  In this first of a series of articles on standard extension algebras we study
standard perverse sheaves on varieties with $\mathbb{G}_m$-actions. Based on
Braden's hyperbolic localisation, we describe their extension algebra
geometrically via a convolution structure on the intersections of attracting
with repelling cells. We introduce a multiplicative structure on open
Richardson varieties which provides a practical way to compose these extensions
in case of flag varieties. For open Richardson varieties in Grassmannians we
construct two explicit cell decompositions, of Gauss- and of Deodhar-type. It
is shown that the latter is a stratification with the same combinatorics as
Deodhar's decomposition. We introduce a calculus of Fukaya diagrams to encode
the geometry of the decompositions. It provides a model for the cohomology of
open Richardson varieties and thus for standard extensions. The calculus is
motivated by the Mak-Smith Fukaya-Seidel category of a natural Lefschetz
fibration and should allow to compute morphism spaces in there. We finally
discuss the relation of our work to extensions of (parabolic) Verma modules in
category O as well as to the computation of R- and R'-polynomials.
