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arXiv · 1510.06606

Hecke algebras for $\mathrm{GL}_n$ over local fields

Valentijn Karemaker
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Abstract

We study the local Hecke algebra $\mathcal{H}_{G}(K)$ for $G = \mathrm{GL}_n$ and $K$ a non-archimedean local field of characteristic zero. We show that for $G = \mathrm{GL}_2$ and any two such fields $K$ and $L$, there is a Morita equivalence $\mathcal{H}_{G}(K) \sim_M \mathcal{H}_{G}(L)$, 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 $G = \mathrm{GL}_n$, there is an algebra isomorphism $\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)$ which is an isometry for the induced $L^1$-norm if and only if there is a field isomorphism $K \cong L$.

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Valentijn Karemaker. 2015-10-22. Hecke algebras for $\mathrm{GL}_n$ over local fields. https://arxiv.org/abs/1510.06606

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