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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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An induction theorem for groups acting on trees
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by Martin H. Weissman
Represent. Theory 23 (2019), 205-212
Published electronically: May 29, 2019


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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Bibliographic Information
  • Martin H. Weissman
  • Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064
  • MR Author ID: 718173
  • Email:
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Represent. Theory 23 (2019), 205-212
  • MSC (2010): Primary 20G25, 20E08, 22E50
  • DOI:
  • MathSciNet review: 3955571