A transversality property of a derivation of the universal enveloping algebra \(U(k)\), for \(SO(n,1)\) and \(SU(n,1)\)
Let \(G_ 0\) be a non-compact connected real semisimple Lie group with finite center, \(K_ 0\) a maximal compact subgroup of \(G_ 0\), \(G_ 0=K_ 0 A_ 0 N_ 0\) an Iwasawa decomposition, \(M_ 0\) the centralizer of \(A_ 0\) in \(K_ 0\), \(g_ 0\), \(k_ 0\), \(n_ 0\), \(m_ 0\) the Lie algebras of \(G_ 0\), \(K_ 0\), \(N_ 0\) and \(M_ 0\) respectively, \(g_ 0=k_ 0\oplus p_ 0\) a Cartan decomposition and \(g\), \(k\), \(n\), \(m\), \(p\) the complexified algebras. To study \(U(g)^ K\) the centralizer of \(K_ 0\) in the universal enveloping algebra of \(g\), B. Kostant suggested to consider the projection map \({\mathcal P}:U(g)\to U(k)\otimes U(a)\) relative to the decomposition \(U(g)=(U(k)\otimes U(a))\oplus U(g)n\). \textit{J. Lepowsky} [Trans. Am. Math. Soc. 208, 193-218 (1975; Zbl 0311.17003)] showed that \({\mathcal P}:U(g)^ K\to U(k)^ M\otimes U(a)\) is an injective anti- homomorphism. The purpose of the paper under review is to show that for \(G_ 0=SO(n,1)\) or \(SU(n,1)\), \(U(k)^ P\cap U(k)m^ +=0\), where \(m^ +\) is a specific subalgebra of \(m\) and \(P\) is a subgroup of \(Ad(k)\) related to \(p\). This result allows to complete the description of \({\mathcal P}(U(g)^ K)\) for \(SO(n,1)\) and \(SU(n,1)\) conjectured by \textit{J. Tirao} [Notas Soc. Mat. Chile 9, No. 1, 53-62 (1990; Zbl 0739.22013)].
scientific article; zbMATH DE number 60081