Abstract
We describe a positive characteristic analogue of the Kazhdan-Lusztig basis for the Hecke algebra of a crystallographic Coxeter system, called the p-canonical basis. Using Soergel calculus, we present an algorithm to calculate this basis.
The p-canonical basis shares strong positivity properties with the Kazhdan-Lusztig basis (similar to the ones described by the Kazhdan-Lusztig positivity conjectures), but it loses many of its combinatorial properties. For this reason, it is much harder to compute the p-canonical basis which is only known in small examples.
Even without explicit knowledge of the p-canonical basis, one may obtain a first approximate description of the multiplicative structure by studying the left, right or two-sided cell preorder with respect to the p-canonical basis. The equivalence classes with respect to these cell preorders lead to the notion of p-cells. Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic.
The first properties of p-cells that we prove are the following:
Left and right p-cells are related by taking inverses, just like for Kazhdan-Lusztig cells. The set of elements with a fixed left descent set decomposes into right p-cells. The right p-cells satisfy a similar parabolic compatibility as Kazhdan-Lusztig right cells. We show that any right p-cell preorder relation in a finite, standard parabolic subgroup WI- induces right p-cell preorder relations in each right WI-coset.
In an attempt to explicitly describe p-cells in finite type A, we study the consequences of the Kazhdan-Lusztig star-operations for the p-canonical basis. We deduce many interesting relations for the structure coefficients of the p-canonical basis and for the base change coefficients between the p-canonical and the Kazhdan-Lusztig basis. These allow us to show that the right star-operations preserve the left cell preorder. Moreover, we explicitly describe the p-cells in finite type A via the Robinson-Schensted correspondence and show that they coincide with Kazhdan-Lusztig cells for all primes p.
A central question is whether Kazhdan-Lusztig cells decompose into p-cells. Based on the star-operations, we can show that the equivalence classes with respect to Vogan's generalized tau-invariant decompose into left p-cells. Garfinkle showed that Vogan's generalized tau-invariant gives a complete invariant of Kazhdan-Lusztig left cells in finite types B and C. From this, we deduce that Kazhdan-Lusztig left cells in finite types B and C decompose into left p-cells for p > 2. We show that in type C3 for p=2, Kazhdan-Lusztig right (resp. two-sided) cells do not decompose into right (resp. two-sided) p-cells. Moreover, we give a criterion that reduces the question about the decomposition of Kazhdan-Lusztig cells to the minimal elements with respect to the weak right Bruhat order.
Recently, Achar, Makisumi, Riche and Williamson proved character formulas for the indecomposable tilting modules of a reductive algebraic group in terms of the p-canonical basis. This further fuels interest in the p-canonical basis because the determination of the tilting characters is a long-standing open problem in modular representation theory. These new character formulas and the geometric Satake equivalence provide two connections between right p-cells in affine Weyl groups and tensor ideals of tilting modules. For this reason, affine Weyl groups of small rank provide intriguing examples of p-cells. We explicitly determine the right p-cell structure in affine types Ã1, Ã2 and partly in C˜2.
