Long-root $\SL_2$ dual pairs in split exceptional groups: Compact types, cone geometry, and theta correspondence

Gordan Savin

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arXiv · 2611.08716

Long-root $\SL_2$ dual pairs in split exceptional groups: Compact types, cone geometry, and theta correspondence

Gordan Savin

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Abstract

The paper establishes a case of Howe duality for an exceptional theta correspondence using the ping-pong of types method. To that end, one needs to compute lifts of types of maximal compact subgroups. A brand new ingredient enters here: The graded module of the minimal representations is the coordinate ring of an affine cone whose projectivization is a Grassmannian. Via the filtered Nakayama lemma the problem of lifting types becomes a problem in algebraic geometry: compute certain linear sections of the Grassmannian.

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BibTeX RIS

Gordan Savin. 2026-11-05. Long-root $\SL_2$ dual pairs in split exceptional groups: Compact types, cone geometry, and theta correspondence. https://arxiv.org/abs/2611.08716

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