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An induction theorem for groups acting on trees


Author: Martin H. Weissman
Journal: Represent. Theory 23 (2019), 205-212
MSC (2010): Primary 20G25, 20E08, 22E50
DOI: https://doi.org/10.1090/ert/526
Published electronically: May 29, 2019
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Abstract: If $ G$ is a group acting on a locally finite tree $ X$, and $ \mathscr {S}$ is a $ G$-equivariant sheaf of vector spaces on $ X$, then its compactly-supported cohomology is a representation of $ G$. Under a finiteness hypothesis, we prove that if $ H_c^0(X, \mathscr {S})$ is an irreducible representation of $ G$, then $ H_c^0(X, \mathscr {S})$ arises by induction from a vertex or edge stabilizing subgroup.

If $ \boldsymbol {\mathrm {G}}$ is a reductive group over a nonarchimedean local field $ F$, then Schneider and Stuhler realize every irreducible supercuspidal representation of $ G = \boldsymbol {\mathrm {G}}(F)$ in the degree-zero cohomology of a $ G$-equivariant sheaf on its reduced Bruhat-Tits building $ X$. When the derived subgroup of $ \boldsymbol {\mathrm {G}}$ has relative rank one, $ X$ is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.


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Additional Information

Martin H. Weissman
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064
Email: weissman@ucsc.edu

DOI: https://doi.org/10.1090/ert/526
Received by editor(s): October 30, 2018
Received by editor(s) in revised form: December 10, 2018
Published electronically: May 29, 2019
Additional Notes: The Simons Foundation Collaboration Grant #426453 supported this work
Article copyright: © Copyright 2019 American Mathematical Society