New directions in representation theory. (Q1931671)

Let \(G\) be a finite Chevalley group with Borel subgroup \(B\) and Weyl group \(W\), and let \(\mathfrak g\) be a semisimple Lie algebra with universal enveloping algebra \(U(\mathfrak g)\). The Iwahori-Hecke algebra \(H(G,B)\) of \(G\) with respect to \(B\) can be thought of as a deformation of the group algebra of \(W\). In a similar way, the \(+\)-part of the quantized enveloping algebra \(U_v^+(\mathfrak g)\) associated with \(\mathfrak g\) can be seen as a deformation of the \(+\)-part of \(U(\mathfrak g)\). The author shows how results about \(H(G,B)\) and \(U_v^+(\mathfrak g)\) can be used to obtain new results about the representation theory of the classical objects \(G\) and \(\mathfrak g\). The article is very well written and gives a nice introduction to these subjects, with a good list of references that invite further reading.