Abstract
In this article we propose a vanishing conjecture for a certain class of \(\ell \)-adic complexes on a reductive group G which can be regarded as a generalization of the acyclicity of the Artin–Schreier sheaf. We show that the vanishing conjecture contains, as a special case, a conjecture of Braverman and Kazhdan on the acyclicity of \(\rho \)-Bessel sheaves (Braverman and Kazhdan in Geom Funct Anal I:237–278, 2002). Along the way, we introduce a certain class of Weyl group equivariant \(\ell \)-adic complexes on a maximal torus called central complexes and relate the category of central complexes to the Whittaker category on G. We prove the vanishing conjecture in the case when \(G={{\text {GL}}}_n\).
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Notes
The group \({\mathrm W}_\chi \) plays an important role in the study of representations of finite reductive groups and character sheaves (see, e.g., [14]).
An question raised by V.Drinfeld according to [10, Section 1.5].
This holds in a more general situation when the neutral component of H is unipotent.
In loc. cit. the authors assumed \(\mathcal {F}\) is irreducible with support a \({\mathrm W}\)-stable sub-torus in T. This is because the proof makes use of the isomorphism \({{\text {Ind}}}_{T\subset B}^G(\mathcal {F})\simeq {{\text {Ind}}}_{T\subset B}^G(w^*\mathcal {F})\) which was constructed only for those \(\mathcal {F}\) satisfying the assumption above (see [4, Theorem 2.5(3)]). Now, with Proposition 3.2, the same argument works for arbitrary \({\mathrm W}\)-equivariant perverse sheaves on T.
Note that \({\mathcal C}(T)_{\ell ,\chi _f}\) is stable under the \({\mathrm W}_{\chi _f}\)-action.
Indeed, we have \(w^*{\mathcal R}_\chi \simeq w^*m_\chi ^*{\mathcal R}^{uni}_\chi \simeq m_{w^{-1}\chi }^*w^*{\mathcal R}^{uni}_\chi \simeq m_{w^{-1}\chi }^*{\mathcal R}^{uni}_{w^{-1} \chi }={\mathcal R}_{w^{-1}\chi }\), where we use the observation that \(w^{-1}{\mathrm W}_\chi w={\mathrm W}_{w^{-1}\chi }\) and hence \(w^*{\mathcal R}^{uni}_\chi \simeq {\mathcal R}^{uni}_{w^{-1}\chi }\).
Note that \({\mathcal R}_\theta \) has finite support and hence the inverse of the Mellin transform is well-defined by (4.3).
As mentioned in loc. cit. the geometry of the conjugation action of Mirabolic has been described by Bernstein in [1].
Note that \(u_{m-1}u_1xu_{m-1}^{-1}=u_1u_{m-1}xu_{m-1}^{-1}\).
There is minor mistake in the computation of \(u_1u_mxu_m^{-1}\) in loc. cit.: is should be \(u_1u_mxu_m^{-1}=\begin{bmatrix} x_{F} &{} y+v_1x_E+v_{m-1}x_E-x_Fv_{m-1}\\ 0 &{} x_{E} \end{bmatrix}\). The same proof goes through after this minor correction.
In fact, the Lemma remains true without the assumption \(G={{\text {GL}}}_n\), see [7, Lemma 7.5].
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Acknowledgements
The paper is inspired by the lectures given by Ginzburg and Ngô on their works [8, 10]. I thank Roman Bezrukavnikov, Tanmay Deshpande, Victor Ginzburg, Sam Gunningham, Augustus Lonergan, David Nadler, Ngô Bao Châu, Lue Pan, and Zhiwei Yun for useful discussions. I thank the anonymous referee for valuable comments. I am grateful for the support of NSF grant DMS-1702337 and DMS-2001257, and the S. S. Chern Foundation.
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Chen, TH. A vanishing conjecture: the \({{\text {GL}}}_n\) case. Sel. Math. New Ser. 28, 13 (2022). https://doi.org/10.1007/s00029-021-00726-2
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DOI: https://doi.org/10.1007/s00029-021-00726-2