The four irreducible characters denoted by $X_{17}, X_{18}, X_{19}, X'_{19}$ in the paper by Chang and Ree were later checked by Lusztig to be precisely the irreducible cuspidal unipotent characters.

If I'm reading the computations correctly in the 1984 Bonn Diplomarbeit by D. Mertens (student of Jantzen, one decomposition of a unipotent Deligne-Lusztig (generalized) character involves six irreducible characters including $X_{18}$ and $X_{19}+X'_{19}$ with opposite signs (plus three other non-cuspidal unipotent characters with coefficients $\pm 1$). It seems likely that this kind of mixed decomposition will occur frequently as the rank increases, though it's unclear whether it has significance.

An obstacle to seeing this much detail is the sheer complexity of writing down explicit decompositions of Deligne-Lusztig characters as $\mathbb{Z}$-linear combinations of ordinary irreducible characters. The papers of Lusztig (including his 1984 monograph) and the 1985 book by Roger Carter contain the machinery needed for this purpose, but the case-by-case computations are nontrivial. Most of the published work of Lusztig and others has emphasized more the theoretical algorithms involved. But the rank 2 groups do serve as a useful laboratory for experimental work.

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A couple of notes about terminology: The reference to a "Frobenius element" $F$ of $G_k$ is out of focus, since the finite group consists of fixed points of a Frobenius morphism $F$ of the ambient algebraic group (such as raising all matrix entries to the $q$th power for $q$ a power of $p$).

Concerning "minisotropic" tori (terminology, like "cuspidal", which I guess comes from the original Harish-Chandra program for representations over real and $p$-adic fields), these are relevant only in reductive groups which are not semisimple---such as general linear groups. Otherwise one can just refer to "anisotropic" tori over $k$.