$\overline{{{\mathbb{Q}}}}_\ell$-valued functions on $G^F$ which are constant on the conjugacy classes of $\Gamma$. (We take $\overline{{{\mathbb{Q}}}}_\ell$ instead of ${{\mathbb{C}}}$ since, in the framework of [@DeLu], [@Lintr], class functions on $\Gamma=G(q)$ are constructed whose values are cyclotomic numbers in $\overline{{{\mathbb{Q}}}}_\ell$.) Given $f,f'\in \mbox{CF}
(\Gamma)$, we denote by $\langle f,f' \rangle=|\Gamma|^{-1} \sum_{g \in 
\Gamma} f(g)\overline{f'(g)}$ the standard scalar product of $f,f'$ where the bar denotes a field automorphism which maps roots of unity to their inverses. Let ${{\operatorname{Irr}}}(\Gamma)$ be the set of irreducible characters of $\Gamma$ over $\overline{{{\mathbb{Q}}}}_\ell$; these form an orthonormal basis of $\mbox{CF}(\Gamma)$ with respect to the above scalar product.

Unipotent character sheaves and almost characters {#sec1}
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Let $p$ be a prime and $k=\overline{{{\mathbb{F}}}}_p$ be an algebraic closure of the field with $p$ elements. Let $G$ be a connected reductive algebraic group over $k$ and assume that $G$ is defined over the finite subfield ${{\mathbb{F}}}_q\subseteq k$, where $q=p^f$ for some $f{\geqslant}1$. Let $F\colon G\rightarrow G$ be the corresponding Frobenius map. Let $B\subseteq G$ be an $F$-stable Borel subgroup and $T\subseteq B$ be an $F$-stable maximal torus. Let $W=N_G(T)/T$ be the corresponding Weyl group. We assume that $F$ acts trivially on $W$ and that $F(t)=t^q$ for all $t\in T$. Then 