Proves a generic categorical local Langlands correspondence for many quasi-split reductive p-adic groups via a fully faithful functor from generic Bernstein blocks to ind-coherent sheaves on L-parameter moduli stacks, generalizing the GL_n case.
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Program to prove categorical local Langlands conjecture via compatibility hypothesis for GL_n, induction on Levi subgroups for general groups, plus new spectral finiteness, duality, and admissible ind-coherent sheaf results.
For n greater than or equal to 3 and sufficiently generic weights, the universal supersingular representation of GL_n(k) is non-admissible and of infinite length.
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A generic categorical local Langlands correspondence for quasi-split reductive groups
Proves a generic categorical local Langlands correspondence for many quasi-split reductive p-adic groups via a fully faithful functor from generic Bernstein blocks to ind-coherent sheaves on L-parameter moduli stacks, generalizing the GL_n case.
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The categorical local Langlands conjecture
Program to prove categorical local Langlands conjecture via compatibility hypothesis for GL_n, induction on Levi subgroups for general groups, plus new spectral finiteness, duality, and admissible ind-coherent sheaf results.
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Non-admissibility of some universal supersingular representations
For n greater than or equal to 3 and sufficiently generic weights, the universal supersingular representation of GL_n(k) is non-admissible and of infinite length.