===== recycled on Sun Mar 22 12:11:25 EDT 2015 by drorbn on Debian-1207 ======
We aim to prove Proposition~\ref{prop:products}, asserting that
$\calA(G\times H)\cong\calA(G)\otimes\calA(H)$. As we shall see, the
proof revolves around the fact that $\bbQ(G\times H)=(\bbQ G)\otimes(\bbQ
H)$ is doubly filtered. Hence we start with some general facts about
doubly-filtered vector spaces.
A double-filtration on a vector space $V$ is a collection of subspaces
$F_{p,q}$ of $V$, with $p,q$ non-negative integers, such that $V=F_{0,0}$
and for every $p,q$, $F_{p,q}\supset F_{p+1,q}$ and $F_{p,q}\supset
F_{p,q+1}$. The associated doubly-graded space of $V$ is defined by
\[ \gr^2V\coloneqq \bigoplus_{p,q}V_{p,q}
\coloneqq \bigoplus_{p,q}\frac{F_{p,q}}{F_{p+1,q}+F_{p,q+1}}.
\]
A double-filtration also defines a single filtration, by setting
$F_n\coloneqq\sum_{p+q=n}F_{p,q}$, and hence a singly-graded
associated space $\gr V=\oplus_nF_n/F_{n+1}$. If $p+q=n$, then
$F_{p,q}\subset F_n$ and $F_{p+1,q}+F_{p,q+1}\subset F_{n+1}$,
and hence there are maps $V_{p,q}\to V_n$, which induce a map
$\alpha_n\colon\bigoplus_{p+q=n}V_{p,q}\to V_n$.
\begin{lemma} The map $\alpha_n$ is an isomorphism. \end{lemma}
\begin{proof} The surjectivity of $\alpha_n$ is obvious. For $s\geq 0$, let
$\alpha_n^s$ be the restriction of $\alpha_n$ to the sum
$V_n^s\coloneqq\bigoplus_{p+q=n,\,p