Coherent Cohomology of Discrete Series

• Coherent cohomology of discrete series is the study of (g,K)-modules with nontrivial cohomology and their role in automorphic vector bundles over Shimura varieties.
• It employs branching laws, minimal K-types, and symmetry breaking operators to decode period maps and establish rationality criteria in automorphic forms.
• Translation functors and vanishing theorems are used to link geometric and arithmetic properties, enabling computation of special values of L-functions in the Langlands program.

Coherent cohomology of discrete series refers to the interplay between the representation-theoretic properties of (g,K)-modules, most notably discrete series representations, and the geometry and arithmetic of automorphic vector bundles and their cohomology on locally symmetric or Shimura varieties. These connections are essential for understanding the arithmetic of automorphic forms, period morphisms, branching laws, and rationality phenomena in the context of the Langlands program.

1. Cohomological Realization of Discrete Series Representations

Discrete series representations of real or p-adic reductive groups arise as square-integrable automorphic representations and are characterized by nonzero (g,K)-cohomology. For real groups G, the cohomology can be computed via relative Lie algebra cohomology: $H^*(\mathfrak{g}, K; \pi \otimes V)$ where $\pi$ is a (g,K)-module, $K$ is a maximal compact subgroup, and $V$ ranges over finite-dimensional algebraic representations. Notably, “cohomological” or “coherent cohomology” representations are those π for which this group is nontrivial; these are linked to automorphic forms contributing to the coherent cohomology of automorphic vector bundles over Shimura varieties.

For classical groups, a minimal K-type (or Blattner K-type) is canonically attached to each discrete series representation, characterized by its highest weight. Salamanca-Riba developed necessary and sufficient conditions for an irreducible K-representation to occur as a minimal K-type of a discrete series representation (Harris, 2012).

The realization of cohomological classes is often via Shimura varieties, using automorphic vector bundles (Harris et al., 21 Sep 2025). For G of Hermitian type, such vector bundles are attached to algebraic representations W; their coherent cohomology, often concentrated in a single degree, admits explicit descriptions in terms of relative Lie algebra cohomology.

2. Branching Laws and Restriction to Subgroups

Central to the arithmetic applications is describing how discrete series representations and their cohomology behave under restriction to subgroups. Consider $G' \subset G$ (e.g., $U(n-1) \subset U(n)$); one investigates the restriction of a discrete series π to $G'$. The nonvanishing of intertwining operators

$\operatorname{Hom}_{G'}(\pi|_{G'}, \pi')$

is governed by intricate combinatorial data (e.g., GGP interleaving signatures), and symmetry breaking operators are uniquely determined (multiplicity one) by these patterns (Harris et al., 21 Sep 2025).

The minimal K-type τ of π restricts to a sum of K'-types. If π' is a discrete series of $G'$ with minimal K'-type τ', there may exist a nonvanishing pairing between τ and τ' if τ' appears in the restriction τ|_{K'} (Harris, 2012). This compatibility is crucial for realizing global period maps and rationality criteria for cohomology classes.

Works such as (Ørsted et al., 2023) and (Weiske, 2021) establish explicit branching laws and compute multiplicities, often drawing analogies with Blattner or Kostant-Heckmann multiplicity formulas, and relate these to periods and cup products in the cohomology of Shimura varieties.

3. Rationality Criteria and Period Relations

A major focus concerns the rationality of coherent cohomology classes. Harris (Harris, 2012) proves that for sufficiently regular discrete series π, a cohomology class $f \in \pi$ is rational if and only if all periods

$$A_y(f)(f') = P_a(T, T')^{-1} \cdot I_{can}(T, T')(f, f') \in \mathbb{Q}$$

where $T$ and $T'$ are the archimedean components of π and π' (with π' in $G'$), $P_a(T,T')$ is the Gross–Prasad period, and $I_{can}(T,T')$ is a canonical invariant pairing, satisfy explicit algebraicity constraints.

These period invariants relate to the nonvanishing of pairings between automorphic forms on $G$ and $G'$, transferred via ergodic and positivity arguments. The main result is that the arithmeticity of cohomology classes can be characterized in terms of the algebraicity of these periods and their compatibility under restriction to subgroups.

Period relations obtained via global cup products in coherent cohomology directly underpin the computation of special values of automorphic L-functions, notably through the Ichino–Ikeda–N. Harris formula (Harris et al., 21 Sep 2025): $$L\left(\frac{1}{2}, BC(\pi) \times BC(\pi')\right) \approx c \; \frac{p(\xi(\pi, \pi'))}{Q(\pi)Q(\pi')}$$ where $Q(\pi)$ and $Q(\pi')$ are period invariants attached to the representations, and $p(\xi(\pi, \pi'))$ is a global automorphic period arising from coherent cohomology cup products.

4. Translation Functors and Symmetry Breaking

Translation functors enable passage between different discrete series representations and their associated cohomological data by tensoring with finite-dimensional representations and projecting onto selected infinitesimal characters (Harris et al., 21 Sep 2025). In the context of branching problems, translation functors are used to “move” from coherent pairs (subject to certain combinatorial signatures) to elementary pairs, where the minimal K-type of π' appears as a direct summand in that of π and cup products in coherent cohomology realize period maps at the geometric and arithmetic levels.

Symmetry breaking operators, elements of $\operatorname{Hom}_{G'}(\pi|_{G'},\pi')$, correspond under the translation framework to cup product or restriction maps between the coherent cohomology groups associated to π and π'. Their uniqueness and nonvanishing reflect deep period relations, with algebraicity properties governed by the branching combinatorics.

5. Arithmetic Applications and Special Values of L-functions

The culmination of these links appears in the explicit connection between coherent cohomology, automorphic periods, and special values of L-functions. For discrete series π and π', and under precise branching constraints, cup products in coherent cohomology yield period integrals that (up to explicit motivic or regulator factors) compute central values of Rankin–Selberg L-functions (Harris et al., 21 Sep 2025).

This framework enables a purely cohomological description of arithmetic invariants classifying automorphic forms, and supports explicit verification of conjectural algebraicity properties (Deligne’s conjecture) for critical L-values. Moreover, the Gorenstein property of the (ordinary) local Hecke algebra is critical in constructing p-adic L-functions varying in Hida families and is verified under general assumptions in coherent cohomology settings (Atanasov et al., 2021).

6. Analytic and Geometric Aspects

Vanishing theorems for higher coherent cohomology, e.g., in the setting of p-adic holomorphic discrete series (Grosse-Klönne, 2014), guarantee that all arithmetic information is concentrated in global sections. These sheaves provide integral structures needed for arithmetic applications and inform the paper of reductions modulo p, compatibility with crystalline and de Rham cohomology, and links to p-adic Hodge theory.

Furthermore, branching and restriction results for representations with nontrivial (g,K)-cohomology (such as the nonzero $\mathfrak{n}$-cohomology for totally degenerate limits of discrete series (Lee, 30 Jul 2025)) provide deep connections to period maps and the Gan–Gross–Prasad conjectures, promising to elucidate the geometric and arithmetic structure of automorphic sheaves at the boundary of discrete series.

7. Summary Table: Key Concepts and Their Cohomological Roles

Concept	Representation Theoretic Realization	Cohomological/Aritmetic Implication
Minimal/Blattner K-types	Characterizes discrete series, central to restriction theory	Determines coherent cohomology generators
Symmetry breaking operators	Intertwining $\pi|_{G'} \to \pi'$	Induce cohomology restriction/cup-product maps
Translation functors	Tensor with finite-dimensional, project onto desired parameter	Relate different pairs of discrete series and cohomology classes
Rationality criteria (Gross–Prasad period)	Period invariants, canonical pairings	Arithmeticity of cohomology, algebraicity of L-values
Diamond operators and étale coverings	Level structure on Shimura varieties	Exactness and Gorenstein properties for Hecke algebras
Vanishing theorems for higher cohomology	Unique realization of discrete series as global sections	Exactness, integral structures, links to p-adic Hodge theory

In summary, coherent cohomology of discrete series representations forms a central theme linking the analytic theory of representations, the arithmetic geometry of Shimura varieties, and the special value formulae of automorphic L-functions. The theory synthesizes deep combinatorial, representation-theoretic, and geometric inputs to provide a comprehensive, functorial picture of rationality, periods, and arithmetic invariants for automorphic forms.