Cuspidal Automorphic Representations

• Cuspidal automorphic representations are irreducible unitary subrepresentations of L²(G(F)\G(𝔸_F)) defined by the vanishing of constant terms along proper parabolics.
• They are characterized by nonzero Fourier/Whittaker coefficients, playing a key role in Langlands correspondences and functorial transfers.
• Explicit constructions using Poincaré series and trace formula techniques yield practical insights into their applications in arithmetic geometry and Galois theory.

A cuspidal automorphic representation is an irreducible, unitary subrepresentation of the Hilbert space of square-integrable automorphic forms on the quotient of an algebraic group by its rational points. These representations form the core of the discrete spectrum in the context of global fields and Lie groups, encoding deep arithmetic and spectral information. They are characterized by the property that all constant terms along proper parabolic subgroups vanish identically, ensuring they do not arise from Eisenstein series or induced forms. Cuspidal automorphic representations can be further distinguished by their local components, genericity (i.e., nonzero Fourier coefficients with respect to unipotent subgroups and characters), and their relationship to global arithmetic and functorial constructions.

1. Definitions and Core Properties

A cuspidal automorphic representation of a connected reductive group $G$ over a global field $F$ is an irreducible constituent of the right-regular representation of $G(\mathbb{A}_F)$ on $L^2(G(F)\backslash G(\mathbb{A}_F))$. Explicitly, it is a subrepresentation $\Pi$ occurring in the subspace of cusp forms—functions $\phi$ such that

$$\int_{U_P(F)\backslash U_P(\mathbb{A}_F)} \phi(ug)\, du = 0$$

for every proper parabolic subgroup $P = MU_P$ and almost all $g$. This vanishing of constant terms is the defining property of cuspidality (Labesse et al., 2018). These representations decompose as restricted tensor products $\Pi \cong \bigotimes'_v \Pi_v$ over local admissible representations. Genericity is often defined via the non-vanishing of Whittaker coefficients,

$$W_\phi(g) = \int_{U(F)\backslash U(\mathbb{A}_F)}\phi(ug)\overline{\psi(u)}\,du,$$

for a unipotent subgroup $U$ and suitable character $\psi$ (Moy et al., 2015).

2. Genericity, Fourier Coefficients, and Existence

Generic cuspidal automorphic representations are those whose matrix coefficients admit non-zero Fourier coefficients along prescribed unipotent subgroups with respect to suitably generic characters. This notion is central for understanding local and global Langlands correspondences, for which the existence of generic cusp forms with prescribed ramification and local types is fundamental (Moy et al., 2015). In the quasi-split case, generic representations correspond to those with non-degenerate Whittaker models. Moy–Muić’s theorem guarantees, under suitable Bernstein component and parabolic-annihilation hypotheses, the existence of $(\psi,U)$-generic cuspidal automorphic representations with precise local and global control (Moy et al., 2015).

3. Functoriality, Central Isogenies, and Restriction/Lifting Theory

When two reductive groups $H$ and $G$ over $F$ are related by a central isogeny of derived subgroups, essential functorial relationships exist between their cuspidal spectra. Labesse–Schwermer prove that the restriction of a cuspidal automorphic representation $\pi$ of $G(\mathbb{A}_F)$ to $H(\mathbb{A}_F)$ always contains a cuspidal automorphic representation of $H$, and—if the morphism is injective—every cuspidal automorphic representation of $H(\mathbb{A}_F)$ can be realized in the restriction of some cuspidal of $G(\mathbb{A}_F)$ (Labesse et al., 2018). This interplay is compatible with expected Langlands functorial correspondences on $L$-groups and controls multiplicity phenomena. Local analogues ensure restriction and induction properties for irreducible admissible representations at every place.

4. Arthur Packets, Parameters, and Classification

Cuspidal automorphic representations of classical groups (including symplectic, orthogonal, and unitary groups) are organized into Arthur packets associated to global Arthur parameters. These parameters are formal sums

$\psi = \bigboxplus_{i=1}^r (\tau_i, b_i)$

with $\tau_i$ irreducible self-dual cuspidal representations of general linear groups, $b_i$ multiplicities, and subject to constraints encoding self-duality and parity (Jiang et al., 2016, Jiang et al., 2015). Generic packets (all $b_i=1$) always contain cuspidal representations, realized concretely via periods and endoscopic constructions. Detailed criteria exist for detecting when a packet contains no cuspidal member, notably via size estimates on the blocks $(\chi, b_i)$ for symplectic groups, where large $b$ always precludes cuspidality (Jiang et al., 2016).

5. Explicit Constructions and Existence Results

Concrete construction of cuspidal automorphic representations is possible using local test functions and global Poincaré series with controlled support and Fourier coefficients. The method covers general semisimple groups, arbitrary unipotent subgroups, and prescribed Bernstein/local types. Poincaré series techniques are used to produce cusp forms with non-vanishing generic Fourier coefficients on both the adelic quotient and congruence subgroups of the real points (Moy et al., 2015). The local–global control ensures representations with specified ramification, genericity, and congruence properties can always be realized under mild hypotheses.

6. Cohomological, Galois, and Arithmetic Aspects

Cuspidal automorphic representations of $\mathrm{GL}_n$ and related groups serve as sources for compatible systems of Galois representations via the cohomology of Shimura varieties and Langlands correspondences. For regular algebraic $\pi$ (not necessarily self-dual), Harris–Lan–Taylor–Thorne construct $\ell$-adic Galois representations matching Hecke eigenvalues at all but finitely many places, verifying local-global compatibility and the correct Hodge-Tate weights (Harris et al., 2014). The absence of a self-duality requirement is enabled by the geometry of unitary similitude Shimura varieties and vanishing theorems for the ordinary locus. Functoriality via exterior powers, tensor products, and symmetric powers leads to a rich spectrum of arithmetic applications (Kim et al., 2014, Bhagwat et al., 2021).

7. Trace Formula, Isolation, and Analysis of the Cuspidal Spectrum

The spectral decomposition of $L^2(G(F)\backslash G(\mathbb{A}_F))$ separates the cuspidal from the continuous spectrum, critical in applications of the Arthur–Selberg trace formula, comparison of orbital integrals, and in deep conjectures such as Gan–Gross–Prasad. Over function fields, Hecke algebra multipliers explicitly constructed via Satake transform isolate the full cuspidal spectrum, kill all Eisenstein series, and enable fine arithmetic applications including period calculations and Weyl laws (Cai et al., 2021). Harder’s theorem on the finiteness of the cusp spectrum and the construction of explicit isolating test functions provide foundational inputs to modern analytic and trace formula techniques.

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Key Papers Referenced:

• Labesse–Schwermer, Central morphisms and Cuspidal automorphic Representations (Labesse et al., 2018)
• Moy–Muić, On Existence of Generic Cusp Forms on Semisimple Algebraic Groups (Moy et al., 2015)
• Jiang–Liu, On cuspidality of global Arthur packets for symplectic groups (Jiang et al., 2016)
• Jiang–Zhang, Arthur Parameters and Cuspidal Automorphic Modules of Classical Groups (Jiang et al., 2015)
• Harris–Lan–Taylor–Thorne, On the Rigid Cohomology of Certain Shimura Varieties (Harris et al., 2014)
• Cai–Xu, Isolation of the Cuspidal Spectrum: the Function Field Case (Cai et al., 2021)
• Kim–Yamauchi, Artin representations for $GL_n$ (Kim et al., 2014)
• Bhagwat–Mondal, Automorphic tensor products and cuspidal cohomology of the ${\rm GL}_4$ (Bhagwat et al., 2021)