We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $\operatorname{SL}_{2}(q)$ for q even over a large enough field of odd characteristics. This article is a continuation of our article Trivial Source Character Tables of $\operatorname{SL}_{2}(q)$, where we considered, in particular, the case in which q is odd in non-defining characteristic.

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

We investigate the tempered representations derived from the principal series of SLℓ(F) and their geometric structure. In particular, we give the parameterization for special representations and prove the tempered part of the Aubert–Baum–Plymen conjecture for the toral cases of SLℓ(F).

In this paper we find the multiplicities $\dim L(\lambda)_{\lambda-\alpha}$ where $\alpha$ is an {\em arbitrary} root and $L(\lambda)$ is an irreducible $SL_n$-module with highest weight $\lambda$. We provide different bases of the corresponding weight spaces and outline some applications to the symmetric groups. In particular we describe certain composition multiplicities in the modular branching rule.

1991 Mathematics Subject Classification: 20C05, 20G05.