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On the classification of unitary highest weight modules

Pavle Pandžić, Ana Prlić, Vladimír Souček, Vít Tuček

TL;DR

The paper provides a direct, elementary proof of the Enright–Howe–Wallach/Jacobsen classification of unitary highest weight modules for the universal covers of $Sp(2n,\mathbb{R})$, $SO^{*}(2n)$, and $SU(p,q)$, grounded in Parthasarathy's Dirac inequality and PRV components. It develops a uniform tensoring framework using the spin module and Weil representations to construct discrete series via PRV products, and analyzes the continuous part along Harish-Chandra lines through a first reduction point. A detailed description of unitary modules with fixed integral or half-integral infinitesimal characters is given, employing line/translation-cone techniques and combinatorial models (Young diagrams) to enumerate unitary parameters. The results yield explicit decompositions of unitary modules as PRV-products of basic building blocks, and reveal a cone-structure for the discrete spectrum, with plans to treat remaining cases in PPSST. Overall, the work provides a streamlined, representation-theoretic route to the full classification and explicit constructions of unitary highest weight modules for these Hermitian symmetric settings.

Abstract

In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$. We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.

On the classification of unitary highest weight modules

TL;DR

The paper provides a direct, elementary proof of the Enright–Howe–Wallach/Jacobsen classification of unitary highest weight modules for the universal covers of , , and , grounded in Parthasarathy's Dirac inequality and PRV components. It develops a uniform tensoring framework using the spin module and Weil representations to construct discrete series via PRV products, and analyzes the continuous part along Harish-Chandra lines through a first reduction point. A detailed description of unitary modules with fixed integral or half-integral infinitesimal characters is given, employing line/translation-cone techniques and combinatorial models (Young diagrams) to enumerate unitary parameters. The results yield explicit decompositions of unitary modules as PRV-products of basic building blocks, and reveal a cone-structure for the discrete spectrum, with plans to treat remaining cases in PPSST. Overall, the work provides a streamlined, representation-theoretic route to the full classification and explicit constructions of unitary highest weight modules for these Hermitian symmetric settings.

Abstract

In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups and . We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.
Paper Structure (6 sections, 24 theorems, 170 equations)

This paper contains 6 sections, 24 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Some preliminary facts
  3. Classification for $\mathfrak{sp}(2n,\mathbb{R})$
  4. Classification for $\mathfrak{su}(p,q)$, $p\leq q$
  5. Classification for $\mathfrak{so}^*(2n)$
  6. Unitary modules for $\mathfrak{sp}(2n,\mathbb{R})$ with fixed integral or half integral infinitesimal character

Key Result

Proposition 1.7

With the above notation, let $F_\mu,F_\nu$ be finite-dimensional $\mathfrak{k}$-modules with highest weights $\mu,\nu$. let $\nu^-$ be the lowest weight of $F_\nu$, and let $\tau=(\mu+\nu^-)^+$. Then $F_\tau$ appears in $F_\mu\otimes F_\nu$, with multiplicity one. Moreover, for any $F_\sigma$ appear with equality attained if and only if $\sigma=\tau$.

Theorems & Definitions (48)

  • Proposition 1.7: PRV
  • Corollary 1.9
  • proof
  • Lemma 1.11
  • Proposition 2.1
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 38 more