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Uniform bounds on the Harish-Chandra characters
Representation Theory · Submitted 3 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For $\textbf{G}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\textbf{G} (F)$, that all irreducible supercuspidal representations of $\textbf{G}(F)$ are compactly induced from an open compact modulo center subgroup.
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The abstract, sentence by sentence
Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element $\gamma $ of $\mathbf{G}(F)$ as $\pi$ varies among supercuspidal representations of $\mathbf{G}(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{{\rm deg}(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $\textbf{G}(F)$ with unitary central character and the formal degree of $\pi$ tends to infinity. For $\textbf{G}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\textbf{G} (F)$, that all irreducible supercuspidal representations of $\textbf{G}(F)$ are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of $\gamma $ ellitpic.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. | background 1.00 | 0.06 | 0.01 | asserted 0.84 |
| s1 | In this paper we study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element $\gamma $ of $\mathbf{G}(F)$ as $\pi$ varies among supercuspidal representations of $\mathbf{G}(F)$. | contribution 0.96 | 0.82 | 0.06 | asserted 0.77 |
| s2 | Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{{\rm deg}(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $\textbf{G}(F)$ with unitary central character and the formal degree of $\pi$ tends to infinity. | background 0.99 | 0.03 | 0.07 | speculative 0.96 |
| s3 | For $\textbf{G}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\textbf{G} (F)$, that all irreducible supercuspidal representations of $\textbf{G}(F)$ are compactly induced from an open compact modulo center subgroup. | result 0.93 | 0.95 | 0.72 | asserted 0.99 |
| s4 | Moreover, we give an explicit upper bound in the case of $\gamma $ ellitpic. | result 0.89 | 0.96 | 0.10 | asserted 1.00 |
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The paper
Original abstract
Let $\mathbf{G}$ be a connected reductive algebraic group over a $p$-adic local field $F$. In this paper we study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element $\gamma $ of $\mathbf{G}(F)$ as $\pi$ varies among supercuspidal representations of $\mathbf{G}(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{{\rm deg}(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $\textbf{G}(F)$ with unitary central character and the formal degree of $\pi$ tends to infinity. For $\textbf{G}$ semisimple we prove that the trace character is uniformly bounded on $\gamma$ under the assumption, which is expected to hold true for every $\textbf{G} (F)$, that all irreducible supercuspidal representations of $\textbf{G}(F)$ are compactly induced from an open compact modulo center subgroup. Moreover, we give an explicit upper bound in the case of $\gamma $ ellitpic.
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