Unipotent representations, theta correspondences, and quantum induction
type: Item
instance of: scholarly article
title: Unipotent representations, theta correspondences, and quantum induction
author: Hongyu He
publication date: 2024-09-13T00:00:00Z
review text: 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…
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 7913532
zbMATH Keywords: orthogonal groups
zbMATH Keywords: metaplectic groups
zbMATH Keywords: unipotent orbit
zbMATH Keywords: unipotent representation
zbMATH Keywords: quantum induction
zbMATH Keywords: theta correspondence
MaRDI profile type: Publication
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Unipotent representations, theta correspondences, and quantum induction (published in)
published in: Memoirs of the American Mathematical Society

Unipotent representations, theta correspondences, and quantum induction (DOI)
DOI: 10.1090/MEMO/1496