Lapointe–Vinet's Jack creation-operator properties

Let $P_\beta^{(\alpha)}$ be a Jack polynomial for a generalized partition $\beta$, let $e_k$ be the elementary symmetric function, and let $F^{(\alpha)}(\kappa)$ be the diagonal operator acting by

$$F^{(\alpha)}(\kappa)P_{(\beta_1,\dots,\beta_N)}^{(\alpha)}=\prod_{i=1}^N\prod_{j=1}^{\infty}(\alpha(\beta_i-j)+\kappa+1-i)P_{(\beta_1,\dots,\beta_N)}^{(\alpha)}.$$

Define $F_{m,\kappa}^{(\alpha)}=F^{(\alpha)}(\kappa)e_mF^{(\alpha)}(\kappa)^{-1}$.

Lapointe–Vinet's Jack-operator conjecture. The creation operators share the following properties:

$$\begin{aligned} \mathrm{(i)}\quad &\tilde B_k^{+(\alpha)}=F^{(\alpha)}(k)e_kF^{(\alpha)}(k)^{-1}=F_{k,k}^{(\alpha)},\\ \mathrm{(ii)}\quad &F_{m,\kappa}^{(\alpha)}=\sum_{|J|=m}x_JD_{J,\kappa-m+1}=e_N^{(m-\kappa)/\alpha}\tilde B_m^{+(\alpha)}e_N^{(\kappa-m)/\alpha},\\ \mathrm{(iii)}\quad &[F_{m,\kappa}^{(\alpha)},F_{n,\kappa}^{(\alpha)}]=0. \end{aligned}$$

These assertions describe the expected Jack-polynomial analogues of the Macdonald creation-operator identities and commutativity relations; the supplied text does not establish them.