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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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An induction theorem for groups acting on trees
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by Martin H. Weissman PDF
Represent. Theory 23 (2019), 205-212 Request permission


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
  • 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