An averaging process for unipotent group actions
    
	Amnon Yekutieli, BGU

Abstract:

Let G be a unipotent algebraic group over a field K of characteristic
0, and let Z be a torsor under G(K). By this I mean, naively, that Z is a 
set endowed with a G(K)-action that's transitive and has trivial stabilizers.
Suppose we are given a sequence z_0, ..., z_q of points in Z, and a
sequence w_0, ..., w_q of scalars in K whose sum is 1 (these are the weights). 
I will explain how to define the weighted average of z_0, ..., z_q in Z. 
This averaging process is very well-behaved (it's functorial and simplicial).

Here is an application. Suppose X is an algebraic variety over K and Z
is a G-torsor on X, in the usual sense, which is locally trivial for the 
Zariski topology. Suppose s_0, ..., s_q are local sections of the torsor Z, 
defined on an open covering of X. The averaging process allows us to construct 
a global simplicial section s of Z. This situation occurs in the theory of 
deformation quantization.