# Groups with isomorphic factors — H3

Let \(k=\mathbb F_2\), \(R=k[t]\), \(A=R^3\), and \(V=k^4\). All tensor products are over \(k\). Put
\[
C=(A\otimes_k A)^{\rm Flip},\quad \Delta(a)=a\otimes a,\quad
D=(A\otimes_k V^*)\oplus C,
\]
where \({\rm Flip}(a\otimes b)=b\otimes a\), and regard the vector spaces as discrete additive groups.
The group \({\rm SL}_3(R)\) acts on \(A\) by its usual action, on \(C\) by the tensor action, and trivially on \(V^*\).

Let \(\widehat D={\rm Hom}_k(D,k)\) with the compact product topology, identified with the character group by \(\chi\mapsto(d\mapsto(-1)^{\chi(d)})\). The \({\rm SL}_3(R)\)-action is \(l\chi=\chi\circ l^{-1}\), where \(l(u,c)=(lu,lc)\).
For any basis \(\varphi_1,\ldots,\varphi_4\) of \(V^*\), set
\[
d_j=(e_1\otimes\varphi_j,0)\quad(1\le j\le4),\qquad
d_5=(0,\Delta(e_1)).
\]
For every \({\rm SL}_3(R)\)-invariant Borel probability measure \(\mu\) on \(\widehat D\),
\[
\mu(\widehat D\setminus\{0\})\le12\sum_{j=1}^5\mu\{\chi:\chi(d_j)=1\}.
\]