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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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Integral structures in the $p$-adic holomorphic discrete series
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by Elmar Grosse-Klönne PDF
Represent. Theory 9 (2005), 354-384 Request permission

Abstract:

For a local non-Archimedean field $K$ we construct ${\mathrm {GL}}_{d+1}(K)$-equivariant coherent sheaves ${\mathcal V}_{{\mathcal O}_K}$ on the formal ${\mathcal O}_K$-scheme ${\mathfrak X}$ underlying the symmetric space $X$ over $K$ of dimension $d$. These ${\mathcal V}_{{\mathcal O}_K}$ are ${\mathcal O}_K$-lattices in (the sheaf version of) the holomorphic discrete series representations (in $K$-vector spaces) of ${\mathrm {GL}}_{d+1}(K)$ as defined by P. Schneider. We prove that the cohomology $H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K})$ vanishes for $t>0$, for ${\mathcal V}_{{\mathcal O}_K}$ in a certain subclass. The proof is related to the other main topic of this paper: over a finite field $k$, the study of the cohomology of vector bundles on the natural normal crossings compactification $Y$ of the Deligne-Lusztig variety $Y^0$ for ${\mathrm {GL}}_{d+1}/k$ (so $Y^0$ is the open subscheme of ${\mathbb P}_k^d$ obtained by deleting all its $k$-rational hyperplanes).
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Additional Information
  • Elmar Grosse-Klönne
  • Affiliation: Mathematisches Institut der Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
  • Email: klonne@math.uni-muenster.de
  • Received by editor(s): October 2, 2004
  • Received by editor(s) in revised form: March 5, 2005
  • Published electronically: April 19, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Represent. Theory 9 (2005), 354-384
  • MSC (2000): Primary 14G22
  • DOI: https://doi.org/10.1090/S1088-4165-05-00259-1
  • MathSciNet review: 2133764