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The cohomology of Deligne–Lusztig varieties for the general linear group

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Abstract

We propose two inductive approaches for determining the cohomology of Deligne–Lusztig varieties in the case of \(G={{\mathrm{GL}}}_n\). The first one uses Demazure compactifications and analyzes the corresponding Mayer–Vietoris spectral sequence. This allows us to give an inductive formula for the Tate twist \(-1\) contribution of the cohomology of a DL-variety. The second approach relies on considering more generally DL-varieties attached to hypersquares in the Weyl group. Here we give explicit formulas for the cohomology of height-one elements.

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Notes

  1. Here the symbol \(\hat{s_i}\) means as usual that \(s_i\) is deleted from the above expression.

  2. Indeed let \(i^G_P \longrightarrow i^G_Q \oplus i^G_R\) be an injective map. Then we may suppose w.l.o.g. that \(i^G_P \longrightarrow i^G_Q\) is injective, as well. We may extend the first map to an injection \(i^G_Q \longrightarrow i^G_Q \oplus i^G_R\) and the graph contains \(i^G_P.\)

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Acknowledgements

I want to thank Olivier Dudas for his numerous remarks on this paper. I am grateful for the invitation to Paris and all the discussions with François Digne and Jean Michel. Finally I thank Roland Huber, Michael Rapoport and Markus Reineke for their support. Dedicated to Michael Rapoport on the occasion of his 65th birthday.

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  1. Fachbereich C - Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaussstr. 20, 42119, Wuppertal, Germany

    Sascha Orlik

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Dedicated to Michael Rapoport on the occasion of his 65th birthday

Appendix

Appendix

Here we give summarizing tables of the cohomology of DL-varieties with respect to Weyl group elements of full support in \({{\mathrm{GL}}}_3\) and \({{\mathrm{GL}}}_4\). We list only representatives of cyclic shift classes.

\({{\mathrm{GL}}}_3\)

\(H^*_c(X(w))\)

(1, 2, 3)

\(j_{(1,1,1)}[-2] \oplus j_{(2,1)}(-1)[-3] \oplus j_{(3)}(-2)[-4]\)

(1, 3)

\(j_{(1,1,1)}[-3] \oplus j_{(3)}(-3)[-6]\)

\({{\mathrm{GL}}}_4\)

\(H^*_c(X(w))\)

(1, 2, 3, 4)

\(j_{(1,1,1,1)}[-3] \oplus j_{(2,1,1)}(-1)[-4] \oplus j_{(3,1)}(-2)[-5] \oplus j_{(4)}(-3)[-6] \)

(1, 2, 4)

\(j_{(1,1,1,1)}[-4] \oplus j_{(2,2)}(-2)[-5] \oplus j_{(4)}(-4)[-8]\)

(1, 3)(2, 4)

\(j_{(1,1,1,1)}[-4] \oplus j_{(2,2)}(-1)[-4] \oplus j_{(2,1,1)}(-2)[-5] \oplus j_{(3,1)}(-2)[-5] \oplus \)

 

\(j_{(2,2)}(-3)[-6] \oplus j_{(4)}(-4)[-8]\)

(1, 3, 2, 4)

\(j_{(1,1,1,1)}[-5] \oplus j_{(2,2)}(-2)[-6] \oplus j_{(2,1,1)}(-2)[-6] \oplus j_{(2,2)}(-3)[-7] \oplus \)

 

\(j_{(3,1)}(-3)[-7] \oplus j_{(4)}(-5)[-10]\)

(1, 4)

\(j_{(1,1,1,1)}[-5] \oplus j_{(2,1,1)}(-1)[-5] \oplus j_{(2,2)}(-2)[-6] \oplus j_{(2,2)}(-3)[-7] \oplus \)

 

\(j_{(3,1)}(-4)[-8] \oplus j_{(4)}(-5)[-10]\)

(1, 4)(2, 3)

\(j_{(1,1,1,1)}[-6] \oplus j_{(2,1,1)}(-2)[-7] \oplus j_{(2,2)}(-3)[-8]^2 \oplus j_{(3,1)}(-4)[-9] \oplus j_{(4)}(-6)[-12]\)

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Orlik, S. The cohomology of Deligne–Lusztig varieties for the general linear group. Res Math Sci 5, 13 (2018). https://doi.org/10.1007/s40687-018-0131-7

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