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Affine quiver Schur algebras and p-adic \({\textit{GL}}_n\)

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Abstract

In this paper we consider the (affine) Schur algebra which arises as the endomorphism algebra of certain permutation modules for the Iwahori–Matsumoto Hecke algebra. This algebra describes, for a general linear group over a p-adic field, a large part of the unipotent block over fields of characteristic different from p. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of \({\text {GL}}_2({\mathbb {Q}}_5)\) in characteristic 3.

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Acknowledgements

We thank Günter Harder, David Helm, Peter Scholze, Shaun Stevens and Torsten Wedhorn for useful discussions on the background material of this paper, Ruslan Maksimau and Andrew Mathas for sharing their insight into Hecke algebras, and the referees for their advice. This work was partly supported by the DFG Grant SFB/TR 45 and EPSRC Grant EP/K011782/1.

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Correspondence to Catharina Stroppel.

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Miemietz, V., Stroppel, C. Affine quiver Schur algebras and p-adic \({\textit{GL}}_n\). Sel. Math. New Ser. 25, 32 (2019). https://doi.org/10.1007/s00029-019-0474-y

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Mathematics Subject Classification

  • 20C08
  • 33D80
  • 20G43
  • 14M15
  • 22E57