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Galois module structure of Deligne--Lusztig varieties for GL3

  • Adithya Sathyanarayana
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Abstract

Let Y̅(w)→X̅(w) be the ramified Galois cover of compactified Deligne--Lusztig varieties. Our objects of study are the Lefschetz numbers for the action of the finite covering group TwF on the coherent cohomology of the structure sheaf of Y̅(w). The Lefschetz numbers for all elements of TwF determine the virtual TwF-representation given by the alternating sum of these cohomology groups. In the two cases of Deligne--Lusztig surfaces of type A2 and 2A2, we give explicit formulas for the Lefschetz numbers for a generator of the stabilizer of an irreducible component. It follows that the equivariant Euler characteristic of the compactification Y̅(w) is not virtually free as a virtual module over the covering group.

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Cite

BibTeXRIS
BibTeX
@misc{sathyanarayana2026galois,
  title = {{Galois module structure of Deligne--Lusztig varieties for $\mathrmGL_3$}},
  author = {Adithya Sathyanarayana},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40335},
  eprint = {2610.40335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40335},
}
RIS
TY  - UNPB
TI  - Galois module structure of Deligne--Lusztig varieties for GL_3
AU  - Adithya Sathyanarayana
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40335
UR  - https://arxiv.org/abs/2610.40335
AB  - Let Y̅(w)→X̅(w) be the ramified Galois cover of compactified Deligne--Lusztig varieties. Our objects of study are the Lefschetz numbers for the action of the finite covering group T^wF on the coherent cohomology of the structure sheaf of Y̅(w). The Lefschetz numbers for all elements of T^wF determine the virtual T^wF-representation given by the alternating sum of these cohomology groups. In the two cases of Deligne--Lusztig surfaces of type A_2 and ^2A_2, we give explicit formulas for the Lefschetz numbers for a generator of the stabilizer of an irreducible component. It follows that the equivariant Euler characteristic of the compactification Y̅(w) is not virtually free as a virtual module over the covering group.
ER  -