Non-Tempered Non-Spherical Representations

• Non-tempered non-spherical representations are unitary representations of reductive groups that lie outside the tempered dual and lack nonzero fixed vectors under maximal compact subgroups.
• They are explicitly constructed via generalized Brylinski–Kostant models, analytic R-groups, and graded Lie algebra techniques that determine their intricate algebraic and analytic properties.
• Their study advances harmonic analysis, Langlands correspondences, and period formulas, offering new insights and counterexamples to classical multiplicity-one theorems.

Non-tempered non-spherical representations are unitary representations of reductive groups, or their associated automorphic, local, or geometric realizations, that lie outside the tempered dual, and are not spherical in the sense of possessing a nonzero fixed vector under a maximal compact subgroup or a maximal parahoric subgroup. These representations occur both locally (in the representation theory of real or p-adic groups) and globally (in the automorphic spectrum), and are inherently connected with advances in geometric representation theory, harmonic analysis on homogeneous spaces, Langlands correspondences, and arithmetic theory. The paper of such representations exposes both the analytic and algebraic boundaries of the unitary dual and has led to new phenomena in branching laws, period integrals, and spectral theory.

1. Foundational Frameworks and Geometric Realizations

Non-tempered non-spherical representations are naturally constructed and classified in several geometric and algebraic frameworks. One significant construction is the generalization of the Brylinski–Kostant model for simple real Lie groups of non-Hermitian type (Achab, 2011). In this approach, unitary irreducible representations are realized as Hilbert spaces of holomorphic functions on certain conical complex varieties, specifically as holomorphic sections of line bundles over conformally compactified Jordan algebras. The explicit model is parametrized by multi-indices $q = (q_1, \ldots, q_s) \in \mathbb{N}^s$, which record the degrees of homogeneity with respect to the factors of the Jordan decomposition.

The traditional minimal—or spherical—representation corresponds to $q = 0$ and yields a unique $K$-fixed vector. Allowing nonzero values of $q$ leads to representations with no nonzero $K$-fixed vectors (non-spherical) and, according to analytic parameters associated with the inner products and reproducing kernels (e.g., weights involving Meijer $G$-functions or generalized hypergeometric functions such as ${}_1F_2$), one includes non-tempered cases. The representations are constructed via a five-step grading of the Lie algebra $$\mathfrak{g} = \mathfrak{p}_{-2} \oplus \mathfrak{p}_{-1} \oplus \mathfrak{p}_0 \oplus \mathfrak{p}_1 \oplus \mathfrak{p}_2$$ with an $\mathfrak{sl}_2$-triple governing the action. Differential operators—generalizing the creation-annihilation formalism—are introduced so that their commutation relations encode the $\mathfrak{sl}_2$ structure, with Theorem 3.4 providing necessary and sufficient analytic conditions for realization.

Thus, such geometric models explicitly produce non-tempered, non-spherical representations via the structure constants, the algebraic properties of the underlying Jordan algebra, and the analytic properties of the Bergman-type Hilbert spaces. The spherical case ($q = 0$) produces a minimal representation, while $q \neq 0$ provides representations where the lowest degree space is strictly larger than one dimensional, and the $K$-invariant is absent or highly constrained.

2. Local Langlands Correspondence and Analytic $R$-Groups

One of the algebraic cornerstones for understanding non-tempered non-spherical representations is the extension of the local Langlands correspondence (LLC) beyond the tempered spectrum (Aubert et al., 2013). In the $p$-adic setting, every irreducible representation of a reductive group $G$ is the unique Langlands quotient of a normalized parabolic induction $I_P(\omega \otimes \chi)$, where $\omega$ is essentially square-integrable and $\chi$ is an unramified (generally non-unitary for the non-tempered case) character. The parametrization and reducibility of such induced representations are controlled by analytic $R$-groups, which generalize Knapp–Stein $R$-groups to the non-tempered setting.

Analytic $R$-groups $R_{\omega \otimes \chi}$ are determined by the stabilizer in the Weyl group $W(M)$ of the inducing data, together with a 2-cocycle arising from normalized intertwining operators. The structure of the twisted group algebra $$\mathbb{C}[R_{\omega \otimes \chi}, \kappa_{\omega \otimes \chi}]$$ dictates the decomposition of $I_P(\omega \otimes \chi)$ into irreducible constituents. The framework remains compatible under unramified twists, thereby enabling a canonical, uniform extension of the LLC from tempered representations ($\chi$ unitary) to all irreducible smooth representations, including those non-tempered and non-spherical.

This analytic $R$-group formalism does not presuppose sphericity; many non-spherical representations—which do not possess nonzero vectors fixed by a maximal compact or Iwahori subgroup—emerge as non-tempered Langlands quotients indexed by the irreducible representations of the associated twisted group algebra. This construction is central to harmonic analysis on $p$-adic groups, especially in the arena removed from the discrete or spherical spectrum.

3. Branching Laws, Gan–Gross–Prasad, and Ext-Analogues

Branching behaviors and period problems offer a fertile ground where non-tempered non-spherical representations are both obstacles and central objects. The local Gan–Gross–Prasad conjecture (GGP), in both non-Archimedean (Chan, 2020, Qadri, 12 Feb 2024) and Archimedean (Chen et al., 6 Jul 2025) settings, predicts criteria for non-vanishing of hom/higher Ext spaces for representations of general linear groups in sequences $GL_n$, $GL_{n-1}$ via the matching (relevance) of their (Arthur) parameters. In the non-tempered scenario, these branching laws are necessarily more subtle and frequently governed by a relevance condition involving the structure of Langlands or Arthur parameters as multisegments or partitions.

Notably, for $p$-adic $GL_n$, Ext-groups between irreducible representations of Arthur type—constructed as products of discrete series and Aubert–Zelevinsky duals—can be nonzero in higher degrees if and only if the pair is "strong Ext relevant": their Arthur parameters match up not only under highest derivatives but also possibly up to duality (Qadri, 12 Feb 2024). In this regime, multiplicity-one fails, and Ext-branching laws rather than pure Hom branching laws become the correct generalization of the classical conjecture.

In the Archimedean context, a related relevance criterion governs the non-vanishing of Rankin–Selberg periods between representations of $GL_{n+1}(\mathbb{R})$ and $GL_n(\mathbb{R})$ (Chen et al., 6 Jul 2025). The work shows that nonzero periods force an explicit relation between their (generalized) SL$_2$-type partitions; specifically, the partitions must be "close"—the difference for each index is at most one. The proof hinges on the properties of annihilator varieties and the transfer of the relevance criterion through micro-local invariants.

Thus, non-tempered, non-spherical representations are mapped to intricate, parameter-dependent branching and period phenomena, often requiring new homological and geometric techniques beyond the reach of spherical or generic cases.

4. Theta Correspondence, Arthur Packets, and Period Formulas

Theta correspondence and the construction of non-tempered packets play a key role in generating and analyzing non-spherical representations, especially when dealing with special orthogonal and unitary groups. Explicit constructions of local and automorphic non-tempered Arthur packets on groups such as SO(3,2) and its inner form SO(4,1) are provided via local and global theta correspondence (Gurevich et al., 2013). The non-tempered Arthur parameters in these cases correspond, for instance, to embeddings of $SL_2(\mathbb{C})$ into Sp$_4$ with nontrivial unipotent content.

Multiplicity formulas, which control the number of representations in the packet with a given global realization, frequently display new features in the non-spherical or non-generic regime; multiplicities can exceed one, and the restriction to subgroups (such as SO(3,1)) shows a departure from classical "multiplicity one," with important analytic implications for period integrals and special values of $L$-functions.

In unitary contexts, non-tempered theta lifts lead to explicit scenarios where conjectural period/L-function relations (Gross–Prasad, Ichino–Ikeda) must be refined: local periods, which converge for tempered representations, become divergent when a constituent is non-tempered. Regularization procedures employ analytic continuation and limit formulas to recover well-defined period formulas, where residues of $L$-functions, rather than central values, control the period (Haan, 2014). This modification is forced both by the analytic properties of the representations and by the lack of sphericity.

5. Explicit Constructions, Examples, and Geometric Structures

Explicit constructions illuminate non-tempered non-spherical representations both in automorphic and geometric contexts:

• Explicit non-tempered, non-spherical cuspidal representations on $O(1,8n+1)$ are constructed via theta liftings from Maass cusp forms, with the archimedean component being tempered but unramified non-archimedean components manifestly non-tempered through their Satake parameters (Li et al., 2018). The associated standard $L$-functions are products of symmetric square $L$-functions and shifted zeta factors, reflecting the arithmetic content of the lift.
• Geometric representation theory provides instances in which representations arising from fundamental groups of surfaces or symmetric spaces generate non-tempered, non-spherical phenomena. For example, non-elementary type-preserving representations of $\pi_1(\Sigma_{0,4})$ into $PSL(2,\mathbb{R})$ with relative Euler class $\pm 1$ give ergodic components in moduli and violate naive Fuchsian/spherical classification—these components are not in the extremal (Teichmüller) sector and every non-peripheral simple closed curve is sent to a hyperbolic matrix, yet the representation is not in the "tempered" class nor is it Fuchsian (Yang, 2014).
• In real rank one symmetric spaces, the multiplicity with which a discrete series representation embeds in smooth and $L^2$ function spaces can differ; for $G/H = SO_0(n,1)/SO_0(n-1,1)$, an irreducible representation can appear with multiplicity two in $C^\infty(G/H)$ but only once in $L^2(G/H)$ (Krötz et al., 2021). This reveals new stratifications in harmonic analysis beyond the Plancherel decomposition and points to latent multiplicities inherent in non-spherical, non-tempered contexts.
• Zariski-dense, non-tempered, non-lattice discrete subgroups in higher rank simple Lie groups, constructed via deformation ("bending") techniques starting from hyperbolic lattices, provide examples where the quasi-regular representation $L^2(\Gamma_0 \backslash G)$ fails to be tempered; the lack of temperedness is governed by the growth indicator of the subgroup, which is quantitatively shown to exceed the decay threshold dictated by property (T) (Fraczyk et al., 25 Oct 2024). These examples embed both geometric and analytic mechanisms for the breakdown of sphericity and temperedness in representation theory of higher rank groups.

6. Analytic Invariants, Reproducing Kernels, and Harmonic Analysis

The analytic structure of non-tempered, non-spherical representations is captured via explicit reproducing kernels, norm formulas, and the analysis of unitary invariants. In geometric models, the reproducing kernel on a Hilbert space of holomorphic functions becomes a generalized hypergeometric function (e.g., ${}_1F_2$), with parameters linked to the representation's non-temperedness and the multi-index $q$ in the graded model (Achab, 2011). The inner product is often realized as a (pseudo-)weighted Bergman norm involving special functions such as the Meijer $G$-function, reflecting precise conditions for unitarity and invariance under maximal compact subgroups.

In period integral and $L$-function contexts, regularization and analytic continuation are needed to define the appropriate periods when integrals diverge for non-tempered representations; the replacement of central $L$-values by residues in formulae for period integrals is a characteristic analytic signature (Haan, 2014). Harmonic analytic criteria, such as temperedness characterized by decay of matrix coefficients, can thus be quantified in terms of growth indicators or decay rates tied to the representation's algebraic parameters (Fraczyk et al., 25 Oct 2024).

This analytic dimension not only offers invariants distinguishing non-tempered, non-spherical representations but directly governs their appearance in the spectral decomposition and their contribution to automorphic periods, harmonic analysis, and the analytic theory of automorphic $L$-functions.

7. Impact and Applications

Non-tempered non-spherical representations are central in multiple active domains:

• They provide explicit counterexamples to naive generalizations of the Ramanujan conjecture and multiplicity-one theorems in higher rank and non-generic settings.
• Their construction and classification guide refinements to the Langlands program, particularly in the paper of packets, endoscopy, and branching laws.
• Applications in automorphic theory—ranging from period formulas to functorial liftings—require a precise understanding of the analytic and algebraic behavior of these representations, including the regularization of divergences and the tracking of parameter-dependent multiplicities.
• In geometric and topological group theory, constructions of Zariski-dense non-tempered subgroups inform the paper of discrete subgroups beyond the arithmetic or lattice case, and their spectral properties provide bridgepoints to ergodic theory and growth phenomena constrained by property (T).

In summary, the paper of non-tempered non-spherical representations illuminates the intricate structure of the unitary dual, reveals rich geometric and analytic behaviors unattainable by spherical/tempered representations, and motivates both the extension and refinement of classic representation-theoretic and automorphic principles.