Abstract. We describe a positive characteristic analogue of the Kazhdan-Lusztig basis of the Hecke algebra of a crystallographic Coxeter system and investigate some of its properties. Using Soergel calculus we describe an algorithm to calculate this basis. We outline some known or expected applications in modular representation theory. We conclude by giving several examples.

We investigate the relation between the Garside normal form for positive braids and the 2-braid group defined by Rouquier. Inspired by work of Brav and Thomas we show that the Garside normal form is encoded in the action of the 2-braid group on a certain categorified left cell module. This allows us to deduce the faithfulness of the 2braid group in finite type. We also give a new proof of Paris' theorem that the canonical map from the generalized braid monoid to its braid group is injective in arbitrary type. 2000 Mathematics subject classification. Primary 20F36, 20J05Section 5 After introducing the important notion of an anchor we prove our main results. AcknowledgementI would like to thank my advisor, Geordie Williamson, for his support and encouragement. I am grateful to Hanno Becker for very valuable discussions. The Hecke algebra and cellsLet (W, S) be a Coxeter system, i.e. W is a group together with a set of generators S admitting a particularly nice presentation: W = ⟨s ∈ S sts . . . ms,t terms = tst . . . ms,t terms , s 2 = 1⟩ where m s,t ⩾ 2 is the order of st for s ≠ t ∈ S. For w ∈ W denote the left (resp. right) descent set of w by L(w) = {s ∈ S sw < w} (resp. R(w) = {s ∈ S ws < w}). Let H (W,S) be the corresponding Hecke algebra over Z[v, v −1 ] which we will also denote by H if there is no danger of confusion. Denote by {H w } w∈W the standard and by {H w } w∈W the Kazhdan-Lusztig basis in Soergel's normalization (see [Soe97]). Since we are not working with the usual normalization, let us state the relations for the standard basis: H 2 s = (v −1 − v)H s + 1 for all s ∈ S, H s H t H s . . . ms,t terms = H t H s H t . . . ms,t terms for all s ≠ t ∈ S.

Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see [JW17]). Our main technical tool are the star-operations introduced by Kazhdan-Lusztig in [KL79] which have interesting numerical consequences for the p-canonical basis. As an application, we explicitely describe p-cells in finite type A (i.e. for symmetric groups) using the Robinson-Schensted correspondence. Moreover, we show that Kazhdan-Lusztig cells in finite types B and C decompose into p-cells for p > 2.

No abstract