$\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields
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| Format: | Preprint |
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2024
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| _version_ | 1866911722060644352 |
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| author | Letellier, Emmanuel Scognamiglio, Tommaso |
| author_facet | Letellier, Emmanuel Scognamiglio, Tommaso |
| contents | In this paper we study the mixed Poincaré polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings$ \left(\mathcal{C}(\mathrm{PGL}_n(\mathbb{F}_q)),Loc(\mathrm{PGL}_n),*\right)$ and $\left(\mathcal{C}(\mathrm{SL}_n(\mathbb{F}_q)), CS(\mathrm{SL}_n),\cdot\right) $ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\mathrm{PGL}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\mathrm{PGL}_n$ and $CS(\mathrm{SL}_n)$ the basis of characteristic functions of Lusztig's character-sheaves on $\mathrm{SL}_n$. Our result reminds us of a non-abelian Fourier transform. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03234 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields Letellier, Emmanuel Scognamiglio, Tommaso Representation Theory Algebraic Geometry In this paper we study the mixed Poincaré polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings$ \left(\mathcal{C}(\mathrm{PGL}_n(\mathbb{F}_q)),Loc(\mathrm{PGL}_n),*\right)$ and $\left(\mathcal{C}(\mathrm{SL}_n(\mathbb{F}_q)), CS(\mathrm{SL}_n),\cdot\right) $ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\mathrm{PGL}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\mathrm{PGL}_n$ and $CS(\mathrm{SL}_n)$ the basis of characteristic functions of Lusztig's character-sheaves on $\mathrm{SL}_n$. Our result reminds us of a non-abelian Fourier transform. |
| title | $\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2412.03234 |