Generic Cartan invariants for Chevalley groups (Q1084183)

Let G be a semi-simple simply connected algebraic group defined over K, the algebraic closure of a finite field of order p. Let \(\sigma\) be an endomorphism of G such that the set of fixed points \(G_{\sigma}\) is finite. Let X be the weight lattice of G, \(X^+\) the set of dominant weights. \(\sigma\) acts also on X and \(X_{\sigma}\) the subset of \(X^+\) consisting of those weights whose coordinates in terms of the fundamental dominant weights are all less than those of \(\sigma\delta\), \(\delta\) being half the sum of the positive roots. For \(\lambda \in X^+\), let \(\lambda_ 0\in X_{\sigma}\) such that \(\lambda \equiv \lambda_ 0 (mod \sigma X)\) and \(\lambda_ 1\) be defined by \(\sigma \lambda_ 1=\lambda -\lambda_ 0\). Let \(\chi\) (\(\lambda)\) and \(\chi_ p(\lambda)\) be the formal characters of the Weyl module and irreducible KG-module of highest weight \(\lambda\), respectively. In a previous paper, the author gives formulas which shows that the \(\chi\) (\(\mu)\) decompose into \(\chi (\lambda_ 1)^{\sigma}\chi_ p(\lambda_ 0)\) as \(\lambda\) varies over \(X^+\) [J. Algebra 82, 255-274 (1983; Zbl 0532.20023)]. These are reformulations of formulas originally given by Jantzen. For \(\lambda\) enough inside \(X^+\) the decomposition pattern becomes uniform. The author modifies these formulas so that the pattern is uniform for all \(\lambda\) by allowing \(\lambda\) and \(\mu\) to range over all of X. He then shows precisely how these invariants correspond to generic Cartan invariants of the finite groups. This gives an alternative and usually easier way of computing these invariants. The rank 2 case can be computed graphically and he gives examples.