On the intertwining differential operators between vector bundles over the real projective space of dimension two

Kubo, Toshihisa; Ørsted, Bent

doi:10.5802/jolt.1421

¹Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto, 612-8577, Japan
²Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark

Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 93-132

Abstract

The main objective of this paper is twofold. One is to classify and construct $\operatorname{SL}(3,\mathbb{R})$-intertwining differential operators between vector bundles over the real projective space $\mathbb{RP}^2$. It turns out that two kinds of operators appear. We call them Cartan operators and PRV operators. The second objective is then to study the representations realized on the kernel of those operators both in the smooth and holomorphic setting. A key machinery is the BGG resolution. In particular, by exploiting some results of Davidson–Enright–Stanke and Enright–Joseph, the irreducible unitary highest weight modules of $\operatorname{SU}(1,2)$ at the (first) reduction points are classified by the image of Cartan operators and kernel of PRV operators.

Article information
Cite this article

Received: 2025-03-27
Accepted: 2025-08-11
Published online: 2026-06-17

DOI: 10.5802/jolt.1421

Classification: 22E46, 17B10
Keywords: intertwining differential operator, generalized Verma module, Cartan component, PRV component, BGG resolution, unitary highest weight module

Author's affiliations:

Kubo, Toshihisa ¹ ; Ørsted, Bent ²

¹ Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto, 612-8577, Japan
² Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark

Kubo, Toshihisa; Ørsted, Bent. On the intertwining differential operators between vector bundles over the real projective space of dimension two. Journal of Lie Theory, Special issue dedicated to the memory of Joseph A. Wolf, Volume 36 (2026) no. 1, pp. 93-132. doi: 10.5802/jolt.1421

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