Typical representations via fixed point sets in Bruhat–Tits buildings
Bibliographic record
Abstract
For a tame supercuspidal representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a connected reductive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not inertially equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The consequence is a set of broadly applicable tools for addressing the branching rules of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the unicity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper G comma pi right-bracket Subscript upper G"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi> π </mml:mi> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">[G,\pi ]_G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -types.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
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".