On infinitesimal $\boldsymbol { \tau }$-isospectrality of locally symmetric spaces
Bibliographic record
Abstract
Abstract Let $(\tau , V_{\tau })$ be a finite dimensional representation of a maximal compact subgroup K of a connected non-compact semisimple Lie group G , and let $\Gamma $ be a uniform torsion-free lattice in G . We obtain an infinitesimal version of the celebrated Matsushima–Murakami formula, which relates the dimension of the space of automorphic forms associated to $\tau $ and multiplicities of irreducible $\tau ^\vee $ -spherical spectra in $L^2(\Gamma \backslash G)$ . This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $\tau $ -isospectrality. Along with this, we prove that the almost equality of $\tau $ -spherical spectra of two lattices assures the equality of their $\tau $ -spherical spectra.
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How this classification was reachedexpand
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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".