When a discrete series representation π\pi of a real reductive Lie group GG is restricted to a symmetric subgroup HH, the unitary restriction decomposes as a direct integral of irreducible representations τ\tau of HH with measurable multiplicities m(τ)m(\tau). Harris, He and Ólafsson (2013) proved the first finite-multiplicity theorem that covers the continuous spectrum of such restrictions, for symmetric pairs satisfying a compact-centralizer covering condition, and posed the general problem: find a criterion, analogous to Kobayashi's discrete-decomposability conditions, that predicts from the data (G,H,π)(G,H,\pi) whether the restriction has only finite multiplicities (continuous spectrum included) or admits infinite multiplicities, and determine the precise boundary between the two regimes. Subsequent work resolved the pair-level part: by Kobayashi–Oshima (2013) and the Kobayashi–Matsuki classification (2014), all irreducible restrictions have finite (respectively uniformly bounded) multiplicity exactly when the pair is real spherical (condition (PP), respectively complex-spherical condition (BB)), and Frahm (2022) proved that the direct-integral multiplicity is dominated by the smooth-category multiplicity, so every discrete series restricts with finite multiplicities on all these pairs. The surviving open core is the triple-level question: for symmetric pairs outside the real-spherical class, determine for which discrete series π\pi the restriction still has finite multiplicities almost everywhere, exhibit when infinite multiplicity on the continuous spectrum actually occurs for a discrete series, and thereby establish the precise finite-infinite boundary.

Background

A basic problem in the representation theory of real reductive groups is the branching problem: given a group GG, a closed subgroup HH, and an irreducible unitary representation π\pi of GG, decompose the restriction π∣H\pi|_H into irreducible representations of HH; here and below KK denotes a maximal compact subgroup of GG. A discrete series representation is an irreducible unitary representation that occurs as a closed irreducible subspace of L2(G)L^2(G) (they exist only when GG has a compact Cartan subgroup); a symmetric subgroup HH is the fixed-point set, up to finite index, of an involutive automorphism of GG, and (G,H)(G,H) is then called a reductive symmetric pair. One works throughout with reductive groups of Harish-Chandra class, the standard class (reductive Lie algebra, finite center of the commutator subgroup, finitely many components) in which Harish-Chandra's representation theory is valid. By the general theory of direct integrals, the restriction decomposes as π∣H≃∫H^⊕m(τ) τ dμ(τ)\pi|_H\simeq\int^{\oplus}_{\widehat H}m(\tau)\,\tau\,d\mu(\tau) with a measurable multiplicity function m(τ)∈N∪{∞}m(\tau)\in\mathbb{N}\cup\{\infty\}; the representations atoms of μ\mu constitute the discrete spectrum and the rest the continuous spectrum. Since HH is non-compact, a continuous spectrum generically appears, and the multiplicities on it can a priori be infinite. Harris, He and Ólafsson introduced a technique of convolving Harish-Chandra Schwartz functions (a class of rapidly decreasing functions on GG adapted to its symmetric-space structure) with distribution characters of tempered representations of HH (tempered meaning weakly contained in the regular representation) to control the continuous spectrum: they proved that if H≃K0×H1H\simeq K_0\times H_1 with K0K_0 compact and the centralizers inside H1∩KH_1\cap K of two natural abelian subspaces (a Cartan subspace a⊂s∩q\mathfrak{a}\subset\mathfrak{s}\cap\mathfrak{q} of the symmetric pair and a maximal abelian b⊂h1∩s\mathfrak{b}\subset\mathfrak{h}_1\cap\mathfrak{s}, in the standard decomposition of the Lie algebra) cover H1∩KH_1\cap K, then the whole direct-integral decomposition of π∣H\pi|_H has finite multiplicities for every discrete series π\pi — the first finiteness theorem covering the continuous spectrum, answering a question of Ørsted–Vargas for Spin(2n,1)⊃Spin(2k)×Spin(2n−2k,1)\mathrm{Spin}(2n,1)\supset\mathrm{Spin}(2k)\times\mathrm{Spin}(2n-2k,1). They also proved, for any closed reductive HH, that the multiplicity function is constant along the continuous parameter ν\nu within each tempered family σ(δ,ν)\sigma(\delta,\nu) of irreducible representations of HH (such families are obtained by parabolic induction from discrete-series data δ\delta of a Levi factor and are parametrized by the continuous variable ν\nu), and they posed the general problem this entry records. Two neighboring finiteness theories set the boundary of what is now known. First, Kobayashi's discrete-decomposability criterion: π∣H\pi|_H splits as an honest direct sum of irreducibles with finite multiplicities (no continuous spectrum at all, called HH-admissibility) when the asymptotic KK-support of π\pi — the closed cone generated by the leading weights of the KK-types of π\pi (the KK-types being the irreducible representations of KK occurring in π\pi) — meets the moment cone of K/(H∩K)K/(H\cap K) (the convex cone generated by the image of the moment map for the KK-action on the compact homogeneous space K/(H∩K)K/(H\cap K)) only at the origin; for discrete series π\pi this cone condition is necessary and sufficient. Second, the pair-level theory of Kobayashi–Oshima: writing Hom⁡H(π∞∣H,τ∞)\operatorname{Hom}_H(\pi^\infty|_H,\tau^\infty) for the space of continuous intertwining operators between the smooth vectors (symmetry breaking operators), the pair (G,H)(G,H) satisfies dimHom⁡H(π∞∣H,τ∞)<∞\operatorname{Hom}_H(\pi^\infty|_H,\tau^\infty)<\infty for all irreducible admissible π,τ\pi,\tau if and only if the pair satisfies the geometric condition (PP), namely that a minimal parabolic subgroup of HH has an open orbit on the real flag variety G/PG/P, with PP a minimal parabolic subgroup of GG (equivalently, the homogeneous space (G×H)/diag H(G\times H)/\mathrm{diag}\,H is real spherical, i.e., contains an open orbit of a minimal parabolic subgroup); the uniform boundedness of these dimensions is equivalent to the complexified condition (BB). Kobayashi–Matsuki completely classified the reductive symmetric pairs satisfying (PP) — essentially the Riemannian pairs, the split-rank-one families (o(p+q,1),o(p)⊕o(q,1))(\mathfrak{o}(p+q,1),\mathfrak{o}(p)\oplus\mathfrak{o}(q,1)) and their unitary, symplectic and f4(−20)\mathfrak{f}_{4(-20)} analogues, the strong-Gelfand-type pairs including (o(p+1,q),o(p,q))(\mathfrak{o}(p+1,q),\mathfrak{o}(p,q)), the group case, and five exceptional cases. Finally, Frahm proved that on the smooth vectors the direct-integral decomposition is pointwise defined, so the unitary multiplicity is bounded by the smooth multiplicity: m(τ)≤dim⁡Hom⁡H(π∞∣H,τ∞)m(\tau)\le\dim\operatorname{Hom}_H(\pi^\infty|_H,\tau^\infty) almost everywhere. Combining these results, on every (PP) symmetric pair every irreducible unitary representation, hence every discrete series, restricts with almost-everywhere finite multiplicity in the full direct-integral decomposition, continuous spectrum included (uniformly bounded on (BB) pairs, which include (O(p,q),O(p−1,q))(O(p,q),O(p-1,q))). What is not covered by any of this is the region strictly between the two anchors: symmetric pairs failing (PP), where Kobayashi–Oshima only produce a possibly non-unitarizable representation with infinite smooth multiplicity, and restrictions that are not discretely decomposable. There the finiteness of the direct-integral multiplicities of a given discrete series — the question Harris–He–Ólafsson raised — remains undetermined.

Problem Statement

Let GG be a real reductive Lie group of Harish-Chandra class, HH a symmetric subgroup, and π\pi a discrete series representation of GG, with direct-integral branching law π∣H≃∫H^⊕m(τ) τ dμ(τ)\pi|_H\simeq\int^{\oplus}_{\widehat H}m(\tau)\,\tau\,d\mu(\tau). Establish a criterion on the triple (G,H,π)(G,H,\pi) — expressed in group-theoretic and representation-theoretic data, analogous to Kobayashi's asymptotic-KK-support criterion for discrete decomposability and to the real-sphericity conditions (PP)/(BB) at the pair level — that is necessary and sufficient for the multiplicity function to be finite μ\mu-almost everywhere, i.e., for the restriction to have finite multiplicities with the continuous spectrum included. Since the pair-level cases are settled (finiteness for all irreducible representations holds exactly on the Kobayashi–Matsuki real-spherical list, and every discretely decomposable restriction is governed by the known cone criterion, which for discrete series is necessary and sufficient), the problem is to treat symmetric pairs outside the (PP) class and restrictions that are not discretely decomposable: determine for which such pairs and which discrete series π\pi the multiplicities nevertheless stay finite almost everywhere, and determine whether infinite multiplicity on the continuous spectrum can occur for a discrete series, thereby delineating the precise boundary between the finite- and infinite-multiplicity regimes.

References

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