Representations of $\text{SL}_n(q)$ of degree $q^{O(n)}$

Question

Every representation of $A_n$ of degree $n^{O(1)}$ is contained in a $O(1)$ tensor power of the defining permutation representation. Is there an analogous result for classical groups of Lie type, say $G = \text{SL}_n(q)$ ($q$ bounded, $n$ large)? The defining action of $G$ on $F_q^n$ gives rise to a complex (permutation) representation $V$ of degree $q^n-1$, and the dual action likewise a representation $W$ also of degree $q^n-1$. Is it true that every representation of degree $q^{O(n)}$ is contained in $V^{\otimes O(1)} \otimes W^{\otimes O(1)}$?

Answer 1

A complete answer to my question is available from recent work on "character level" of Guralnick, Larsen, and Tiep: see [1] for linear and unitary groups and [2] for symplectic and orthogonal groups.

First of all, a correction to my question. There is no need to include the representation $W$, because it is isomorphic to $V$. Indeed, both have character $g\mapsto q^{\dim \ker (g-1)}$ (the permutation representations are not permutation-isomorphic, however).

GLT actually define (in the connected cases, $\mathrm{SL}_n(q)$, $\mathrm{SU}_n(q)$, $\mathrm{Sp}_n(q)$, $\Omega_n^{(\pm)}(q)$, for simplicity) the level of an irreducible character $\chi$ to be the smallest $\ell$ such that $\chi$ is contained in $\psi^\ell$, where $\psi$ is

• the permutation character $\tau : g \mapsto q^{\dim \ker(g-1)}$ in the linear and orthogonal cases,
• the reducible Weil character $\zeta : g \mapsto (-1)^n (-q)^{\dim \ker(g-1)}$ in the unitary case (note that $\zeta^2 = (q^2)^{\dim \ker (g-1)}$ is the permutation character in this case),
• $\tau + \zeta$ in the symplectic even-characteristic case,
• $\omega + \omega^*$ in the symplectic odd-characteristic case, where $\omega$ and $\omega^*$ are the reducible Weil characters of $\mathrm{Sp}_n(q)$ (such that $\{\tau, \zeta\} = \{\omega^2, \omega\omega^*\}$).

For this notion of level, GLT prove exactly what I asked for.

(Closer to the spirit of my question would be, say, "permutation level" $\ell_\text{perm}$, which would be as above with just $\psi = \tau$ in all cases. I don't know whether this is equivalent, i.e., whether $\zeta,\omega,\omega^*$ are contained in $\tau^{O(1)}$.)

[1] Guralnick, Robert M.; Larsen, Michael; Tiep, Pham Huu, Character levels and character bounds, ZBL07158138.

[2] Guralnick, Robert M.; Larsen, Michael; Tiep, Pham Huu, Character levels and character bounds. II.