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Frobenius semisimplicity for convolution morphisms

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Abstract

This article concerns properties of mixed \(\ell \)-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson–Bernstein–Deligne–Gabber is valid over finite fields. We prove our conjecture for (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields. As a crucial tool, we develop a new schematic theory of big cells for loop groups. With suitable reformulations, the main results are valid over any algebraically closed ground field.

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Notes

  1. We have normalized the intersection complex \(\mathcal {IC}_X\) of an integral variety X so that if X is smooth, then \(\mathcal {IC}_X \cong {\overline{\mathbb Q}_\ell }_X\); this is not a perverse sheaf; the perverse sheaf counterpart is \(IC_X=\mathcal {IC}_X [\dim X]\).

  2. Equivalently, \(\mathcal {F}\) is pure of weight w and each \(\mathcal {H}^i(\mathcal F)\) is pointwise pure of weight \(w+i\) in the sense of  [3, p.  126].

  3. The reason for using \(t^{-1}\) instead of t here is to make (3.4) and (3.5) true.

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Acknowledgements

We gratefully acknowledge discussions with Patrick Brosnan, Pierre Deligne, Xuhua He, Robert Kottwitz, Mircea Mustaţă, George Pappas, Timo Richarz, Jason Starr, Geordie Williamson, and Zhiwei Yun.

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Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Mark Andrea de Cataldo

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    Thomas J. Haines

  3. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    Li Li

Authors
  1. Mark Andrea de Cataldo
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  2. Thomas J. Haines
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  3. Li Li
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Corresponding author

Correspondence to Thomas J. Haines.

Additional information

The research of M. A. de Cataldo was partially supported by NSF Grant DMS-1301761 and by a Grant from the Simons Foundation (\(\#\)296737 to Mark Andrea de Cataldo). The research of T. Haines was partially supported by NSF Grant DMS-1406787. The research of L. Li was partially supported by the Oakland University URC Faculty Research Fellowship Award.

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de Cataldo, M.A., Haines, T.J. & Li, L. Frobenius semisimplicity for convolution morphisms. Math. Z. 289, 119–169 (2018). https://doi.org/10.1007/s00209-017-1946-4

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