The Lafforgue variety and irreducibility of induced representations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913325446594560 |
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| author | Psaromiligkos, Kostas I. |
| author_facet | Psaromiligkos, Kostas I. |
| contents | We construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative unital algebra $R$ over any field $k$, provided that the center $Z(R)$ is finitely generated and $R$ is finitely generated as a $Z(R)$-module. Our main technical tool is a generalization of the Hilbert scheme for non-commutative algebras, which may be of independent interest.
Applying our construction in the case of Hecke algebras of Bernstein components, we derive a characterization for the irreducibility of induced representations in terms of the vanishing of a generalized discriminant on the Bernstein variety. We explicitly compute the discriminant in the case of an Iwahori-Hecke algebra of a split reductive $p$-adic group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11834 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Lafforgue variety and irreducibility of induced representations Psaromiligkos, Kostas I. Representation Theory Algebraic Geometry 20C08 20C08 20C08 We construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative unital algebra $R$ over any field $k$, provided that the center $Z(R)$ is finitely generated and $R$ is finitely generated as a $Z(R)$-module. Our main technical tool is a generalization of the Hilbert scheme for non-commutative algebras, which may be of independent interest. Applying our construction in the case of Hecke algebras of Bernstein components, we derive a characterization for the irreducibility of induced representations in terms of the vanishing of a generalized discriminant on the Bernstein variety. We explicitly compute the discriminant in the case of an Iwahori-Hecke algebra of a split reductive $p$-adic group. |
| title | The Lafforgue variety and irreducibility of induced representations |
| topic | Representation Theory Algebraic Geometry 20C08 20C08 20C08 |
| url | https://arxiv.org/abs/2211.11834 |