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Published online by Cambridge University Press: 11 November 2024
We compute the Jantzen filtration of a $\mathcal {D}$-module on the flag variety of
$\operatorname {\mathrm {SL}}_2(\mathbb {C})$. At each step in the computation, we illustrate the
$\mathfrak {sl}_2(\mathbb {C})$-module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the
$\mathcal {D}$-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson and Bernstein.
Communicated by Oded Yacobi