The extended Weyl group $\widetilde{W}(D_5^{(1)})$ as an extension of   KNY's birational representation of $\widetilde{W}(A_1^{(1)}\times A_3^{(1)})$
  We study the birational representation of $\wt{W}(A_1^{(1)}\times A_3^{(1)})$ proposed by Kajiwara-Noumi-Yamada (KNY) in the case of $m=2$ and $n=4$. It is shown that the equation can be lifted to an automorphism of a family of $A_3^{(1)}$ surfaces and therefore the group of Cremona isometries is $\wt{W}(D_5^{(1)})$ ($\supset \wt{W}(A_1^{(1)}\times A_3^{(1)})$). The equation can be decomposed into two mappings which are conjugate to the $q$-$P_{VI}$ equation. It is also shown that the subgroup of Cremona isometries which commute with the original translation is isomorphic to $\mz \times \wt{W}(A_3^{(1)}) \times \wt{W}(A_1^{(1)})$. 
Tomoyuki Takenawa