Center of the affine $\mathfrak{gl}_{n|1}$ at the critical level and pseudo-differential operators
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| Format: | Preprint |
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2026
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| _version_ | 1866908798125342720 |
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| author | Adamović, Dražen Feigin, Boris Nakatsuka, Shigenori |
| author_facet | Adamović, Dražen Feigin, Boris Nakatsuka, Shigenori |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21850 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Center of the affine $\mathfrak{gl}_{n|1}$ at the critical level and pseudo-differential operators Adamović, Dražen Feigin, Boris Nakatsuka, Shigenori Representation Theory Mathematical Physics Quantum Algebra 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. |
| title | Center of the affine $\mathfrak{gl}_{n|1}$ at the critical level and pseudo-differential operators |
| topic | Representation Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2601.21850 |