Subag's Jantzen quotient conjecture

Let σ\sigma be a real structure of an algebraic family of Harish–Chandra pairs (g,K)(\boldsymbol{\mathfrak{g}},\boldsymbol{K}), let FM~(g,K)μ\mathcal{F}\in\widetilde{\mathcal{M}}(\boldsymbol{\mathfrak{g}},\boldsymbol{K})_{\mu}, and let [α:β]RP1[\alpha:\beta]\in\mathbb{R}\mathbb{P}^{1}. When a nonzero rational intertwining operator from F\mathcal{F} to its σ\sigma-twisted dual exists, it determines a Jantzen filtration, and J~μ,[α:β](F)\widetilde{J}_{\mu,[\alpha:\beta]}(\mathcal{F}) denotes the unique Jantzen quotient containing μ\mu. The notation Jμ,[α:β](F)J_{\mu,[\alpha:\beta]}(\mathcal{F}) denotes the unique composition factor containing μ\mu. Subag's Jantzen quotient conjecture. For every [α:β]RP1[\alpha:\beta]\in\mathbb{R}\mathbb{P}^{1} and every FM~(g,K)μ\mathcal{F}\in\widetilde{\mathcal{M}}(\boldsymbol{\mathfrak{g}},\boldsymbol{K})_{\mu} for which F[α:β]\mathcal{F}|_{[\alpha:\beta]} is reducible,

J~μ,[α:β](F)Jμ,[α:β](F).\widetilde{J}_{\mu,[\alpha:\beta]}(\mathcal{F})\simeq J_{\mu,[\alpha:\beta]}(\mathcal{F}).

The claim would provide a method for computing the composition factor used in the correspondence from Jantzen filtrations; it is stated as an additional conjecture and is not resolved in the paper.

Sources & referencesView supporting material

Primary source

Eyal Subag, “The algebraic Mackey-Higson bijections”, arXiv:1706.05616 (2017).

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