On the extended Shi arrangement of type $B_{\mathrm{\ell }}$

Motivated by Kawanoue’s construction of a basis for the derivation module of the cone of the extended Catalan arrangement of type $B_{\mathrm{\ell }}$, we construct an explicit basis for the derivation module of $\mathbf{c}Shi(B_{\mathrm{\ell }},m)$. Our construction generalizes the previously known case $m=1$ to $m\ge 1$. It also provides a partial answer to a question of Abe and Terao concerning explicit constructions of simple-root bases.