## Unipotent representations as a categorical centre

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- Represent. Theory
**19**(2015), 211-235 Request permission

## Abstract:

Let $G(F_q)$ be the group of rational points of a split connected reductive group $G$ over the finite field $F_q$. In this paper we show that the category of representations of $G(F_q)$ which are finite direct sums of unipotent representations in a fixed two-sided cell is equivalent to the centre of a certain monoidal category of sheaves on the flag manifold of $G\times G$. We also consider a version of this for nonsplit groups.## References

- A. A. Beĭlinson, J. Bernstein, and P. Deligne,
*Faisceaux pervers*, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR**751966** - Roman Bezrukavnikov, Michael Finkelberg, and Victor Ostrik,
*Character $D$-modules via Drinfeld center of Harish-Chandra bimodules*, Invent. Math.**188**(2012), no. 3, 589–620. MR**2917178**, DOI 10.1007/s00222-011-0354-3 - P. Deligne and G. Lusztig,
*Representations of reductive groups over finite fields*, Ann. of Math. (2)**103**(1976), no. 1, 103–161. MR**393266**, DOI 10.2307/1971021 - Ben Elias and Geordie Williamson,
*The Hodge theory of Soergel bimodules*, Ann. of Math. (2)**180**(2014), no. 3, 1089–1136. MR**3245013**, DOI 10.4007/annals.2014.180.3.6 - Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik,
*On fusion categories*, Ann. of Math. (2)**162**(2005), no. 2, 581–642. MR**2183279**, DOI 10.4007/annals.2005.162.581 - George Lusztig,
*Characters of reductive groups over a finite field*, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR**742472**, DOI 10.1515/9781400881772 - George Lusztig,
*Character sheaves. II, III*, Adv. in Math.**57**(1985), no. 3, 226–265, 266–315. MR**806210**, DOI 10.1016/0001-8708(85)90064-7 - George Lusztig,
*Character sheaves. II, III*, Adv. in Math.**57**(1985), no. 3, 226–265, 266–315. MR**806210**, DOI 10.1016/0001-8708(85)90064-7 - George Lusztig,
*Character sheaves. IV*, Adv. in Math.**59**(1986), no. 1, 1–63. MR**825086**, DOI 10.1016/0001-8708(86)90036-8 - George Lusztig,
*Coxeter groups and unipotent representations*, Astérisque**212**(1993), 191–203. - G. Lusztig,
*Character sheaves on disconnected groups. VII*, Represent. Theory**9**(2005), 209–266. MR**2133758**, DOI 10.1090/S1088-4165-05-00278-5 - G. Lusztig,
*Character sheaves on disconnected groups. IX*, Represent. Theory**10**(2006), 353–379. MR**2240705**, DOI 10.1090/S1088-4165-06-00315-3 - G. Lusztig,
*Character sheaves on disconnected groups. X*, Represent. Theory**13**(2009), 82–140. MR**2495562**, DOI 10.1090/S1088-4165-09-00348-3 - G. Lusztig,
*Cells in affine Weyl groups and tensor categories*, Adv. Math.**129**(1997), no. 1, 85–98. MR**1458414**, DOI 10.1006/aima.1997.1645 - G. Lusztig,
*Hecke algebras with unequal parameters*, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR**1974442**, DOI 10.1090/crmm/018 - G. Lusztig,
*On certain varieties attached to a Weyl group element*, Bull. Inst. Math. Acad. Sin. (N.S.)**6**(2011), no. 4, 377–414. MR**2907958** - G. Lusztig,
*Restriction of a character sheaf to conjugacy classes*, Bull. Soc. Math. Roum., Tome**58**(2015), no. 3, pp. 297–309. - G. Lusztig,
*Truncated convolution of character sheaves*, Bull. Inst. Math. Acad. Sin. (N.S.)**10**(2015), no. 1, 1–72. MR**3309291** - G. Lusztig,
*Non-unipotent character sheaves as a categorical centre*, arxiv: 1506.04598. - Michael Müger,
*From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors*, J. Pure Appl. Algebra**180**(2003), no. 1-2, 159–219. MR**1966525**, DOI 10.1016/S0022-4049(02)00248-7 - Wolfgang Soergel,
*Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen*, J. Inst. Math. Jussieu**6**(2007), no. 3, 501–525 (German, with English and German summaries). MR**2329762**, DOI 10.1017/S1474748007000023

## Additional Information

**G. Lusztig**- Affiliation: Department of Mathematics, M.I.T., Cambridge, Massachusetts 02139
- MR Author ID: 117100
- Email: gyuri@math.mit.edu
- Received by editor(s): December 5, 2014
- Received by editor(s) in revised form: August 26, 2015
- Published electronically: October 28, 2015
- Additional Notes: Supported in part by National Science Foundation grant 1303060.
- © Copyright 2015 American Mathematical Society
- Journal: Represent. Theory
**19**(2015), 211-235 - MSC (2010): Primary 20G99
- DOI: https://doi.org/10.1090/ert/468
- MathSciNet review: 3416310