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Tits Groups of Iwahori-Weyl Groups and Presentations of Hecke Algebras

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Abstract

Let G be a connected reductive group over a non-archimedean local field F and I be an Iwahori subgroup of G(F). Let \(I_n\) is the n-th Moy-Prasad filtration subgroup of I. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with \(I_n\)-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups G that split over an unramified extension of F. The first main result of this paper is a presentation of the Hecke algebra \(\mathcal {H}(G(F),I_n)\), generalizing the previous work of Iwahori-Matsumoto on the affine Hecke algebras. For split \(GL_n\), Howe gave a refined presentation of the Hecke algebra \(\mathcal {H}(G(F),I_n)\). To generalize such a refined presentation to other groups requires the existence of some nice lifting of the Iwahori-Weyl group W to G(F). The study of a certain nice lifting of W is the second main motivation of this paper, which we discuss below. In 1966, Tits introduced a certain subgroup of \(G(\textbf{k})\) for any algebraically closed field \(\textbf{k}\), which is an extension of the finite Weyl group \(W_0\) by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of the elements in the finite Weyl group. The “Tits group” \(\mathcal {T}\) for the Iwahori-Weyl group W is a certain subgroup of G(F), which is an extension of the Iwahori-Weyl group W by an elementary abelian 2-group. The second main result of this paper is a construction of Tits group \(\mathcal {T}\) for W when G splits over an unramified extension of F. As a consequence, we generalize Howe’s presentation to such groups. We also show that when G is ramified over F, such a group \(\mathcal {T}\) of W may not exist.

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Acknowledgements

The authors would like to thank T. Haines, G. Lusztig and M.-F. Vigneras for useful discussions. R.G.  would like to thank the Infosys foundation for their support through the Young Investigator award. X.H. is partially supported by a start-up grant and by funds connected with Choh-Ming Chair at CUHK, and by Hong Kong RGC grant 14300220. We thank the referee for the detailed and useful comments and suggestions.

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  1. Department of Mathematics, Indian Institute of Science, Bengaluru, Karnataka, 560012, India

    Radhika Ganapathy

  2. The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, China

    Xuhua He

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Ganapathy, R., He, X. Tits Groups of Iwahori-Weyl Groups and Presentations of Hecke Algebras. Transformation Groups (2023). https://doi.org/10.1007/s00031-023-09810-7

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