Unipotent representations, theta correspondences, and quantum induction (Q6605403)

In this book the author explains a construction of unitary unipotent representations of orthogonal and metaplectic real reductive groups. This is achieved by using compositions (quantum induction) of even number of theta correspondences and the quantum induction (detailed in Chapter 4), that give irreducible representations when the unipotent orbits are special.\N\NLet \(G\) be a connected semisimple group, \(\mathfrak g = \mathrm{Lie}(G)\) its Lie algebra, \(U(\mathfrak g)\) the corresponding universal enveloping algebra, \(\mathrm{Ann}(\pi) \subset U(\mathfrak g)\) the annihilator of a representation \(\pi\) and \(\mathcal V(\mathrm{Ann}(\pi)) \subset \mathfrak g_\mathbb{C}^*\) the associated variety, stable under the action of the adjoint group \(G_\mathbb{C}^{ad} = \mathrm{Ad}(G)\) in the complex dual space \(\mathfrak{g}_\mathbb{C}^*\) of the complexified Lie algebra \(\mathfrak{g}_\mathbb{C}\). The representation \(\pi\) is irreducible when the associated variety \(\mathcal V(\mathrm{Ann}(\pi))\) is the closure of a single (coadjoint) nilpotent orbit in \(\mathfrak{g}_\mathbb{C}^*\) therefore, to any representation \(\pi\) in the unitary package \(\Pi_u(G)\) one associates a unique complex nilpotent orbit \(\mathcal O\). Conversely the unitary package \(\Pi_u(G)\) is a finite disjoint union of packages \(\Pi_\mathcal O(\mathcal O)\).\N\NThe local theta correspondence \(\theta\) is a one-to-one correspondence between certain infinitesimal equivalence classes of irreducible representations of the dual pair and it induces the correspondence between the associated varieties.\N\NThe main results of the book are explained in Theorem 1.1: for symplectic groups \(G = Sp_{2n}(n)\) or orthogonal groups \(O(p,q)\) with \(p+q\) even, the special rigid nilpotent orbit \(\mathcal O_\mathbf{d}\) of \(G_\mathbb{C}\) parametrized by the partition \(\mathbf d\) corresponding to a Young diagram and its transpose \(\mathbf d^t = (m_1 > m_2 > \dots >m_k )\) then all \(m_i, 1\leq i\leq k\) are even and there is a finite set \(\mathcal N(\mathcal O)\) of irreducible unitary representations of \(G\) such that for any representation \(\pi \in \mathcal N(\mathcal O)\) the associated variety of \(\pi\) is the closure \(cl(\mathcal O_\mathbf{d})\) with exact \(k\)-tuples of half-sum of positive roots characters of \(\mathfrak g\). \N\NTo any special nilpotent orbit \(\mathcal O_\mathbf{D}\) in \(\mathcal O_\mathbf{d}\) there exists (Chapter 7) an irreducible unitary representation \(\pi\) of \(G\) such that \(\mathcal V(\mathrm{Ann}(\pi)) = cl(\mathcal O_\mathbf{d})\) (Theorem 1.2).