PreprintCC BY 4.0arXiv
Automorphic C*-algebras of reductive groups
Abstract
Let G be a reductive group over a number field F. We define C*aut(G(𝔸)), the automorphic C*-algebra of G, to be the C*-algebraic image of the full automorphic representation of G(𝔸) on L2(G(F)backslash G(𝔸)). Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective *-homomorphism C*aut(G(𝔸)) ⟶ ⨁[M,σ] KC0(AM̂) ( IndPG C0(AM̂,HM,σ) )W(G,M,σ), where [M,σ] ranges over the associate classes of discrete Levi data. When G is GL(n) or an inner form of it, we prove that this map is an isomorphism of C*-algebras.
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BibTeX
@misc{yang2026automorphic,
title = {{Automorphic $C^*$-algebras of reductive groups}},
author = {Jun Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35543},
eprint = {2609.35543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35543},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Automorphic C^*-algebras of reductive groups
AU - Jun Yang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35543
UR - https://arxiv.org/abs/2609.35543
AB - Let G be a reductive group over a number field F. We define C^*_aut(G(𝔸)), the automorphic C^*-algebra of G, to be the C^*-algebraic image of the full automorphic representation of G(𝔸) on L^2(G(F)backslash G(𝔸)). Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective *-homomorphism C^*_aut(G(𝔸)) ⟶ ⨁_[M,σ] K_C_0(A_M̂) ( Ind_P^G C_0(A_M̂,H_M,σ) )^W(G,M,σ), where [M,σ] ranges over the associate classes of discrete Levi data. When G is GL(n) or an inner form of it, we prove that this map is an isomorphism of C^*-algebras.
ER -