The Derived l-Modular Unipotent Block of p-adic GLn

Berry, Rose (2025) The Derived l-Modular Unipotent Block of p-adic GLn. Doctoral thesis, University of East Anglia.

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Abstract

Representations of p-adic groups have deep applications to number theoretic questions via the conjectured Langlands correspondence. While the complex representation theory is well-understood in a wide variety of cases, the l-modular theory for l ̸= p is still largely unsolved. For the case of GLn, a block decomposition is known, as is a description of the irreducible representations in each block, but the full structure of the blocks remains open. Recent developments in the categorification of the Langlands correspondence have suggested that it is in fact the study of the derived category that is of central interest.

We obtain, for the derived unipotent l-modular block Dbfg(H1(G)) of G = GLn(F) for a p-adic field F, an explicit classical generator V . In the process, we also obtain an analogous result in the case of G = GLn(k) for k a finite field. The proof proceeds in two parts. Firstly, we show that another representation Q, which plays a key role in the underived l-modular representation theory, is a classical generator. This requires establishing various finiteness properties for the unipotent block B1(G), namely that it is Noetherian and possesses a certain subcategory B′1(G) of finite global dimension. Secondly, we relate the two classical generators using the theory of irreducible l-modular representations of GLn(k).

Using this classical generator, we give a (triangulated, linear) equivalence from Dbfg(H1(G)) to the perfect complexes over a dg Schur algebra. This is a derived l-modular analogue for the result in the complex setting that the unipotent block is equivalent to modules over the Iwahori-Hecke algebra. We conclude with a composition formula for the dg Schur algebra.

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Item Type:	Thesis (Doctoral)
Faculty \ School:	Faculty of Science > School of Engineering, Mathematics and Physics
Depositing User:	Kitty Laine
Date Deposited:	04 Nov 2025 13:27
Last Modified:	04 Nov 2025 13:27
URI:	https://ueaeprints.uea.ac.uk/id/eprint/100893
DOI:

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