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Journal of the American Mathematical Society

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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
Published electronically: October 5, 2021


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:
  • Jonathan Wang
  • Affiliation: Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • MR Author ID: 912751
  • Email:
  • 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:
  • MathSciNet review: 4433079