  We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on
the flag variety $G/B$ of a complex reductive group $G$, with coefficients in
an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the
projective cover of the skyscraper sheaf in terms of a "multiplicative"
coinvariant algebra, and then establish an equivalence of categories between
projective (or tilting) objects in this category and a certain category of
"Soergel modules" over this algebra. We also obtain a description of the
derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$,
$T\subset B$ are the unipotent radical and the maximal torus), as a monoidal
category, in terms of coherent sheaves on the formal neighborhood of the base
point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where
$T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$.
