A. Ocneanu has observed that a mysterious orbifold phenomenon, an analogue of the orbifold subfactors D. E. Evans and I had studied in [13], occurs in the system of the $M_\infty$-$M_\infty$ bimodules of the asymptotic inclusion $M\vee(M'\cap M_\infty)\subset M_\infty$ of the Jones subfactor $N\subset M$ of type $A_{2n+1}$. The construction of the system of $M_\infty$-$M_\infty$ bimodules from the original $M$-$M$ bimodules is known to be a subfactor/paragroup analogue of the quantum double construction of Drinfel$'$d. The subfactor $M\vee (M'\cap M_\infty)\subset M_\infty$ is called the asymptotic inclusion of the original subfactor $N\subset M$.
We have shown that this is a general phenomenon and identified some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type $A$ of Wenzl. That is, we study their asymptotic inclusions and show that the $M_\infty$-$M_\infty$ bimodules are described by certain orbifolds (with ghosts in the sense of Ocneanu) for $SU(3)_{3k}$ in [25]. We have actually computed several examples of the (dual) principal graphs of the asymptotic inclusions.
As a corollary of the identification of Ocneanu's orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of $D_{2n}$, $n>2$. It is expected that this braiding coincides with the ones constructed by Ocneanu and Turaev-Wenzl recently by different methods.
These studies are based on Ocneanu's paragroup theory. We have prepared a book manuscript [24] on this theory and related topics. Many basics of this theory had been unpublished, and this is the first self-contained treatment of the theory.