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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Intersection complexes and unramified $L$-factors
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by Yiannis Sakellaridis and Jonathan Wang
J. Amer. Math. Soc. 35 (2022), 799-910
DOI: https://doi.org/10.1090/jams/990
Published electronically: October 5, 2021

Abstract:

Let $X$ be an affine spherical variety, possibly singular, and $\mathsf L^+X$ its arc space. The intersection complex of $\mathsf L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified $L$-functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and $L$-monoids. In this paper, we compute this intersection complex for the large class of those spherical $G$-varieties whose dual group is equal to $\check G$, and the stalks of its nearby cycles on the horospherical degeneration of $X$. We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional $\check G$-representation determined by the set of $B$-invariant valuations on $X$. We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of $\mathsf L^+X$ as a ratio of local $L$-values for a large class of spherical varieties.
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Bibliographic Information
  • Yiannis Sakellaridis
  • Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
  • MR Author ID: 796283
  • ORCID: 0000-0003-3924-286X
  • Email: sakellar@jhu.edu
  • Jonathan Wang
  • Affiliation: Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • MR Author ID: 912751
  • Email: jwang4@perimeterinstitute.ca
  • Received by editor(s): February 8, 2021
  • Received by editor(s) in revised form: July 14, 2021
  • Published electronically: October 5, 2021
  • Additional Notes: The first author was supported by NSF grants DMS-1801429 and DMS-1939672, and by a stipend to the IAS from the Charles Simonyi Endowment. The second author was supported by NSF grant DMS-1803173
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 35 (2022), 799-910
  • MSC (2020): Primary 22E57, 11F67; Secondary 14D24, 14M27, 43A85
  • DOI: https://doi.org/10.1090/jams/990
  • MathSciNet review: 4433079