Algebraic constructions of first-reduction-point and exceptional unitary highest weight modules

The 1983 classification of unitary highest weight modules over Hermitian symmetric Lie groups by Enright-Howe-Wallach and Jakobsen has recently been re-proved by elementary means (Dirac inequality plus PRV tensoring) for the classical families sp(2n,R), so*(2n), su(p,q), and, by a 2025 sequel, for the remaining families so(2,n), e6(-14), e7(-25); the unitary set has also been described at fixed infinitesimal character for all six families in the integral case. The sequel resolves the principal deferred item of the source paper; the surviving open core of the source paper's further-work program is: (i) explicit algebraic constructions of the first-reduction-point unitary modules in the so*(2n) subcases where unitarity is currently certified only by a continuity argument on the Shapovalov form; and (ii) elementary constructions of the discrete-part unitary modules in the exceptional cases e6(-14) and e7(-25), where the authors state that PRV products do not realize them and theta correspondence is used instead. Extending the fixed-infinitesimal-character description to the non-integral characters of su(p,q) and so*(2n) is a further optional direction beyond the sources' stated program, not part of the open core.

Background

Let $G$ be a connected simple Lie group of Hermitian type, meaning that the symmetric space $G/K$ ($K$ a maximal compact subgroup) carries a $G$-invariant complex structure. On the Lie algebra level this is equivalent to the complexified Lie algebra decomposing as $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}^+\oplus\mathfrak{p}^-$ with $\mathfrak{p}^\pm$ abelian, and the simple real Lie algebras of Hermitian type are exactly $\mathfrak{sp}(2n,\mathbb{R})$, $\mathfrak{so}^*(2n)$, $\mathfrak{su}(p,q)$, $\mathfrak{so}(2,n)$, $\mathfrak{e}_{6(-14)}$ (the Lie algebra of the exceptional symmetric space E III) and $\mathfrak{e}_{7(-25)}$ (E VII). A $(\mathfrak{g},K)$-module (Harish-Chandra module) is called a highest weight module if it is generated by a weight vector annihilated by all positive root spaces of $\mathfrak{g}$; its highest weight $\lambda$ is then dominant and integral for $\mathfrak{k}$, and the module is the unique irreducible quotient $L(\lambda)$ of the generalized Verma module $N(\lambda)=U(\mathfrak{g})\otimes_{U(\mathfrak{k}\oplus\mathfrak{p}^+)}F_\lambda$, where $F_\lambda$ is the irreducible $\mathfrak{k}$-module of highest weight $\lambda$. Unitary highest weight modules include the holomorphic discrete series and small representations such as the Segal-Shale-Weil (metaplectic) representation, and they are basic building blocks in the harmonic analysis of Hermitian symmetric spaces. The infinitesimal character, represented by the Harish-Chandra parameter $\Lambda=\lambda+\rho$ modulo the Weyl group ($\rho$ being half the sum of the positive roots), is the standard invariant by which irreducible modules are organized.

Which $L(\lambda)$ are unitary was classified in 1983 by Enright-Howe-Wallach and independently by Jakobsen. The answer is organized along lines: fixing the highest weight, one studies the line $\lambda_0+z\zeta$ parameterized by $z\in\mathbb{R}$, where $\zeta$ is the element orthogonal to the compact roots normalized against the maximal noncompact root. For $z$ sufficiently negative, $N(\lambda_0+z\zeta)$ is a holomorphic discrete series module, irreducible and unitary; the smallest $a$ for which $N(\lambda_0+a\zeta)$ is reducible is the first reduction point, and the unitary points on the line form a continuous part $z<a$ together with finitely many discrete points, one of which may sit at $z=a$. Enright-Howe-Wallach proved unitarity on the continuous part and at $z=a$ by a continuity argument on the Shapovalov form (the invariant Hermitian form on $N(\lambda)$, positive definite for $z<a$ and positive semidefinite at $z=a$ by passage to the limit), and constructed the discrete modules via theta correspondence, i.e. the transfer of representations between the two members of a dual pair of subgroups realized on a common Weil representation.

The source paper, by Pandžić-Prlić-Souček-Tuček, re-proves the classification for $\mathfrak{sp}(2n,\mathbb{R})$, $\mathfrak{so}^*(2n)$ and $\mathfrak{su}(p,q)$ by elementary means. Its two main tools are, first, the Dirac inequality of Parthasarathy, a necessary condition for unitarity of the form $\|(\lambda-s)^++\rho_{\mathfrak{k}}\|^2\ge\|\lambda+\rho\|^2$ for every Schmid $K$-submodule with lowest weight $-s$ of the symmetric algebra $S(\mathfrak{p}^-)$ (here $\rho_{\mathfrak{k}}$ is half the sum of the positive compact roots and $(\cdot)^+$ denotes the $\mathfrak{k}$-dominant rearrangement), used at full strength through refined versions proved in the authors' Dirac-inequality papers; and second, PRV products, the technique going back to Parthasarathy-Ranga Rao-Varadarajan of realizing extreme $K$-types $F$ inside tensor products of unitary modules and taking the generated submodule $U(\mathfrak{p}^-)F$, which is then an irreducible unitary highest weight module. Starting from basic modules obtained from the Weil representation, this tensoring construction produces all discrete unitary modules in most cases. The same authors, with Savin, completed the remaining families $\mathfrak{so}(2,n)$, $\mathfrak{e}_{6(-14)}$ and $\mathfrak{e}_{7(-25)}$ in 2025, and the unitary set has since been described at fixed infinitesimal character for all six families in the integral case. What the elementary account still lacks is the subject of the problem below: in certain $\mathfrak{so}^*(2n)$ subcases the first-reduction-point modules are certified unitary only by the continuity argument (Lemma 1.11 of the source paper), and in the exceptional cases the discrete modules are reached only through theta correspondence. The non-integral infinitesimal characters of $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$ are also untreated; that extension is noted below as an optional direction beyond the sources' stated program, not as part of the open core. For $SU(1,2)$, Kubo and Ørsted have recently exhibited the shape such completions can take, realizing the first-reduction-point modules as kernels and images of explicit intertwining differential operators built from Bernstein-Gelfand-Gelfand (BGG) resolutions, i.e. algebraic resolutions of highest weight modules by direct sums of Verma-type modules.

Problem Statement

Complete the constructive, elementary account of unitary highest weight modules for the Hermitian symmetric groups, removing the remaining non-algebraic steps. The classification itself (which $L(\lambda)$ are unitary) is closed, so the question is not which modules are unitary but whether each unitary module can be produced and certified by a direct algebraic construction. Concretely, with the notation of the Background:

1. For the $\mathfrak{so}^*(2n)$ lines $\lambda_0+z\zeta$ in the subcases where the tensoring/PRV technique of the source paper produces all discrete unitary points except the first reduction point $z=a$ — of which the Case-1 lines with $\lambda_1>\lambda_2$, on which the unique discrete point is the endpoint of the continuous part, are the instances the source singles out, the target being the whole class rather than these instances alone — give a direct algebraic construction of $L(\lambda_0+a\zeta)$, together with a proof of unitarity and irreducibility that does not pass through the continuity argument of Lemma 1.11 of the source paper.

2. Give an elementary algebraic construction of the discrete-part unitary highest weight modules of the exceptional Hermitian groups $E_{6(-14)}$ and $E_{7(-25)}$, extending or replacing the tensoring/PRV program of the source papers — basic modules from Segal-Shale-Weil/Weil representations, tensor products and PRV products. The sequel classification (arXiv:2412.06317, Section 1) states explicitly that PRV products do not construct the discrete part in these cases; the existing certifications — the $E_6$ dual-pair exceptional theta correspondence of Section 6 of that paper and, for $E_7$, the Enright-Howe-Wallach second-tensor-power argument cited there — are the current non-elementary state of the art and do not provide such a construction.

Parts 1 and 2 above jointly constitute one objective: a fully constructive, algebraic account of the unitary highest weight modules for all six Hermitian symmetric families, with no step certified only by continuity or theta correspondence.

Optional extension (not part of the completion criterion): the 2026 fixed-infinitesimal-character treatment of $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$ deliberately restricts to integral infinitesimal characters, with a stated rationale, and announces no follow-up. Describing the unitary set at the non-integral infinitesimal characters of these two families is a meaningful further direction, but solving it is not required to close the problem.

References

1. Pavle Pandžić, Ana Prlić, Vladimír Souček, Vít Tuček, 'On the classification of unitary highest weight modules', Japanese Journal of Mathematics (2026), DOI 10.1007/s11537-026-2451-5; arXiv:2305.15892, https://arxiv.org/abs/2305.15892.
2. Pavle Pandžić, Ana Prlić, Gordan Savin, Vladimír Souček, Vít Tuček, 'On the classification of unitary highest weight modules in the exceptional cases', Journal of Algebra 684 (2025) 524-562, DOI 10.1016/j.jalgebra.2025.07.018; arXiv:2412.06317, https://arxiv.org/abs/2412.06317.
3. Pavle Pandžić, Ana Prlić, Vladimír Souček, Vít Tuček, 'Unitary highest weight modules for $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$ with fixed integral infinitesimal character', arXiv:2604.20566 (2026), https://arxiv.org/abs/2604.20566.
4. Toshihisa Kubo, Bent Ørsted, 'On the intertwining differential operators between vector bundles over the real projective space of dimension two', arXiv:2503.22323 (2025), https://arxiv.org/abs/2503.22323.
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6. H. P. Jakobsen, 'Hermitian symmetric spaces and their unitary highest weight modules', Journal of Functional Analysis 52 (1983) 385-412, DOI 10.1016/0022-1236(83)90076-9, https://doi.org/10.1016/0022-1236(83)90076-9.
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10. Pavle Pandžić, Ana Prlić, Vladimír Souček, Vít Tuček, 'Dirac inequality for highest weight Harish-Chandra modules I', Mathematical Inequalities and Applications 26 (2023) 233-265, DOI 10.7153/mia-2023-26-17, https://doi.org/10.7153/mia-2023-26-17.