Character sheaves on neutrally solvable groups

Let $G$ be an algebraic group over an algebraically closed field $\mathtt{k}$ of characteristic $p>0$. In this paper we develop the theory of character sheaves on groups $G$ such that their neutral connected components $G^\circ$ are solvable algebraic groups. For such algebraic groups $G$ (which we call neutrally solvable) we will define the set $\operatorname{CS}(G)$ of character sheaves on $G$ as certain special (isomorphism classes of) objects in the category $\mathscr{D}_G(G)$ of $G$-equivariant $\overline{\mathbb{Q}}_\ell$-complexes (where we fix a prime $\ell\neq p$) on $G$. We will describe a partition of the set $\operatorname{CS}(G)$ into finite sets known as $\mathbb{L}$-packets and we will associate a modular category $\mathscr{M}_L$ with each $\mathbb{L}$-packet $L$ of character sheaves using a truncated version of convolution of character sheaves. In the case where $\mathtt{k}=\overline{\mathbb{F}}_q$ and $G$ is equipped with an $\mathbb{F}_q$-Frobenius $F$ we will study the relationship between $F$-stable character sheaves on $G$ and the irreducible characters of (all pure inner forms of) $G^F$. In particular, we will prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the "trace of Frobenius" functions associated with $F$-stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by $F$-stable $\mathbb{L}$-packets. Moreover, we will prove that the block in this transition matrix corresponding to any $F$-stable $\mathbb{L}$-packet $L$ can be described as the crossed S-matrix associated with the auto-equivalence of the modular category $\mathscr{M}_L$ induced by $F$.