Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 21 min ago
PreprintCC0arXiv

Tate cohomology of unipotent cuspidal representations of finite groups of Lie type

  • Sabyasachi Dhar
Open at sourcePDFDOICite

Abstract

Let p be a prime and k be an algebraic closure of the finite field with p elements. Let G be a connected reductive algebraic group over k, and let F:Gβ†’ G be a Frobenius morphism, defining an 𝔽q-rational structure on G, where q is a power of p. Let β„“ be a prime different from p and β„“β‰  2. Let Ξ  be an β„“-adic cuspidal unipotent representation of GFβ„“ such that Π≃ Ξ Οƒ, where Οƒ is a generator the Galois group Gal(𝔽qβ„“/𝔽q). In this paper, we prove the irreducibility of the zeroth Tate cohomology group HΜ‚0(Οƒ,β„’) and vanishing of the first Tate cohomology group HΜ‚1(Οƒ,β„’) for any GFβ„“rtimesGal(𝔽qβ„“/𝔽q)-stable lattice β„’ in Ξ , using the multiplicity-one property of a progenerator constructed via the generalized Gelfand-Graev representations and Harish-Chandra induction.

Related

  1. Special unipotent representations of real classical groups: construction and unitarity
    Dan Barbasch, Jiajun Ma, Binyong Sun +1
    Journal of the American Mathematical SocietyPeer-reviewedLicence not statedFull text23 citedAlgebra & number theoryMathematical physicsOpenAlex
  2. Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups
    Qingchun Hao, Yang Yang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Fixed point properties and cohomology of Banach representations of arithmetic groups
    Saar Bader, Shaked Bader, Uri Bader +1
    arXivPreprintarXiv licenceFull textAlgebra & number theoryGeometry & topologyarXiv
  4. Residual finiteness and cuspidal cohomology of Picard modular surfaces
    Richard M. Hill
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. On the cuspidal cohomology of Iwahori congruence subgroups of SL(3, β„€)
    Zachary Porat
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{dhar2026tate,
  title = {{Tate cohomology of unipotent cuspidal representations of finite groups of Lie type}},
  author = {Sabyasachi Dhar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15037},
  eprint = {2610.15037},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15037},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Tate cohomology of unipotent cuspidal representations of finite groups of Lie type
AU  - Sabyasachi Dhar
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15037
UR  - https://arxiv.org/abs/2610.15037
AB  - Let p be a prime and k be an algebraic closure of the finite field with p elements. Let G be a connected reductive algebraic group over k, and let F:Gβ†’ G be a Frobenius morphism, defining an 𝔽_q-rational structure on G, where q is a power of p. Let β„“ be a prime different from p and β„“β‰  2. Let Ξ  be an β„“-adic cuspidal unipotent representation of G^F^β„“ such that Π≃ Ξ ^Οƒ, where Οƒ is a generator the Galois group Gal(𝔽_q^β„“/𝔽_q). In this paper, we prove the irreducibility of the zeroth Tate cohomology group HΜ‚^0(Οƒ,β„’) and vanishing of the first Tate cohomology group HΜ‚^1(Οƒ,β„’) for any G^F^β„“rtimesGal(𝔽_q^β„“/𝔽_q)-stable lattice β„’ in Ξ , using the multiplicity-one property of a progenerator constructed via the generalized Gelfand-Graev representations and Harish-Chandra induction.
ER  -