Some applications of CHEVIE to the theory of algebraic groups. (Q2919608)

The main goal of this article is to provide three examples of how the computer algebra package CHEVIE has played a key role in recent developments in the theory of algebraic groups and their representation theory. The package has been used directly to verify special cases of theorems as well as indirectly through examples which have suggested new general results.NEWLINENEWLINE Let \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\) of prime characteristic \(p\), and let \(W\) denote the associated Weyl group. The first application of CHEVIE discussed is the recent work of \textit{G. Lusztig} [Represent. Theory 15, 494-530 (2011; Zbl 1263.20045)]. In that work, under the assumption that the prime \(p\) is good relative to the root system of \(G\), Lusztig constructed a surjective map from conjugacy classes of \(W\) to unipotent classes of \(G\). In addition to explaining the result and the role played by CHEVIE, the author also provides some computations suggesting that the result may also hold in bad characteristic. Lusztig's proof involves the notion of an excellent element in a conjugacy class. In the more general setting of \(W\) being a finite Coxeter group, the author presents a new result on the general existence of excellent elements.NEWLINENEWLINE The second example is in the context of irreducible characters of \(W\), information about which plays a key role in the representation theory of \(G\). In much earlier work, \textit{G. Lusztig} [Indag. Math. 41, 323-335 (1979; Zbl 0435.20021)], Lusztig introduced the notion of \(a\)-invariants associated to irreducible characters of \(W\). These were originally defined through the use of an associated Iwahori-Hecke algebra. In this paper, the author shows how \(a\)-invariants can be defined directly in terms of the irreducible characters themselves. The author then discusses Lusztig's families of irreducible characters.NEWLINENEWLINE The last example presented is in connection with Green functions on a finite group of Lie type \(G^F\) (the fixed points under the Frobenius morphism \(F\)). The author begins by discussing an algorithm on irreducible characters of \(W\) which can be implemented in CHEVIE. The author gives examples of this and then discusses the connection with Green functions. Finally, the author discusses how this all plays a role in Lusztig's aforementioned map on conjugacy classes, and conjectures a map from conjugacy classes of \(W\) (in the general setting of an arbitrary finite Coxeter group) to Lusztig's families of irreducible characters of \(W\).