Two dual pair methods in the study of generalized Whittaker models for irreducible highest weight modules (Q2751530)

Let \(G\) be a connected simple linear group of Hermitian type with Lie algebra \(\mathfrak{g}_0\) and \(K\) a maximal compact subgroup of \(G\). Let \(\tau\) be an irreducible finite-dimensional representation of \(K\) and \(L(\tau)\) the corresponding irreducible \((\mathfrak{g},K)\)-module. In this note Yamashita gives an overview of his algebraic and geometric approach to the study of generalized Whittaker models for \(L(\tau)\). Especially, in the final Section 5 he gives the complete embedding scheme of the generalized Whittaker models for \(L(\tau)\) into the generalized Gelfand-Graev representation \((\Gamma_m,C^\infty(G;\eta_m))\) of \(G\) attached to the nilpotent \(G\)-orbit \({\mathcal{O}'}_m\), \(0\leq m\leq r=\text{ rank}G\), employing two kinds of dual pair methods. The first dual pair method -- the kernel theorem -- yields that \(\text{Hom}_{\mathfrak{g},K}(L(\tau),C^\infty(G;\eta_m))\) is the kernel \(\mathcal{Y}(\tau)\) of a differential operator of gradient type, and the second dual pair method -- the oscillator representation of the reductive dual pair \((G,G')\) -- describes the structure of \(\mathcal{Y}(\tau)\) more explicitly.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].