Center of the affine $\mathfrak{gl}_{n|1}$ at the critical level and pseudo-differential operators

We prove that the center of the affine Lie algebra $\widehat{\mathfrak{gl}}_{n|1}$ at the critical level is generated by the coefficients in the expansion of the pseudo-differential operator $(\partial_z-u_1(z))\cdots (\partial_z-u_n(z))(\partial_z+u_{n+1}(z))^{-1}$ taking values in the Cartan subalgebra. This is an affine analogue of the Harish-Chandra isomorphism in the finite case. The key ingredient of the proof is the identification of the center with the Heisenberg coset of the regular W-superalgebra of $\mathfrak{gl}_{n|1}$ at the critical level, whose associated graded algebra is realized as the affine supersymmetric polynomials. Based on this, we derive a character formula for the center, which coincides with the generating function of plane partitions with a pit condition. We also prove that the Heisenberg coset at generic levels has a similar interpretation in terms of pseudo-differential operators that deform the one at the critical level.