  Naisse and Vaz defined an extension of KLR algebras to categorify Verma
modules. We realise these algebras geometrically as convolution algebras in
Borel-Moore homology. For this we introduce Grassmannian-Steinberg quiver flag
varieties. They generalize Steinberg quiver flag varieties in a non-obvious
way, reflecting the diagrammatics from the Naisse-Vaz construction. Using
different kind of stratifications we provide geometric explanations of the
rather mysterious algebraic and diagrammatic basis theorems.
  A geometric categorification of Verma modules was recently found in the
special case of $\mathfrak{sl}_2$ by Rouquier. Rouquier's construction uses
coherent sheaves on certain quasi-map spaces to flag varieties (zastavas),
whereas our construction is implicitly based on perverse sheaves. Both should
be seen as parts (on dual sides) of a general geometric framework for the
Naisse-Vaz approach.
  We first treat the (substantially easier) $\mathfrak{sl}_2$ case in detail
and construct as a byproduct a geometric dg-model of the nil-Hecke algebras.
The extra difficulties we encounter in general require the use of more
complicated Grassmannian-Steinberg quiver flag varieties. Their definition
arises from combinatorially defined diagram varieties which we assign to each
Naisse-Vaz basis diagram. Our explicit analysis here might shed some light on
categories of coherent sheaves on more general zastava spaces studied by
Feigin-Finkelberg-Kuznetsov-Mirkovi\'c and Braverman, which we expect to occur
in a generalization of Rouquier's construction away from $\mathfrak{sl}_2$.
