The Kazhdan-graded structure conjecture for finite W-superalgebras

Let ${\frak {g}}$ be a Lie superalgebra with reductive even part ${\frak {g}}_{\bar 0}$, let $W_\chi$ be its finite W-superalgebra, and let $Gr_KW_\chi$ denote its Kazhdan-graded algebra. If $\dim({\frak {g}}_{-1})_{\bar 1}$ is even, then

$$Gr_KW_\chi\simeq S({\frak {g}}^\chi),$$

and if $\dim({\frak {g}}_{-1})_{\bar 1}$ is odd, then

$$Gr_KW_\chi\simeq S({\frak {g}}^\chi)\otimes {\Bbb C}[\xi],$$

where ${\Bbb C}[\xi]$ is the exterior algebra generated by one element $\xi$. Kazhdan-graded structure conjecture. The preceding isomorphisms hold under the stated parity assumptions. This describes the expected associated graded structure of the finite W-superalgebra; the supplied text gives no resolution status beyond the assertion itself.