Kobayashi's conjecture on admissible restrictions: does $G'$-admissibility imply $K'$-admissibility?

Let $G$ be a real reductive Lie group and $G'\subset G$ a $\theta$-stable closed reductive subgroup with maximal compact subgroup $K'=G'\cap K$. For an irreducible unitary representation $\pi$ of $G$, the restriction $\pi|_{G'}$ is called $G'$-admissible when it splits into a discrete Hilbert direct sum of irreducible unitary representations of $G'$ with finite multiplicities. Kobayashi proved (Invent. Math. 1994) that $K'$-admissibility of $\pi|_{K'}$ forces $G'$-admissibility of $\pi|_{G'}$, and conjectured the converse (Conjecture D, 2000). The conjecture, shown by Zhu and Liang (2010) to be equivalent to the equivalence of $G'$-admissibility, $K'$-admissibility, and $\mathfrak{g}'$-admissibility of the underlying Harish-Chandra module, remains open in general. It is known for discrete series representations (the square-integrable irreducible representations; Duflo-Vargas) and under auxiliary hypotheses on infinitesimal characters (Zhu-Liang). An affirmative answer would upgrade Kobayashi's symplectic cone criterion to a necessary and sufficient condition for admissible restrictions.

Background

For a real reductive Lie group $G$ (the standard class of Lie groups in Harish-Chandra's representation theory, e.g. $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$, or the metaplectic group $Mp(n,\mathbb{R})$) with Lie algebra $\mathfrak{g}$, fix a Cartan involution $\theta$ (an involutive automorphism whose fixed-point set is a maximal compact subgroup) and let $K$ be the corresponding maximal compact subgroup. By Harish-Chandra's admissibility theorem, every irreducible unitary representation $\pi$ of $G$ is $K$-admissible: its restriction to $K$ decomposes as a direct sum of irreducible representations of $K$ (called $K$-types), each occurring with finite multiplicity. For a non-compact reductive subgroup $G'\subset G$ no such theorem holds: the restriction $\pi|_{G'}$ decomposes in general as a direct integral over the unitary dual $\widehat{G'}$ (a branching law), possibly with continuous spectrum and infinite multiplicities. Kobayashi introduced the framework of admissible restrictions: $\pi|_{G'}$ is $G'$-admissible if it decomposes as a discrete Hilbert direct sum $\pi|_{G'}\simeq\bigoplus m(\tau)\tau$ of irreducible unitary representations $\tau$ of $G'$ with finite multiplicities $m(\tau)$. In this case the restriction can be studied algebraically through Harish-Chandra modules (the $(\mathfrak{g},K)$-module $\pi_K$ of $K$-finite vectors, which determines $\pi$): one says $\pi|_{G'}$ is $\mathfrak{g}'$-discretely decomposable if $\pi_K$ decomposes into a direct sum of irreducible $(\mathfrak{g}',K')$-modules, and $\mathfrak{g}'$-admissible if moreover all multiplicities in this decomposition are finite, where $\mathfrak{g}'=\mathrm{Lie}(G')$ and $K'=G'\cap K$ is a maximal compact subgroup of $G'$ when $G'$ is $\theta$-stable. Assume throughout that $G'$ is $\theta$-stable, closed, and reductive. Three fundamental results organize the subject. First (Kobayashi, Invent. Math. 117, 1994, Theorem 1.2), if $\pi|_{K'}$ is $K'$-admissible — every $K'$-type occurs with finite multiplicity — then $\pi|_{G'}$ is automatically $G'$-admissible, so the restriction has no continuous spectrum and finite multiplicities. Second (Kobayashi, Invent. Math. 131, 1998, Part III of the same series, Theorem 4.2), for the cohomologically induced modules $A_{\mathfrak{q}}(\lambda)$ (Zuckerman derived functor modules attached to a $\theta$-stable parabolic $\mathfrak{q}$, with $\lambda$ in the weakly fair range, a standard cohomological range condition) over reductive symmetric pairs, $\mathfrak{g}'$-discrete decomposability is equivalent to $K'$-admissibility. Third, the modern cone criterion (Kobayashi, Pure Appl. Math. Q. 2021, completing a program announced at ICM 2002): an admissible $(\mathfrak{g},K)$-module $X$ of finite length is $K'$-admissible if and only if $\mathrm{AS}_K(X)\cap C_K(K')=\{0\}$, where $\mathrm{AS}_K(X)$ is the asymptotic $K$-support (the limit cone of the highest weights of occurring $K$-types) and $C_K(K')$ is the momentum cone of the Hamiltonian $K$-action on the cotangent bundle $T^*(K/K')$; combined with the first result this gives a checkable sufficient condition for $G'$-admissibility. What is missing is the converse: nothing known excludes the possibility that a unitary restriction decomposes discretely with finite multiplicities while some $K'$-type occurs with infinite multiplicity across the infinitely many $G'$-constituents. Kobayashi conjectured in 2000 (Conjecture D in his Adv. Stud. Pure Math. 26 survey) that this cannot happen, and Zhu and Liang (C. R. Math. Acad. Sci. Paris 2010) proved that the conjecture is equivalent to the equality of the three admissibility notions: their Theorem 1.3 shows $\mathfrak{g}'$-admissibility implies $K'$-admissibility in general, using Kostant's theorem that the center of the universal enveloping algebra of $\mathfrak{g}'$ acts finitely — the infinitesimal character of an irreducible module being the character by which this center acts — so that the infinitesimal characters of the $\mathfrak{g}'$-constituents lie in translates of a fixed infinitesimal character by weights of $\mathfrak{g}/\mathfrak{g}'$, together with a Casimir inequality.

Problem Statement

Let $G$ be a real reductive Lie group, $\theta$ a Cartan involution, $K$ the fixed maximal compact subgroup, and $G'\subset G$ a $\theta$-stable closed reductive subgroup with maximal compact subgroup $K'=G'\cap K$. Let $\pi$ be an irreducible unitary representation of $G$ whose restriction is $G'$-admissible, that is,

$$\pi|_{G'}\simeq\bigoplus_{\tau\in\widehat{G'}} m(\tau)\,\tau \quad\text{(Hilbert direct sum)}$$

with all multiplicities $m(\tau)$ finite. Must $\pi|_{K'}$ be $K'$-admissible, i.e. must every irreducible representation $V$ of the compact group $K'$ occur with finite multiplicity, $\dim\mathrm{Hom}_{K'}(V,\pi|_{K'})<\infty$? Equivalently, are the three notions — $G'$-admissibility of the unitary restriction, $K'$-admissibility, and $\mathfrak{g}'$-admissibility of the Harish-Chandra module $\pi_K$ (the $(\mathfrak{g},K)$-module of $K$-finite vectors of $\pi$) as a $(\mathfrak{g}',K')$-module, i.e. decomposability of $\pi_K$ into a direct sum of irreducible $(\mathfrak{g}',K')$-modules with finite multiplicities — equivalent for every such triple $(G,G',\pi)$, without any auxiliary hypothesis such as $\mathfrak{q}$-relatedness of infinitesimal characters (pairwise differences of the occurring infinitesimal characters lying in the lattice of $\mathfrak{g}/\mathfrak{g}'$-weights) or reality of the occurring infinitesimal characters?

References

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