Multiplicity free actions of quantum groups and generalized Howe duality
The theory of tensor invariants for the action of classical groups has been extended and unified by \textit{R. Howe} [Trans. Am. Math. Soc. 313, 539-570 (1989; Zbl 0674.15021)]. In particular, he proved a duality theorem for a pair of commuting subalgebras. Howe later gave a treatment of classical invariant theory involving multiplicity free actions of reductive groups on algebraic varieties [The Schur Lectures (1992), Bar Ilan Univ. (1995; Zbl 0821.00011), \textit{I. Piatetski-Shapiro} and \textit{S. Gelbart}, Editors]. A pair of mutually centralizing closed reductive subgroups of a symplectic group is called a dual reductive pair. Howe conjectured a bijection involving inverse images of a dual reductive pair in the double cover of the symplectic group. This was proved by Howe over an Archimedean field [J. Am. Math. Soc. 2, No.~3, 535--552 (1989; Zbl 0716.22006)] and by \textit{J. L. Waldsburger} over a non-Archimedean field [Isr. Math. Conf. Proc. 2, 267--324 (1989; Zbl 0722.22009)]. Howe duality for quantum groups has been studied by various authors. In the paper under review, the authors study multiplicity free actions of quantized univeral enveloping algebras on associative algebras and the generalized Howe dualities arising from them. Let \(\mathfrak g\) and \({\mathfrak g}'\) be distinct Lie algebras from among \(so_{2n},so_{2n+1},sp_{2n}\) and \(gl_n\). They first construct a module algebra over the Hopf algebra \(U_q{\mathfrak g}\otimes U_g({\mathfrak g}')\) which decomposes as an infinite direct sum of irreducible submodules, each appearing with multiplicity one. This leads to a generalized Howe duality between \(U_q({\mathfrak g})\) and \(U_q({\mathfrak g}')\). This generalizes results of \textit{R. B. Zhang} for the orthogonal and symplectic Lie algebras [Proc. Am. Math. Soc. 131, No.~9, 2681--2692 (2003; Zbl 1068.17005)]. The case of \(U_q(sp_{2n})\) and \(U_q(gl_n)\) is also considered in the context of quantum harmonic representations.
scientific article; zbMATH DE number 2057651