Modularity for $\mathcal{W}$-algebras and affine Springer fibres
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916188612722688 |
|---|---|
| author | Shan, Peng Xie, Dan Yan, Wenbin |
| author_facet | Shan, Peng Xie, Dan Yan, Wenbin |
| contents | We construct a bijection between admissible representations for an affine Lie algebra $\mathfrak{g}$ at boundary admissible levels and $\mathbb{C}^\times$ fixed points in homogeneous elliptic affine Springer fibres for the Langlands dual affine Lie algebra $\mathfrak{g}^\vee$. Using this bijection, we relate the modularity of the characters of admissible representations to Cherednik's Verlinde algebra construction coming from double affine Hecke algebras. Finally, we show that the expected behaviors of simple modules under quantized Drinfeld-Sokolov reductions are compatible with the reductions from affine Springer fibres to affine Spaltenstein varieties. This yields (modulo some conjectures) a similar bijection for irreducible representations of $\mathcal{W}$-algebras, as well as an interpretation for their modularity properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00760 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modularity for $\mathcal{W}$-algebras and affine Springer fibres Shan, Peng Xie, Dan Yan, Wenbin Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra We construct a bijection between admissible representations for an affine Lie algebra $\mathfrak{g}$ at boundary admissible levels and $\mathbb{C}^\times$ fixed points in homogeneous elliptic affine Springer fibres for the Langlands dual affine Lie algebra $\mathfrak{g}^\vee$. Using this bijection, we relate the modularity of the characters of admissible representations to Cherednik's Verlinde algebra construction coming from double affine Hecke algebras. Finally, we show that the expected behaviors of simple modules under quantized Drinfeld-Sokolov reductions are compatible with the reductions from affine Springer fibres to affine Spaltenstein varieties. This yields (modulo some conjectures) a similar bijection for irreducible representations of $\mathcal{W}$-algebras, as well as an interpretation for their modularity properties. |
| title | Modularity for $\mathcal{W}$-algebras and affine Springer fibres |
| topic | Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2404.00760 |