Metaplectic covers of p-adic groups and quantum groups at roots of unity
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Bibliographic record
Abstract
We describe the structure of the Whittaker or Gelfand–Graev module on an n-fold metaplectic cover of a p-adic group G at both the Iwahori and spherical level. Following a conjecture of Gaitsgory and Lurie, we express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of G. To do so, we introduce an algebro-combinatorial model for our p-adic Whittaker modules and develop for them a Kazhdan–Lusztig theory involving several generic parameters. These parameters can either be specialized to Gauss sums to recover the p-adic theory or to the grading parameter in the representation theory of quantum groups. As applications, we deduce geometric Casselman–Shalika type results for metaplectic covers as well as a variant of Savin’s local Shimura type correspondences at the Whittaker level.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it