Blattner-type multiplicity formula for discretely decomposable restrictions of Zuckerman modules $A_{\mathfrak q}(\lambda)$ to symmetric subgroups (Ørsted–Speh conjecture)

When a Zuckerman derived functor module $A_{\mathfrak q}(\lambda)$ of a real reductive group $G$ restricts discretely decomposably to a symmetric subgroup $H$, Ørsted and Speh (2008) conjectured that the branching law has an explicit algebraic shape: there is a well-aligned pair of $\theta$-stable parabolic subalgebras $(\mathfrak p, \mathfrak p^H)$, related to $(\mathfrak q, \mathfrak q^H)$ by the compact Weyl group, such that every irreducible $H$-summand $V$ is isomorphic to $A_{\mathfrak p^H}(\mu_V)$ for some character $\mu_V$ of the Levi subgroup of $\mathfrak p^H$ (or, when $\mu_V$ lies outside the fair range and $A_{\mathfrak p^H}(\mu_V)$ is reducible, to an irreducible constituent of it) and the multiplicity of each $H$-type is given by an alternating Blattner-type formula involving Lie algebra homology of $\mathfrak u\cap\mathfrak h$ and symmetric powers of $\mathfrak u\cap\mathfrak s$. The conjecture is proved by the authors only for specific $SL(4,\mathbb R)$ restrictions (to subgroups isomorphic to $Sp(2,\mathbb R)$ and $GL(2,\mathbb C)$, all multiplicities one), and Oshima (2015) established the corresponding general upper bound — every constituent embeds into some $A_{\mathfrak q'}(\lambda')$ — together with the associated-variety equality, but the exact multiplicity formula is open in general. The problem is to prove or refute the conjecture for all symmetric pairs and all $K\cap H$-admissible $A_{\mathfrak q}(\lambda)$ in the well-aligned setup of the Problem Statement, with any answer consistent with the classical Blattner formula at $H=K$ and the established $SL(4,\mathbb R)$ cases.

Background

A central problem in the representation theory of real reductive Lie groups is the branching problem: decompose the restriction of an irreducible unitary representation of a group $G$ to a closed subgroup $H$. Unlike the compact case, for a non-compact reductive subgroup such a restriction typically has continuous spectrum and infinite multiplicities, and no general theory of branching laws exists.

The representations at issue here are the Zuckerman derived functor modules $A_{\mathfrak q}(\lambda)$, also called cohomologically induced modules. For a real reductive group $G$ with maximal compact subgroup $K$ and Cartan involution $\theta$ (the involution giving the Cartan decomposition $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$), a $\theta$-stable parabolic subalgebra $\mathfrak q=\mathfrak l\oplus\mathfrak u$ of the complexified Lie algebra $\mathfrak g$ is one whose Levi factor $\mathfrak l$ is stabilized by $\theta$; from a unitary character $\lambda$ of the Levi subgroup $L$ one obtains a $(\mathfrak g,K)$-module $A_{\mathfrak q}(\lambda)$ by derived-functor cohomological induction (the standard reference is the book of Knapp and Vogan). These modules include the discrete series and many of the known isolated unitary representations; $A_{\mathfrak q}(\lambda)$ is guaranteed irreducible and unitary when $\lambda$ lies in the good or fair range of parameters, and may be reducible outside it.

Take $H$ to be a symmetric subgroup, i.e. the connected fixed-point set of an involution $\sigma$ of $G$ (so $\mathfrak g=\mathfrak h\oplus\mathfrak s$, the $\pm1$ eigenspaces of $\sigma$). A theorem of Kobayashi (building on his 1994 Inventiones paper and its sequels) gives a discrete/continuous alternative: the restriction of $A_{\mathfrak q}(\lambda)$ to $H$ decomposes as a direct sum of irreducible unitary representations of $H$ exactly when $A_{\mathfrak q}(\lambda)$ is $K\cap H$-admissible (every $K\cap H$-type occurs with finite multiplicity), a condition testable by a micro-local criterion on associated varieties. In this discretely decomposable case, writing $K^H=K\cap H$, an irreducible $(\mathfrak h,K\cap H)$-module $V$ with $\operatorname{Hom}_{(\mathfrak h,K^H)}(V,A_{\mathfrak q})\neq 0$ is called an $H$-type of $A_{\mathfrak q}$, and $\dim\operatorname{Hom}_{(\mathfrak h,K^H)}(V,A_{\mathfrak q})$ is its multiplicity.

When $H$ is the maximal compact subgroup $K$, the restriction is always discretely decomposable and the multiplicities of $K$-types in $A_{\mathfrak q}(\lambda)$ are given by the Blattner formula, an explicit alternating sum (Blattner's conjecture for discrete series was proved by Hecht and Schmid; the generalization to $A_{\mathfrak q}(\lambda)$ is in Knapp–Vogan). Ørsted and Speh, in their study of the small unitary representation $A_{\mathfrak q}$ of $SL(4,\mathbb R)$ (the module with the infinitesimal character of the trivial representation, realized also as a residual automorphic representation of $GL(4)$), computed its restriction to the symmetric subgroups $H_1\simeq Sp(2,\mathbb R)$ and $H_1'\simeq GL(2,\mathbb C)$: both restrictions are multiplicity-one direct sums of modules cohomologically induced from $\mathfrak q\cap\mathfrak h$, while the restriction to two other symmetric subgroups $H_2$, $H_2'$ has continuous spectrum. Their key geometric notion is that of well-aligned parabolics: a point $x_0$ in the compact Cartan $T^H$ of $H$ defines simultaneously a $\theta$-stable parabolic $\mathfrak q$ of $\mathfrak g$ and $\mathfrak q^H=\mathfrak q\cap\mathfrak h$ of $\mathfrak h$; pairs $(\mathfrak p,\mathfrak p^H)$ obtained from a base point $y_0$ $W_K$-conjugate to $x_0$ (where $W_K$ is the compact Weyl group) are called related to $(\mathfrak q,\mathfrak q^H)$, and up to $H$-conjugacy there are at most $|W_K/W_{K\cap H}|$ of them. Based on their $SL(4,\mathbb R)$ calculations they formulated a general conjecture, whose $H=K$ specialization recovers the classical Blattner formula.

Subsequent work has clarified the landscape but not settled the conjecture. Kobayashi's program organizes branching problems into stages (Stage A: when is the restriction discretely decomposable; Stage B: find the explicit branching law; Stage C: construct the intertwining operators). Kobayashi and Oshima completed a Stage-A-type classification of the symmetric pairs for which discretely decomposable restrictions exist at all, and Oshima, using $D$-module localization of cohomological induction on complex flag varieties, proved that whenever $A_{\mathfrak q}(\lambda)|_H$ is discretely decomposable there is an injective $(\mathfrak h,K^H)$-map of $A_{\mathfrak q}(\lambda)$ into a direct sum of modules $A_{\mathfrak q'}(\lambda')$ with explicit Blattner-type multiplicities — an upper bound, not yet an equality — so that in particular every irreducible constituent is a submodule of some cohomologically induced module of $H$ (with $\lambda'$ possibly outside the weakly fair range).

Problem Statement

Let $G$ be a connected linear semisimple real Lie group with maximal compact subgroup $K$ and Cartan involution $\theta$, and let $H$ be the connected fixed-point set of an involution $\sigma$ of $G$ which commutes with $\theta$ (a symmetric subgroup; as in the source paper, $K^H=K\cap H$ is then maximal compact in $H$ and the compact Cartan subgroup $T^H$ of $H$ satisfies $T^H=T\cap H$), with complexified Lie algebras $\mathfrak g=\mathfrak h\oplus\mathfrak s$, the $\pm1$ eigenspaces of $\sigma$. Let $A_{\mathfrak q}(\lambda)$ be the Zuckerman derived functor module attached to a $\theta$-stable parabolic subalgebra $\mathfrak q=\mathfrak l\oplus\mathfrak u$ of $\mathfrak g$ and a unitary character $\lambda$ of the Levi subgroup $L$, and suppose $A_{\mathfrak q}(\lambda)$ is $K\cap H$-admissible, so that its restriction to $H$ is discretely decomposable (a direct sum of irreducible unitary representations of $H$, by Kobayashi's theorem). Suppose further that $(\mathfrak q,\mathfrak q^H)$ is well aligned in the sense of the source paper: a fixed point $x_0$ in the compact Cartan subgroup $T^H$ of $H$ defines simultaneously the $\theta$-stable parabolic subalgebra $\mathfrak q$ of $\mathfrak g$ and $\mathfrak q^H=\mathfrak q\cap\mathfrak h$, a $\theta$-stable parabolic subalgebra of $\mathfrak h$, and such a pair $(\mathfrak q,\mathfrak q^H)$ is called well aligned. Write $L$ and $L^H$ for the centralizers of $x_0$ in $G$ and in $H$, the Levi subgroups of $\mathfrak q$ and $\mathfrak q^H$ (so $L^H=L\cap H$). Call a well-aligned pair $(\mathfrak p,\mathfrak p^H)$ related to $(\mathfrak q,\mathfrak q^H)$ if it is defined by a point $y_0\in T^H$ conjugate to $x_0$ by the compact Weyl group $W_K$ (so $A_{\mathfrak p}\cong A_{\mathfrak q}$); up to $H$-conjugacy there are at most $|W_K/W_{K\cap H}|$ related pairs.

Ørsted–Speh conjecture: prove or refute that there exists a related well-aligned pair $(\mathfrak p,\mathfrak p^H)$ such that:

1. every $H$-type $V$ of $A_{\mathfrak q}(\lambda)$ — i.e. every irreducible $(\mathfrak h,K\cap H)$-module with $\operatorname{Hom}_{(\mathfrak h,K^H)}(V,A_{\mathfrak q})\neq 0$ — is isomorphic to $A_{\mathfrak p^H}(\mu_V)$ for some character $\mu_V$ of the Levi subgroup of $\mathfrak p^H$ (or, when $\mu_V$ lies outside the fair range and $A_{\mathfrak p^H}(\mu_V)$ is reducible, to an irreducible constituent of it); and

2. the multiplicity of $V$ is given by the Blattner-type formula

$$\dim\operatorname{Hom}_{\mathfrak h,K^H}(V,\,A_{\mathfrak q}) =\sum_i\sum_j (-1)^{s-j}\,\dim\operatorname{Hom}_{L\cap H}\big(H_j(\mathfrak u\cap\mathfrak h,\,V),\;S^i(\mathfrak u\cap\mathfrak s)\otimes\mathbb C_{\lambda_H}\big),$$

where $\mathfrak u$ is the nilradical of $\mathfrak q$, $s=\dim(\mathfrak u\cap\mathfrak k)$, $H_j$ denotes Lie algebra homology, $S^i$ the $i$-th symmetric power, and $\mathbb C_{\lambda_H}$ is the one-dimensional representation of $L^H$ defined by $\lambda\otimes\wedge^{\mathrm{top}}(\mathfrak u\cap\mathfrak s)$, the source paper's convention; the Hom spaces on the right are understood in the admissible $(\mathfrak l\cap\mathfrak h,\,L^H\cap K)$-module sense, as in the source paper's worked cases (equivalently, the left side may be written $\dim\operatorname{Hom}_{\mathfrak h,K^H}(V,A_{\mathfrak p})$ since $A_{\mathfrak p}\cong A_{\mathfrak q}$).

An answer must hold in the stated generality — all symmetric pairs $(G,H)$ and all $K\cap H$-admissible $A_{\mathfrak q}(\lambda)$ in the well-aligned setup above, including the cases admitted by the Kobayashi–Oshima classification — and must be consistent with the two established boundary cases: the classical Blattner formula when $H=K$, and the multiplicity-one branching laws of $SL(4,\mathbb R)$ to $Sp(2,\mathbb R)$ and $GL(2,\mathbb C)$ computed by Ørsted and Speh.

References

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