are known when the characteristic p of k is much bigger than the Coxeter number of G, by Henning Andersen, Jens Jantzen, and Wolfgang Soergel (proving Lusztig's conjecture in that case). Their character formula for p large is based on the Kazhdan–Lusztig polynomials, which are combinatorially complex. For any prime p, Simon Riche and Geordie Williamson conjectured the irreducible characters of a reductive group in terms of the p-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable. Non-split reductive groups As discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the classical groups are: Every nondegenerate quadratic form q over a field k determines a reductive group G = SO(q). Here G is simple if q has dimension n at least 3, since is isomorphic to SO(n) over an algebraic closure . The k-rank of G is equal to the Witt index of q (the maximum dimension of an isotropic subspace over k). So the simple group G is split over k if and only if q has the maximum possible Witt index, . Every central simple algebra A over k determines a reductive group G = SL(1,A), the kernel of the reduced norm on the group of units A* (as an algebraic group over k). The degree of A means the square root of the dimension of A as a k-vector space. Here G is simple if A has degree n at least 2, since is isomorphic to SL(n) over . If A has index r (meaning that A is isomorphic to the matrix algebra Mn/r(D) for a division algebra D of degree r over k), then the k-rank of G is (n/r) − 1. So the simple group G is split over k if and only if A is a matrix algebra over k. As a result, the problem of classifying reductive groups over k essentially includes the problem of classifying all quadratic forms over k or all central simple algebras over k. These problems are easy for k algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions. A reductive group over a field k is called isotropic if it has k-rank greater than 0 (that is,