Elementary and Coherent Pairs

• Elementary and coherent pairs are discrete series representations defined by compatibility of interleaving decorations and minimal K-type branching.
• They exhibit a one-dimensional symmetry breaking operator that ensures multiplicity one restrictions between unitary group representations.
• Translation functors transform coherent pairs into elementary ones, enabling isomorphisms in cohomological restriction and advancing automorphic period studies.

The concepts of elementary and coherent pairs appear in diverse mathematical contexts, notably in the theory of group and ring coherence, orthogonal polynomials, the topology and combinatorics of surfaces, and—specifically in the context of (Harris et al., 21 Sep 2025)—the representation theory of real reductive groups, the cohomology of Shimura varieties, and arithmetic applications to special values of automorphic $L$-functions. This article focuses on the detailed structure and significance of elementary and coherent pairs of discrete series representations for unitary groups and their applications to symmetry breaking, translation functors, restriction of cohomology, and automorphic periods.

1. Elementary and Coherent Pairs of Discrete Series Representations

In the representation theory of real reductive groups, an irreducible discrete series representation $\pi$ of $G = U(p,q)$ is parameterized by a Harish–Chandra parameter $\lambda$ and a combinatorial “decoration” $\delta$ in a finite set $\mathrm{Deco}(p,q)$. For the subgroup $G^\prime = U(p-1,q)$, a representation $\pi^\prime$ is similarly parameterized by $\lambda^\prime$ and $\delta^\prime$ in $\mathrm{Deco}(p-1,q)$. The main focus is the pair $(\pi, \pi^\prime)$ and the space of symmetry breaking operators,

$$\operatorname{Hom}_{G^\prime}(\pi|_{G^\prime}, \pi^\prime).$$

A coherent pair is defined as one for which this Hom space is nonzero and whose signatures $[\delta \delta^\prime]$ (derived from the interleaving of decorations) satisfy strong compatibility conditions prescribed in the Gan–Gross–Prasad (GGP) theory. Explicitly, the refined coherent condition is established via a reduction or deletion procedure on the interleaving pattern that ultimately reduces to a single symbol $+$. Proposition 4.10 (Harris et al., 21 Sep 2025) and its equivalents provide various formulations, including the existence of a $K^\prime$-equivariant inclusion between associated finite-dimensional representations.

An elementary pair is a coherent pair with the extra property that the minimal $K$-type (Blattner $K$-type) $\mu_\lambda$ of $\pi$ restricts to the minimal $K^\prime$-type $\mu_{\lambda^\prime}$ of $\pi^\prime$ with multiplicity one: $$\operatorname{Hom}_{K^\prime}(\mu_\lambda|_{K^\prime}, \mu_{\lambda^\prime}) \ne 0.$$ The minimal $K$-type is given by $$\mu_\lambda = \lambda - \rho_\mathfrak{g} + \rho(\delta)$$, where $\rho_\mathfrak{g}$ is half the sum of positive roots and $\rho(\delta)$ encodes additional combinatorial information from the decoration.

2. Symmetry Breaking Operators and Branching Problems

The central object of paper is the symmetry breaking operator: $S \in \operatorname{Hom}_{G'}(\pi|_{G'}, \pi').$ For the unitary groups being considered, the local Gan–Gross–Prasad conjecture (now theorem) ensures that this space is at most one-dimensional and non-zero exactly when the interleaving pattern $[\delta \delta^\prime]$ is compatible.

Such operators identify the discrete series constituents appearing in the restriction $\pi|_{G'}$ and, crucially, induce pullback maps between automorphic vector bundles and their cohomology in global settings.

In the case of an elementary pair, the operator $S$ not only exists but has the property that its induced map between (coherent) cohomology groups is an isomorphism in suitable degree, reflecting a perfect branching of cohomology analogous to the representation-theoretic branching law.

3. Translation Functors and Their Role

Translation functors provide a mechanism to transfer symmetry breaking properties across parameters by the process of tensoring with finite-dimensional representations and projecting to certain generalized eigenspaces of the center of the enveloping algebra.

Given finite-dimensional representations $F$ (for $G$) and $F'$ (for $G'$) with highest weights $\mu$ and $\mu'$, respectively, and projections $P_{\tau+\mu}$ onto eigenspaces of central characters, the translation functor provides a commutative diagram linking representations $(\pi, \pi')$ and $(\sigma, \sigma')$ as: $$\psi_\tau^{\tau+\mu}(-) = P_{\tau+\mu}(P_\tau(-) \otimes F),$$ with analogous constructions for $G'$. Through such operations, any coherent pair can be linked to an elementary pair, since translation by suitable finite-dimensional representations preserves the coherent interleaving signature and canonical properties of symmetry breaking.

This structure is critical, as not every coherent pair is elementary, but any coherent pair appears as a translation of some elementary pair, facilitating the analysis of restriction phenomena and the passage between geometric and representation-theoretic viewpoints.

4. Cohomological Restriction in Shimura Varieties

The theory extends to automorphic forms and the geometry of Shimura varieties attached to unitary groups. For an automorphic discrete series representation $\pi$ with associated finite-dimensional $K$-type $W$, the automorphic vector bundle $[\mathbb{W}_\pi]$ lives on the corresponding Shimura variety $S(H, Y_V)$. The inclusion $(H', Y_{V'}) \hookrightarrow (H, Y_V)$ at the group level yields a canonical map between Shimura varieties, giving rise to a restriction morphism in coherent cohomology: $$R: H^q_{\mathcal{O}}(S(H, Y_V), [\mathbb{W}_\pi]) \longrightarrow H^q_{\mathcal{O}}(S(H', Y_{V'}), [\mathbb{W}_{\pi'}]).$$ For elementary pairs, this restriction is nontrivial and, in appropriate settings, an isomorphism (Theorem 4.16 (Harris et al., 21 Sep 2025)). The translation functor enables one to deduce the non-vanishing of cup product or period maps in situations where the local pair is merely coherent (by first translating to an elementary one).

This compatibility between local harmonic analysis (symmetry breaking) and global geometry (restriction of automorphic cohomology) underpins significant advances in understanding automorphic period integrals.

5. Applications to Periods and Special Values of L-functions

The arithmetic consequences of elementary and coherent pairs are realized through the paper of period integrals and critical values of $L$-functions, particularly those of Rankin–Selberg type for base changed representations $GL(n+1)\times GL(n)$. The Ichino–Ikeda–N. Harris formula relates the central value $L(1/2, \Pi \times \Pi')$ to period integrals of automorphic forms: $$\mathcal{P}(f, f') = (\text{constant}) \cdot \mathcal{L}^S\left(\frac12, \Pi \times \Pi'\right) \times (\text{product of local factors}),$$ where $f$ and $f'$ are test vectors in appropriate cohomology groups. When the archimedean components $(\pi, \pi')$ form an elementary or coherent pair, the analytic period is non-vanishing and can be explicitly related to motivic periods (as suggested by Deligne’s conjecture on special values).

Period ratios $P(\pi, \pi')$, calculated via cup product pairings in coherent cohomology, are shown to match the algebraic part of $L$-values up to explicit, well-understood transcendental constants, tightly coupling representation theory, geometry, and arithmetic.

6. Technical Formulations and Classification Criteria

For clarity, Table 1 outlines the classification and structural results:

Pair Type	Symmetry Breaking Operator	Minimal K-type Branching	Cohomological Restriction
Coherent pair	Exists, $\dim=1$	Not necessarily multiplicity one	Nonzero, but not always an isomorphism
Elementary pair	Exists, $\dim=1$	$\mu_{\lambda'}$ in $\mu_{\lambda}|_{K'}$ ($\dim=1$)	Isomorphism in degree $q$

The parameter conditions for coherence are provided via explicit combinatorial rules on the signature $[\delta\delta']$, including reduction via deletion of adjacent $+-$ or $-+$ pairs.

7. Significance and Broader Context

The introduction and detailed classification of elementary and coherent pairs for discrete series representations lead to a substantial unification of representation theory, Hodge theory, and the arithmetic of automorphic $L$-values. The translation functor method provides a canonical mechanism to link local symmetry breaking and cup product phenomena in arithmetic geometry, enabling precise control of restriction maps on cohomology and period integrals.

These developments have direct implications for the paper of special values of motive-related $L$-functions and their conjectural properties, most notably the expected algebraicity conjectures of Deligne. The representation-theoretic framework for elementary/coherent pairs thus forms an essential bridge between analytic, automorphic, and motivic approaches to modern number theory.