Inverse Satake isomorphism and change of weight
Bibliographic record
Abstract
Let <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> be any 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. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K subset-of upper G"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K\subset G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be any special parahoric subgroup and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V comma upper V prime"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">V,V’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be any two irreducible smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper F Subscript p Baseline With bar left-bracket upper K right-bracket"> <mml:semantics> <mml:mrow> <mml:mover> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo stretchy="false">[</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathbb {F}_p}[K]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules. The main goal of this article is to compute the image of the Hecke bimodule <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E n d Subscript ModifyingAbove double-struck upper F Sub Subscript p Subscript With bar left-bracket upper K right-bracket Baseline left-parenthesis c minus upper I n d Subscript upper K Superscript upper G Baseline upper V comma c minus upper I n d Subscript upper K Superscript upper G Baseline upper V prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>End</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo stretchy="false">[</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msub> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>c</mml:mi> <mml:mo> − </mml:mo> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> <mml:msubsup> <mml:mi>d</mml:mi> <mml:mi>K</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo> − </mml:mo> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> <mml:msubsup> <mml:mi>d</mml:mi> <mml:mi>K</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {End}_{\overline {\mathbb {F}_p}[K]}(c-Ind_K^G V, c-Ind_K^G V’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the generalized Satake transform and to give an explicit formula for its inverse, using the pro- <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> Iwahori Hecke algebra 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> . This immediately implies the “change of weight theorem” in the proof of the classification of mod <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> irreducible admissible representations 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> in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:s
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".