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%  An Averaging Process for Unipotent Group Actions
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%  Amnon Yekutieli
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% {An Averaging Process for Unipotent 
% Group Actions}
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\begin{document}

%% top matter for notes %%
\mbox{}
\vspace{2cm}

\begin{center}
{\large \bfseries An Averaging Process for Unipotent 
Group Actions} 

\vspace{1ex}
Lecture Notes\\
written 19 June 2008

\vspace{3ex}
Amnon Yekutieli\\
Ben Gurion University, ISRAEL

{\small http://www.math.bgu.ac.il/$\sim$amyekut/lectures}
\end{center}

\vspace{2ex}
%% ** end of top matter for notes **


% \begin{frame}
% \titlepage
% \end{frame}

%\begin{frame}
Here is the plan of my lecture: 

\begin{enumerate}
\item[1.] Introduction
\item[2.] Exponential Map
\item[3.] First Approximation of the Average
\item[4.] Improved Formula for the Average
\item[5.] More Sophisticated Geometry
\item[6.] An Arithmetic Application
\item[7.] An Application to Deformation Quantization 
\end{enumerate}
%\end{frame}

\section{Introduction}
%\begin{frame}
%\sect{1. Introduction}
%\pause

In this talk I will explain a geometric result that is a 
by-product of research on deformation quantization \cite{Ye1}. 
This could be useful for other applications. Full details can be found
in \cite{Ye2}.
%\pause

Let $\K$ be a field of characteristic $0$, and let $G$
be a linear (i.e.\ affine) algebraic group $G$ over $\K$.
%\pause

For a commutative 
$\K$-algebra $A$, we write $G(A)$ for the group of $A$-valued 
points of $G$. 
%\pause

For instance if $G = \mrm{GL}_{n, \K}$, then 
$G(A) = \mrm{GL}_{n}(A)$, the group of invertible matrices with 
entries in $A$.
%\end{frame}  

%\begin{frame}
Here is a naive definition of a {\em $G(\K)$-torsor}:
it is a set $Z$, endowed with an action of the group $G(\K)$, which is
transitive and has 
trivial stabilizers. 
%\pause

Thus for every point $z \in Z$ the map 
$G(\K) \to Z$, $g \mapsto g \cdot z$, is bijective.
%\pause

By a {\em weight sequence} we mean a sequence 
\[ \bsym{w} = (w_0, \ldots, w_q) \in \K^{q+1} \]
such that 
\[ \sum_{i=0}^q w_i = 1 . \]
%\end{frame}

%\begin{frame}
Suppose $G$ is a vector space, i.e.\ 
$G \cong \mbf{A}^n_{\K}$ with its additive group structure. 
%\pause

Let $Z$ be a $G(\K)$-torsor. 
%\pause

Given a sequence $\bsym{z} = (z_0, \ldots, z_q)$ of points in $Z$,
and a weight sequence $\bsym{w}$, we have the usual weighted 
average
\[ \opn{wav}_{G, \bsym{w}}(\bsym{z}) \in Z . \]
%\pause

Let me recall the formula. To simplify notation, in this case
we will use addition for the operation in $G$ and the action on $Z$.
%\pause

Choose any base point $z \in Z$, and define 
$g_i \in G(\K)$ to be the unique element such that
\[ z_i = g_i + z . \] 
%\end{frame}

%\begin{frame}
Using the isomorphism $G \cong \mbf{A}^n_{\K}$
we get elements 
\[ w_i g_i \in G(\K) , \]
and then we let
\[ \opn{wav}_{G, \bsym{w}}(\bsym{z}) := 
\Bigl( \sum_{i=0}^q w_i g_i \Bigr) + z  \in Z . \]
%\pause

Since $\sum_{i=0}^q w_i = 1$, this is independent of the base point 
$z$.  
%\end{frame}

%\begin{frame}
We want to generalize this construction to {\em unipotent groups}.
%\pause

Let $\bar{\K}$ be an algebraic closure of $\K$.
An element $g \in G(\bar{\K})$ is called {\em unipotent} if for 
any morphism of groups $\rho : G \to \mrm{GL}_{n, \K}$,
the matrix $\rho(g) \in \mrm{GL}_n(\bar{\K})$ is unipotent (i.e.\ 
its only eigenvalue is $1$). 
%\pause

The group $G$ is called unipotent of all elements of 
$G(\bar{\K})$ are unipotent. See \cite{Ho} for details.
%\pause

Here is equivalent characterization: $G$ is unipotent iff it is 
isomorphic to a closed subgroup of the upper triangular 
matrix group inside $\mrm{GL}_{m, \K}$, for some $m$.
%\pause

Thus, for any $\K$-algebra $A$ we get
\[ G(A) \subset 
\left[ \begin{matrix}
1 & A & A & \cdots & A \\
0 & 1 & A & \cdots & A \\
0 & 0 & 1 & \cdots & A \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{matrix} \right]
\subset \mrm{GL}_{m}(A) .   \]
%\end{frame}

%\begin{frame}
\begin{exa}
$G := \mbf{A}^n_{\K}$ is unipotent. 
%\pause

A possible embedding is
\[ G(A) \cong 
\left[ \begin{matrix}
1 & A & A & \cdots & A \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{matrix} \right]
\subset \mrm{GL}_{m}(A)  \]
for $m = n+1$. 
%\pause

In fact, any abelian unipotent
group is isomorphic to $\mbf{A}^n_{\K}$, for some $n$. 
\end{exa}
%\pause

\begin{rem}
If $\K = \mbb{R}$ and $G$ is unipotent, then $G(\mbb{R})$ is a simply
connected nilpotent Lie group. And vice versa.
\end{rem}
%\end{frame}

%\begin{frame}
Let $G$ be a unipotent group, and let $Z$ be a $G(\K)$-torsor. 
%\pause

Again we are given a sequence of points 
$\bsym{z} = (z_0, \ldots, z_q)$ in $Z$, and a weight sequence
$\bsym{w}$. 
%\pause

We want to construct a weighted average
$\opn{wav}_{G, \bsym{w}}(\bsym{z}) \in Z$.
%\pause

A couple of obvious requirements of such a construction are:
%\pause

\begin{enumerate}
\item[1.] Functoriality: suppose $G \to G'$ is a morphism of groups, 
with $G'$ also unipotent; $Z'$ is a torsor under $G'(\K)$; 
\[ f : Z \to Z' \] 
is a $G(\K)$-equivariant function; and
\[ \bsym{z}' := (f(z_0), \ldots, f(z_q)). \] 
Then
\[ f(\opn{wav}_{G, \bsym{w}}(\bsym{z})) = 
\opn{wav}_{G', \bsym{w}}(\bsym{z}') . \]
\end{enumerate}
%\end{frame}

%\begin{frame}
\begin{enumerate}
\item[2.] Simpliciality. This will be defined later, but here is an 
instance: suppose $w_q = 0$. Then
\[ \opn{wav}_{G, \bsym{w}}(\bsym{z})
 =
 \opn{wav}_{G, (w_0, \ldots, w_{q-1})}(z_0, \ldots, z_{q-1}) .
\]
%\pause

\item[3.] Symmetry: the average is invariant under simultaneous
permutation of the sequences $\bsym{w}$ and $\bsym{z}$.
\end{enumerate}
%\end{frame}



\section{Exponential Map}
%\begin{frame}
%\sect{2. Exponential Map}
%\pause

As before, $\K$ is a field of characteristic $0$, and $G$ is a
unipotent linear algebraic group over $\K$.
%\pause

Let $\mfrak{g}$ be the Lie algebra of $G$. This is a Lie algebra 
over $\K$, and also a scheme: $\mfrak{g} \cong \mbf{A}^n_{\K}$, 
where $n = \opn{dim} G$. 
%\pause

There is an isomorphism of schemes

\negskip{1.5}
\[ \opn{exp}_G : \mfrak{g} \iso G \]

\negskip{1}
called the {\em exponential map}. Its inverse is $\opn{log}_G$.
%\pause

The exponential map is functorial; namely if $G'$ is another
unipotent group, and $\phi : G \to G'$ is a morphism of groups, then
the diagram
\[ \UseTips
\xymatrix  @C=7ex @R=5ex  {
\mfrak{g}
\ar[r]^{\opn{exp}_G}
\ar[d]_{\mrm{d} \phi}
& G 
\ar[d]^{\phi} \\
\mfrak{g}'
\ar[r]^{\opn{exp}_{G'}}
& G' 
} \]
is commutative. 

The functor $G \mapsto \mfrak{g}$ is an equivalence from
the category of unipotent groups to the category
of nilpotent Lie algebras over $\K$. 

The multiplication in $G$ is expressible via the CBH formula:
\[ \opn{exp}_G(\gamma_1) \cdot \opn{exp}_G(\gamma_2) = 
\opn{exp}_G(F(\gamma_1, \gamma_2)) , \]
where
\[ F(\gamma_1, \gamma_2) = \gamma_1 + \gamma_2 + 
\smfrac{1}{2} [\gamma_1, \gamma_2] + \cdots \]
is a universal ``commutator power series''.

See \cite{Ho} for details. 
%\end{frame}

%\begin{frame}
\begin{exa}
For a unipotent closed subgroup 
$G \subset \mrm{GL}_{n, \K}$, the Lie algebra 
$\mfrak{g}(\K)$ is a subalgebra of 
$\mfrak{gl}_n(\K) = \mrm{M}_n(\K)$ consisting 
of nilpotent matrices. 
%\pause

For any element $\gamma \in \mfrak{g}(\K)$,
the formula for $\opn{exp}_G(\gamma)$ is 
\[ \opn{exp}_G(\gamma) = \sum_{i \geq 0}
\smfrac{1}{i!} \gamma^i \in G(\K) \subset
\mrm{GL}_{n}(\K) . \]
\end{exa}
%\pause

%\end{frame}





\section{First Approximation of the Average}
%\begin{frame}
%\sect{3. First Approximation of the Average}
%\pause

With $G$ a unipotent group 
as before, let $Z$ be a $G(\K)$-torsor.
%\pause

Suppose $\bsym{w} = (w_0, \ldots, w_q)$ 
is a weight sequence, and $\bsym{z} = (z_0, \ldots, z_q)$ 
is a sequence of points in $Z$. 
%\pause

Choose some base point $z \in Z$, and define elements $g_i \in G(\K)$
by the rule 
\[ z_i  = g_i \cdot z . \]
%\pause

For any $i$ let
\[ \gamma_i := \opn{log}_G(g_i) . \]
%\end{frame}

%\begin{frame}
Using the vector space structure 
of $\mfrak{g}(\K)$ define
\[ \gamma := \sum_{i=0}^q w_i \gamma_i \in \mfrak{g}(\K). \]
%\pause

Next let 
\[ g := \opn{exp}_G(\gamma) \in G(\K) . \] 
%\pause

The first candidate for the weighed average is
\[ \opn{wav}_{G, \bsym{w}}(\bsym{z}) := g \cdot z \in Z . \] 
%\end{frame}

%\begin{frame}
But there is a problem here: this ``average'' depends on the 
choice of base point $z$! 
%\pause

A calculation shows that this dependence 
is in the form of commutators. 
%\pause

Observe that in the abelian case 
$G \cong \mbf{A}^n_{\K}$, the exponential map is 
actually an isomorphism of groups

\negskip{1}
\[ \opn{exp}_G : (\mfrak{g}, +) \iso (G, \cdot) . \]
%\pause

\negskip{1}
This explains why the problem of defining the average does not
arise in this case.
%\end{frame}


\section{Improved Formula for the Average}
%\begin{frame}
%\sect{4. Improved Formula for the Average}
%\pause

Continuing with the previous setup, 
we will do something a little bit strange. 
%\pause

There are $q+1$ preferred 
base points $z_0, \ldots, z_q \in Z$. We shall perform the 
averaging process with respect to each of these base points. 
%\pause

Thus, for fixed weight sequence $\bsym{w}$, we will obtain $q+1$
possibly distinct ``averages'' $z'_0, \ldots, z'_q$. 
%\pause

The explicit formulas are
\[ z'_i := \opn{exp}_G \Bigl( \sum_{j=0}^q 
w_j \opn{log}_G(g_{i,j}) \Bigr) \cdot z_i , \]
where
$g_{i,j} \in G(\K)$ are the elements satisfying 
\[ g_{i,j} \cdot z_i = z_j . \]
%\end{frame}

%\begin{frame}
This can be viewed as an operation
\[ \opn{wsym}_{G, \bsym{w}} : 
\begin{cases}
Z^{q+1} \to Z^{q+1} \\
(z_0, \ldots, z_q) \mapsto (z'_0, \ldots, z'_q) . 
\end{cases} \]
%\pause

I claim that if we iterate the operation 
$\opn{wsym}_{G, \bsym{w}}$ enough times, we end up with a 
{\em constant sequence}. 
%\pause

The reason is that the
group $G$ has a filtration by closed normal subgroups
\[ G = G_0 \supset G_1 \supset \cdots \supset G_d = \{ 1 \} \]
such that $G_{i} / G_{i+1}$ is abelian,
and the conjugation action of $G$ on $G_{i} / G_{i+1}$ is trivial. 
%\pause

Therefore, if the points
$z_0, \ldots, z_q$ are all in the same $G_i$-orbit for some $i$, 
then $z'_0, \ldots, z'_q$ are in the same $G_{i+1}$-orbit. 
%\end{frame}

%\begin{frame}
Thus, starting with a sequence of points 
\[ \bsym{z} = (z_0, \ldots, z_q) , \]
we get 
\[ \opn{wsym}_{G, \bsym{w}}^d(\bsym{z}) = 
(z, \ldots, z)  \]
for some $z \in Z$. 
%\pause

Now we can define the correct weighted average
\[ \opn{wav}_{G, \bsym{w}}(\bsym{z}) := z . \]
%\pause

It is easy to see that this average enjoys the required properties 
(functoriality, simpliciality and symmetry). 
%\end{frame}


\section{More Sophisticated Geometry}
\label{sec5}
%\begin{frame}
%\sect{5. More Sophisticated Geometry}
%\pause

The same idea can be dressed up in fancier language. 
%\pause

Let $\K$ be a field, and let $G$ be a linear algebraic group over
$\K$. 
%\pause

By {\em $G$-torsor} (or principal homogeneous space) 
we mean a finite type $\K$-scheme $Z$, with group action

\negskip{1.5}
\[ \mu : G \times Z \to Z , \] 

\negskip{1}
such that 
\[ \mu \times \mrm{p}_2 : G \times Z \to Z \times Z \]
is an isomorphism of schemes. 
%\pause

It follows that for any $\K$-algebra $A$ for which 
$Z(A) \neq \emptyset$, the set $Z(A)$ is a $G(A)$-torsor in the naive
sense.
%\end{frame}

%\begin{frame}
For $q \in \mbb{N}$ let $\bsym{\Delta}^q$ be the 
{\em combinatorial $q$-dimensional simplex}. 
%\pause

Its set of $p$-dimensional faces
$\bsym{\Delta}^q_p$ consists of all sequences
$(i_0, \ldots, i_p)$ satisfying
\[ 0 \leq i_0 \leq \cdots \leq i_p \leq q . \]
%\pause

In particular, for $p = 0$ we have the set of vertices
\[ \bsym{\Delta}^q_0 = \{ 0, \ldots, q \} . \]
%\pause

\negskip{1}
Hence the set
\[ Z(\K)^{\bsym{\Delta}^q_0} = 
\opn{Hom}_{\cat{Set}} \bigl( \bsym{\Delta}^q_0, Z(\K) \bigr)  \]
is nothing but the set of sequences $(z_0, \ldots, z_q)$ in 
$Z(\K)$.
%\end{frame}

%\begin{frame}
The {\em geometric $q$-dimensional simplex} is the scheme
\[ \bsym{\Delta}^q_{\K} :=
\opn{Spec}\, \K [t_0, \ldots, t_q]\, /\, (t_0 + \cdots + t_q - 1) . \]
%\pause

A point in 
$\bsym{w} \in \bsym{\Delta}^q_{\K}(\K)$ 
is precisely a weight sequence. 
%\pause

We denote by
\[ \opn{Hom}_{\msf{Sch} / \K}(\bsym{\Delta}^q_{\K}, Z) \]
the set of scheme morphisms. 
%\end{frame}

%\begin{frame}
Now assume $\opn{char} \K = 0$ and $G$ is unipotent. 
%\pause

The construction in the previous section can be geometrized, to 
yield a function
\begin{equation} \label{eqn1}
\opn{wav}_{G} : 
Z(\K)^{\bsym{\Delta}^q_0} \to
\opn{Hom}_{\msf{Sch} / \K}(\bsym{\Delta}^q_{\K}, Z) .
\end{equation}
%\pause

The relation to the operation $\opn{wav}_{G, \bsym{w}}$ of the
previous section is as follows.
%\pause

Take a sequence of points
$\bsym{z} \in Z(\K)^{\bsym{\Delta}^q_0}$. 
This gives rise to a morphism of $\K$-schemes
\[ \opn{wav}_{G}(\bsym{w}) : 
\bsym{\Delta}^q_{\K} \to Z . \]
%\pause

\negskip{1}
Now take a weight sequence
$\bsym{w} \in \bsym{\Delta}^q_{\K}(\K)$.
We get a point
\[ \opn{wav}_{G}(\bsym{z})(\bsym{w}) \in Z(\K) . \]
%\pause

\negskip{1}
It is easy to see that
\[ \opn{wav}_{G}(\bsym{z})(\bsym{w}) = 
\opn{wav}_{G, \bsym{w}}(\bsym{z}) \in Z(\K) . \]
%\end{frame}

%\begin{frame}
We can make things even more sophisticated.
Suppose $X$ is some 
$\K$-scheme, and $\pi : Z \to X$ is a $G$-torsor over $X$. 
%\pause

For any $X$-scheme $Y$, the set of $Y$-valued 
sections of $\pi$ is of course 
\[ \opn{Hom}_{\msf{Sch} / X}(Y, Z) . \]
%\pause

\negskip{1.5}
The averaging operation now becomes a function
\begin{equation} \label{eqn2}
\opn{wav}_{G} : 
\opn{Hom}_{\msf{Sch} / X}(Y, Z)^{\bsym{\Delta}^q_0} \to
\opn{Hom}_{\msf{Sch} / X}(\bsym{\Delta}^q_{\K} \times Y, Z) .
\end{equation}
%\end{frame}

%\begin{frame}
Here is an illustration with $q = 1$: we are given morphisms
\[ \sigma_0, \sigma_1 : Y \to Z \]

\begin{figure}
\includegraphics[scale=0.4, bb = 14 14 613 427]{pic1.eps}
\end{figure}
%\end{frame}

%\begin{frame}
The averaging process interpolates them into a $1$-dimensional family
of morphisms
\[ \sigma := \opn{wav}_{G}(\sigma_0, \sigma_1)
: \Delta^1_{\K} \times Y \to Z \]

\begin{figure}
\includegraphics[scale=0.4, bb = 14 14 613 427]{pic2.eps}
\end{figure}
%\end{frame}

%\begin{frame}
The function $\opn{wav}_{G}$ enjoys the functoriality and 
symmetry explained earlier. 
%\pause

Regarding simpliciality: as $q$ varies, the two sets 
appearing in equation (\ref{eqn2}) become simplicial sets, and the 
function $\opn{wav}_{G}$ becomes a map of simplicial sets.

%\pause
Note that formula (\ref{eqn1}) above is the special case
$X = Y = \opn{Spec} \K$.
%\end{frame}




\section{An Arithmetic Application}
%\begin{frame}
%\sect{6. An Arithmetic Application}
%\pause


Here is an application of the averaging process to an arithmetic
problem. The idea is due to Z. Reichstein. 
%\pause

Suppose $G$ is a linear algebraic group over a field $\K$, and let
$Z$ be a $G$-torsor. 
%\pause

In case the set of rational points 
$Z(\K)$ is nonempty, then any point $z \in Z(\K)$ defines an
isomorphism of $G$-schemes $G \cong Z$, namely 
$g \mapsto g \cdot z$. Hence one says that the torsor $Z$ is {\em
trivial}. 
%\pause

It is easy to find nontrivial torsors. Here is an example. 
%\end{frame}

%\begin{frame}
\begin{exa}
Take $\K := \mbb{R}$, and let $G$ be the non-split $1$-dimensional
torus. 
%\pause

Let's recall that as an $\mbb{R}$-scheme  
\[ G = \opn{Spec}\, \mbb{R} [s, t]\, /\, (s^2 + t^2 -1) . \]
%\pause

Thus $G$ is a closed subscheme of the plane
$\mbf{A}^2_{\mbb{R}}$. 
The set of points $G(\mbb{R})$ is nothing but the unit circle inside
the real plane:
\[ G(\mbb{R}) \subset \mbf{A}^2(\mbb{R}) = \mbb{R}^2 
\cong \mbb{C} . \]
%\pause

The group structure on $G$ is such that $G(\mbb{R})$ becomes a
subgroup of the multiplicative group $\mbb{C}^{\times}$.
\end{exa}
%\end{frame}

%\begin{frame}
For the torsor $Z$ we take the scheme
\[ Z := \opn{Spec}\, \mbb{R} [s, t]\, /\, (s^2 + t^2 + 1) , \]
with an obvious $G$-action. 
%\pause

The set
$Z(\mbb{R})$, which is the circle of radius $-1$, is then empty.
Hence $Z$ is a nontrivial torsor. 
(Of course the torsor $Z$ becomes trivial over $\mbb{C}$.)
%\pause

\bigskip
Note that the group $G$ in the example above is not unipotent. 
%\pause

There are also examples of nontrivial torsors under unipotent groups
in
positive characteristics.
%\end{frame}

%\begin{frame}
However it is well-known that if $G$ is a unipotent group and 
$\opn{char} \K = 0$, then any $G$-torsor is trivial. This is usually
proved using Galois cohomology. 
%\pause

Here is a new proof using averaging. So assume $\opn{char} \K = 0$,
$G$ is a unipotent group over $\K$, and $Z$ is a 
$G$-torsor. We want to show that $Z(\K)$ is nonempty. 
%\pause

Choose any closed point $z \in Z$. 
%\pause

There is some finite Galois 
extension $L$ of $\K$ such that $z \in Z(L)$. 
%\pause

Let us denote by 
$\Gamma$ the Galois group of $L / \K$. Then $\Gamma$ acts on the set
$Z(L)$, and 
\[ Z(L)^{\Gamma} = Z(\K) . \]
%\end{frame}

%\begin{frame}
Let us denote by 
$z_0 = z, z_1, \ldots, z_q$ the distinct $\Gamma$-conjugates of $z$ 
inside $Z(L)$. We get a sequence
\[ \bsym{z} = (z_0, \ldots, z_q) \in Z(L)^{\bsym{\Delta}^q_0} . \]
%\pause

\negskip{1}
Consider the weight sequence
\[ \bsym{w} := (\smfrac{1}{q+1}, \ldots, \smfrac{1}{q+1}) \in
\bsym{\Delta}^q_{\K}(\K) . \]
%\pause

\negskip{1}
From formula (\ref{eqn2}), with 
$X := \opn{Spec} \K$ and $Y := \opn{Spec} L$,  we obtain an element
\[ z' := \opn{wav}_{G}(\bsym{z})(\bsym{w}) \in Z(L) . \]
%\pause

\negskip{1}
It is not hard to check, using the functoriality and symmetry of 
$\opn{wav}_{G}$, that $z'$ is $\Gamma$-invariant. This 
implies that $z' \in Z(\K)$, as desired.
%\end{frame}




\section{An Application to Deformation Quantization}
%\begin{frame}
%\sect{7. An Application to Deformation Quantization}
%\pause

Here is the original application. 
%\pause

Let $\K$ be a field of characteristic $0$, let $X$ be a
$\K$-scheme, and let $G$ be a pro-unipotent group. 
%\pause

As in Section \ref{sec5}, suppose $\pi : Z \to X$ is a $G$-torsor
over $X$. 
%\pause

Assume 
\[ X = \bigcup_{i = 0}^m U_{i} \]
is an open covering, and for every $i$ we are given a section
\[ \sigma_{i} : U_{i} \to Z \] 
of $\pi$. 
%\end{frame}

%\begin{frame}
Consider a sequence of indices
\[ \bsym{i} = (i_0, \ldots, i_q) \in \bsym{\Delta}^m_q , \]
and let
\[ U_{\bsym{i}} := U_{i_0} \cap \cdots \cap U_{i_q} . \]
%\pause

Each of the sections 
\[ \sigma_{i_0}, \ldots, \sigma_{i_q} : 
U_{\bsym{i}} \to Z \]
is an element of $\opn{Hom}_{\msf{Sch} / X}(U_{\bsym{i}}, Z)$.
%\end{frame}

%\begin{frame}
The averaging process of formula (\ref{eqn2}) works also for
pro-unipotent groups (by passing to the limit), so we get a
morphism 
\[ \sigma_{\bsym{i}} :=
\opn{wav}_{G}(\sigma_{i_0}, \ldots, \sigma_{i_q}) :
\bsym{\Delta}^q_{\K} \times U_{\bsym{i}} \to Z . \]
%\pause

This is a family of sections $U_{\bsym{i}} \to Z$ interpolating 
between the various sections 
$\sigma_{i_0}, \ldots, \sigma_{i_q}$.
%\pause

As $\bsym{i}$ varies the morphisms $\sigma_{\bsym{i}}$
form a {\em simplicial section} of $\pi : Z \to X$.
%See Figure \ref{fig1}.
%\end{frame}

%\begin{frame}
%\begin{figure}
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& \quad &
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%
% \caption{Simplicial section, $q = 1$. 
% We start with sections over two open sets $U_{0}$ and $U_{1}$ 
% in the left picture; and we pass to a simplicial 
% section $\sigma_{(0,1)}$ on the right.} 
%\label{fig1}
%\end{figure}

\negskip{3}
Simplicial section, $q = 1$. 
We start with sections over two open sets $U_{0}$ and $U_{1}$ 
in the left picture; and we pass to a simplicial 
section $\sigma_{(0,1)}$ on the right.
%\end{frame}

%\begin{frame}
Next consider the power series algebra
\[ \K[[t_1, \ldots, t_n]] . \]
%\pause

\negskip{2}
Its automorphism group is a semi-direct product
\[ \mrm{GL}_n(\K) \ltimes G(\K) , \]
where $G$ is a pro-unipotent group.
%\pause

Suppose $X$ is a smooth $n$-dimensional scheme $X$ over $\K$.
There is an infinite dimensional bundle
\[ \pi : Z \to X \]
called the {\em bundle of formal coordinate systems}.
%\pause

This construction goes back to Gelfand-Kazhdan, and is a 
key method in deformation quantization (after Fedosov and
Kontsevich). Cf.\ \cite{Ko} and \cite{Ye1}.
%\end{frame}

%\begin{frame}
Now $Z$ turns out to be a torsor under the group 
$\mrm{GL}_{n, \K} \ltimes G$ above. 
%\pause

Let $\bar{Z}$ be the quotient bundle $Z / \mrm{GL}_{n, \K}$.
%\pause

Locally the bundle $\bar{Z}$ has 
sections, and a modification of the averaging process described 
above yields a global simplicial section of $\bar{Z}$. 
%\pause

Using this simplicial section we can prove an algebro-geometric 
analogue of Kontsevich's result on deformation quantization.
%\pause

% \begin{center}
% {\red $\sim$ \ END \ $\sim$}
% \end{center}
% %\end{frame}



\begin{thebibliography}{Ye1}
\bibitem[Ho]{Ho} G. Hochschild, 
    ``Basic Theory of Algebraic Groups and Lie Algebras,''
    Springer, 1981.
\bibitem[Ko]{Ko} M. Kontsevich,
    Deformation quantization of Poisson manifolds,
    Lett.\ Math.\ Phys.\ {\bf 66} (2003), no.\ 3, 157-216.
\bibitem[Ye1]{Ye1} A.\ Yekutieli, 
	Deformation Quantization in Algebraic Geometry,
	Adv.\ Math.\ {\bf 198} (2005), 383-432.
	Erratum: Adv.\ Math.\ {\bf 217} (2008), 2897-2906.
\bibitem[Ye2]{Ye2} A.\ Yekutieli, 
	An Averaging Process for Unipotent Group Actions,
	Representation Theory {\bf 10} (2006), 147-157.
\bibitem[Ye3]{Ye3} A.\ Yekutieli, 
	Mixed Resolutions and Simplicial Sections,
	Israel J. Math.\ {\bf 162} (2007), 1-27.
\end{thebibliography}


\end{document}
