Theta-lifting and geometric quantization for \(\mathrm {GL}(n,\mathbb C)\) (Q1758138)

Consider a complex reductive Lie group \(G\), and let \((\chi, V_\chi)\) be an irreducible representation of the finite group \(G(\lambda)/G(\lambda)_0\). The Vogan conjecture on quantization asserts that there is attached to \(\chi\) an irreducible unitary representation \(\pi(\lambda,\chi)\) of \(G\). The space of \(K\)-finite vectors of \(\pi(\lambda,\chi)\) is isomorphic to the space of the algebraic sections of a certain bundle \(\mathcal{V}_\chi\). The author studies the Vogan conjecture in the case of \(GL(n,\mathbb{C})\). In this case, he obtains a relationship between the induction of orbits and the theta lifting of representations. The obtained representations can be lifted from the trivial representation of small groups.