Hecke algebras for GL n over local fields

Valentijn Karemaker

doi:10.1007/s00013-016-0974-3

Hecke algebras for GL _n over local fields

Valentijn Karemaker^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We study the local Hecke algebra H_G(K) for G= GL _n and K a non-archimedean local field of characteristic zero. We show that for G= GL ₂ and any two such fields K and L, there is a Morita equivalence H_G(K) ∼ _MH_G(L) , by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for G= GL _n, there is an algebra isomorphism H_G(K) ≅ H_G(L) which is an isometry for the induced L¹-norm if and only if there is a field isomorphism K≅ L.

Original language	English
Pages (from-to)	341-353
Number of pages	13
Journal	Archiv der Mathematik
Volume	107
Issue number	4
DOIs	https://doi.org/10.1007/s00013-016-0974-3
Publication status	Published - 1 Oct 2016

Keywords

Hecke algebras
Local fields

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abstract = "We study the local Hecke algebra HG(K) for G= GL n and K a non-archimedean local field of characteristic zero. We show that for G= GL 2 and any two such fields K and L, there is a Morita equivalence HG(K) ∼ MHG(L) , by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for G= GL n, there is an algebra isomorphism HG(K) ≅ HG(L) which is an isometry for the induced L1-norm if and only if there is a field isomorphism K≅ L.",

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AB - We study the local Hecke algebra HG(K) for G= GL n and K a non-archimedean local field of characteristic zero. We show that for G= GL 2 and any two such fields K and L, there is a Morita equivalence HG(K) ∼ MHG(L) , by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for G= GL n, there is an algebra isomorphism HG(K) ≅ HG(L) which is an isometry for the induced L1-norm if and only if there is a field isomorphism K≅ L.

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KW - Local fields

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ER -

Hecke algebras for GL _n over local fields

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