The coarse quotient for affine Weyl groups and pseudo-reflection groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911394178269184 |
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| author | Gannon, Tom |
| author_facet | Gannon, Tom |
| contents | We study the coarse quotient $\mathfrak{t}^*//W^{\text{aff}}$ of the affine Weyl group $W^{\text{aff}}$ acting on a dual Cartan $\mathfrak{t}^*$ for some semisimple Lie algebra. Specifically, we classify sheaves on this space via a "pointwise" criterion for descent, which says that a $W^{\text{aff}}$-equivariant sheaf on $\mathfrak{t}^*$ descends to the coarse quotient if and only if the fiber at each field-valued point descends to the associated GIT quotient. We also prove the analogous pointwise criterion for descent for an arbitrary finite group acting on a vector space. Using this, we show that an equivariant sheaf for the action of a finite pseudo-reflection group descends to the GIT quotient if and only if it descends to the associated GIT quotient for every pseudo-reflection, generalizing a recent result of Lonergan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_00175 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The coarse quotient for affine Weyl groups and pseudo-reflection groups Gannon, Tom Representation Theory Algebraic Geometry We study the coarse quotient $\mathfrak{t}^*//W^{\text{aff}}$ of the affine Weyl group $W^{\text{aff}}$ acting on a dual Cartan $\mathfrak{t}^*$ for some semisimple Lie algebra. Specifically, we classify sheaves on this space via a "pointwise" criterion for descent, which says that a $W^{\text{aff}}$-equivariant sheaf on $\mathfrak{t}^*$ descends to the coarse quotient if and only if the fiber at each field-valued point descends to the associated GIT quotient. We also prove the analogous pointwise criterion for descent for an arbitrary finite group acting on a vector space. Using this, we show that an equivariant sheaf for the action of a finite pseudo-reflection group descends to the GIT quotient if and only if it descends to the associated GIT quotient for every pseudo-reflection, generalizing a recent result of Lonergan. |
| title | The coarse quotient for affine Weyl groups and pseudo-reflection groups |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2206.00175 |