A geometric realization of the principal block of quantum group modules at roots of unity is established as microsheaves on an affine Springer fiber, together with a wild-ramification geometric Langlands equivalence to coherent sheaves on the dual Springer resolution.
arXiv:2310.07006
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Under standard assumptions the GK-dimension of Hecke eigenspaces in mod p cohomology of D^x for inner forms D of GL2 over totally real fields unramified at p is computed, including the division-algebra-at-p case, and the same arguments simplify a prior theorem for the matrix-algebra case.
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Affine Springer fiber and the small quantum group
A geometric realization of the principal block of quantum group modules at roots of unity is established as microsheaves on an affine Springer fiber, together with a wild-ramification geometric Langlands equivalence to coherent sheaves on the dual Springer resolution.
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Gelfand--Kirillov dimension and mod $p$ cohomology for inner forms of $\mathrm{GL}_2$
Under standard assumptions the GK-dimension of Hecke eigenspaces in mod p cohomology of D^x for inner forms D of GL2 over totally real fields unramified at p is computed, including the division-algebra-at-p case, and the same arguments simplify a prior theorem for the matrix-algebra case.