## On endomorphism algebras of Gelfand-Graev representations

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- by Tzu-Jan Li PDF
- Represent. Theory
**27**(2023), 80-114

## Abstract:

For a connected reductive group $G$ defined over $\mathbb {F}_q$ and equipped with the induced Frobenius endomorphism $F$, we study the relation among the following three $\mathbb {Z}$-algebras: (i) the $\mathbb {Z}$-model $\mathsf {E}_G$ of endomorphism algebras of Gelfand-Graev representations of $G^F$; (ii) the Grothendieck group $\mathsf {K}_{G^\ast }$ of the category of representations of $G^{\ast F^\ast }$ over $\overline {\mathbb {F}_q}$ (Deligne-Lusztig dual side); (iii) the ring $\mathsf {B}_{G^\vee }$ of the scheme $(T^\vee /\!\!/ W)^{F^\vee }$ over $\mathbb {Z}$ (Langlands dual side). The comparison between (i) and (iii) is motivated by recent advances in the local Langlands program.## References

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## Additional Information

**Tzu-Jan Li**- Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), 4 place Jussieu, 75252 Paris cedex 05, France
- Email: tzu-jan.li@imj-prg.fr
- Received by editor(s): December 10, 2021
- Received by editor(s) in revised form: June 7, 2022, and June 22, 2022
- Published electronically: April 26, 2023
- Additional Notes: This project received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754362.
- © Copyright 2023 Copyright by Tzu-Jan Li
- Journal: Represent. Theory
**27**(2023), 80-114 - MSC (2020): Primary 20C33
- DOI: https://doi.org/10.1090/ert/627
- MathSciNet review: 4580514