The enigma of the missing invariants on the nilradical of a Borel
The paper under review deals with the open problem to determine the semicentre \(Sz({\mathfrak p})\) of the enveloping algebra \(U({\mathfrak p})\) of a parabolic subalgebra \(\mathfrak p\) of a complex semisimple Lie algebra \(\mathfrak g\) and, in particular, to show that it is a polynomial algebra. In [\textit{F. Fauquant-Millet} and \textit{A. Joseph}, Transform. Groups 6, No. 2, 125--142 (2001; Zbl 1018.17008)] it was shown that the quantum analogue \(Sz_q({\mathfrak p})\) in \(U_q({\mathfrak p})\) is a polynomial algebra. Surprisingly, the standard specialization \(q=1\) does not recover the whole \(Sz({\mathfrak p})\) from \(Sz_q({\mathfrak p})\). In the present paper the author uses an elementary construction due to Kostant to recover most of the generators of \(Sz({\mathfrak b})\), where \(\mathfrak b\) is the standard Borel subalgebra. An extension of the construction gives all the generators except for one generator in type \(E_8\).
scientific article; zbMATH DE number 2153819