The Geometrical Lemma for Smooth Representations in Natural Characteristic
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914644418887680 |
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| author | Heyer, Claudius |
| author_facet | Heyer, Claudius |
| contents | The Geometrical Lemma is a classical result in the theory of (complex) smooth representations of $p$-adic reductive groups, which helps to analyze the parabolic restriction of a parabolically induced representation by providing a filtration whose graded pieces are (smaller) parabolic inductions of parabolic restrictions. In this article, we establish the Geometrical Lemma for the derived category of smooth mod $p$ representations of a $p$-adic reductive group.
As an important application we compute higher extension groups between parabolically induced representations, which in a slightly different context had been achieved by Hauseux assuming a conjecture of Emerton concerning the higher ordinary parts functor. We also compute the (cohomology functors of the) left adjoint of derived parabolic induction on principal series and generalized Steinberg representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Geometrical Lemma for Smooth Representations in Natural Characteristic Heyer, Claudius Representation Theory Number Theory 11F85, 18G80, 20G25 The Geometrical Lemma is a classical result in the theory of (complex) smooth representations of $p$-adic reductive groups, which helps to analyze the parabolic restriction of a parabolically induced representation by providing a filtration whose graded pieces are (smaller) parabolic inductions of parabolic restrictions. In this article, we establish the Geometrical Lemma for the derived category of smooth mod $p$ representations of a $p$-adic reductive group. As an important application we compute higher extension groups between parabolically induced representations, which in a slightly different context had been achieved by Hauseux assuming a conjecture of Emerton concerning the higher ordinary parts functor. We also compute the (cohomology functors of the) left adjoint of derived parabolic induction on principal series and generalized Steinberg representations. |
| title | The Geometrical Lemma for Smooth Representations in Natural Characteristic |
| topic | Representation Theory Number Theory 11F85, 18G80, 20G25 |
| url | https://arxiv.org/abs/2303.14721 |