Chen, Zhe

Green Functions and Higher Deligne-Lusztig Characters

Doc. Math. 23, 2027-2041 (2018)
DOI: 10.25537/dm.2018v23.2027-2041
Communicated by Dan Ciubotaru

Summary

We give a generalisation of the character formula of Deligne-Lusztig representations from the finite field case to the truncated formal power series case. Motivated by this generalisation, we give a definition of Green functions for these local rings, and prove some basic properties along the lines of the finite field case, like a summation formula. As an application we show that the higher Deligne-Lusztig characters and Gérardin's characters agree at regular semisimple elements.

Mathematics Subject Classification

20G99, 14F20, 20C15

Keywords/Phrases

Deligne-Lusztig theory, reductive group schemes, Green functions, discrete valuation rings

References

  • 1. Michel Broué and Jean Michel. Blocs et séries de Lusztig dans un groupe réductif fini. J. Reine Angew. Math., 395:56--67, 1989. zbl 0654.20048; MR0983059.
  • 2. Roger W. Carter. Finite groups of Lie type. Wiley Classics Library. John Wiley & Sons Ltd., Chichester, 1993. Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication. MR1266626.
  • 3. Zhe Chen. Generic character sheaves on reductive groups over a finite ring. Preprint 2016. arxiv 1604.02016.
  • 4. Zhe Chen. On generalised Deligne--Lusztig constructions. 2017. PhD Thesis. Durham University.
  • 5. Zhe Chen. On the inner products of some Deligne--Lusztig-type representations. Accepted by International Mathematics Research Notices, 2018. DOI 10.1093/imrn/rny195; arxiv 1711.10208.
  • 6. Zhe Chen and Alexander Stasinski. The algebraisation of higher Deligne--Lusztig representations. Selecta Math. (N.S.), 23(4):2907--2926, 2017. DOI 10.1007/s00029-017-0349-z; zbl 06796869; MR3703469; arxiv 1604.01615.
  • 7. Pierre Deligne and George Lusztig. Representations of reductive groups over finite fields. Ann. of Math. (2), 103(1):103--161, 1976. DOI 10.2307/1971021; zbl 0336.20029; MR0393266.
  • 8. François Digne and Jean Michel. Foncteurs de Lusztig et caractères des groupes linéaires et unitaires sur un corps fini. J. Algebra, 107(1):217--255, 1987. DOI 10.1016/0021-8693(87)90087-1; zbl 0622.20034; MR0883883.
  • 9. François Digne and Jean Michel. Representations of finite groups of Lie type, volume 21 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991. zbl 0815.20014; MR1118841.
  • 10. Paul Gérardin. Construction de séries discrètes $p$-adiques. Lecture Notes in Mathematics, Vol. 462. Springer-Verlag, Berlin-New York, 1975. DOI 10.1007/BFb0082161; zbl 0302.22002; MR0396859.
  • 11. George Lusztig. Representations of finite Chevalley groups, volume 39 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8--12, 1977. zbl 0418.20037; MR0518617.
  • 12. George Lusztig. Some remarks on the supercuspidal representations of $p$-adic semisimple groups. In Automorphic forms, representations and $L$-functions, Part 1, pages 171--175. Amer. Math. Soc., 1979. zbl 0421.22009; MR0546595.
  • 13. George Lusztig. Representations of reductive groups over finite rings. Represent. Theory, 8:1--14, 2004. DOI 10.1090/S1088-4165-04-00232-8; zbl 1085.20029; MR2048585; arxiv math/0208037.
  • 14. Alexander Stasinski. Unramified representations of reductive groups over finite rings. Represent. Theory, 13:636--656, 2009. DOI 10.1090/S1088-4165-09-00350-1; zbl 1220.20043; MR2558788; arxiv 0808.1351.

Affiliation

Chen, Zhe
Department of Mathematics, Shantou University, Guangdong, China

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