<em>p</em>-Kazhdan-Lusztig Theory

Abstract

We describe a positive characteristic analogue of the Kazhdan-Lusztig basis for the Hecke algebra of a crystallographic Coxeter system, called the <em>p</em>-canonical basis. Using Soergel calculus, we present an algorithm to calculate this basis.<br /> The <em>p</em>-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 <em>p</em>-canonical basis which is only known in small examples.<br /> Even without explicit knowledge of the <em>p</em>-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 <em>p</em>-canonical basis. The equivalence classes with respect to these cell preorders lead to the notion of <em>p</em>-cells. Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic.<br /> The first properties of <em>p</em>-cells that we prove are the following:<br /> Left and right <em>p</em>-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 <em>p</em>-cells. The right <em>p</em>-cells satisfy a similar parabolic compatibility as Kazhdan-Lusztig right cells. We show that any right <em>p</em>-cell preorder relation in a finite, standard parabolic subgroup W_I- induces right <em>p</em>-cell preorder relations in each right W_I-coset.<br /> In an attempt to explicitly describe <em>p</em>-cells in finite type A, we study the consequences of the Kazhdan-Lusztig star-operations for the <em>p</em>-canonical basis. We deduce many interesting relations for the structure coefficients of the <em>p</em>-canonical basis and for the base change coefficients between the <em>p</em>-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 <em>p</em>-cells in finite type A via the Robinson-Schensted correspondence and show that they coincide with Kazhdan-Lusztig cells for all primes <em>p</em>. <br /> A central question is whether Kazhdan-Lusztig cells decompose into <em>p</em>-cells. Based on the star-operations, we can show that the equivalence classes with respect to Vogan's generalized tau-invariant decompose into left <em>p</em>-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 <em>p</em>-cells for <em>p</em> > 2. We show that in type C₃ for <em>p</em>=2, Kazhdan-Lusztig right (resp. two-sided) cells do not decompose into right (resp. two-sided) <em>p</em>-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.<br /> Recently, Achar, Makisumi, Riche and Williamson proved character formulas for the indecomposable tilting modules of a reductive algebraic group in terms of the <em>p</em>-canonical basis. This further fuels interest in the <em>p</em>-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 <em>p</em>-cells in affine Weyl groups and tensor ideals of tilting modules. For this reason, affine Weyl groups of small rank provide intriguing examples of <em>p</em>-cells. We explicitly determine the right <em>p</em>-cell structure in affine types Ã₁, Ã₂ and partly in C^˜₂.