[Paper Review] Conical Distributions on the Space of Flat Horocycles

Fulton B. Gonzalez|ArXiv.org|Apr 9, 2009

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper classifies conical distributions on the space of flat horocycles in symmetric spaces associated with noncompact semisimple Lie groups. It proves that for rank-one symmetric spaces, the space of conical distributions in each generic eigenspace of the algebra of $G_0$-invariant differential operators is one-dimensional, and provides an explicit characterization of all such distributions via $M'$-invariant distributions on $K/M$. The key result establishes a complete classification when $\dim\mathfrak{a} = 1$.

ABSTRACT

Let $G_0=K\ltimes\mathfrak p$ be the Cartan motion group associated with a noncompact semisimple Riemannian symmetric pair $(G, K)$. Let $\frak a$ be a maximal abelian subspace of $\mathfrak p$ and let $\p=\a+\q$ be the corresponding orthogonal decomposition. A flat horocycle in $\p$ is a $G_0$-translate of $\q$. A conical distribution on the space $Ξ_0$ of flat horocycles is an eigendistribution of the algebra $\mathbb D(Ξ_0)$ of $G_0$-invariant differential operators on $Ξ_0$ which is invariant under the left action of the isotropy subgroup of $G_0$ fixing $\q$. We prove that the space of conical distributions belonging to each generic eigenspace of $\mathbb D(Ξ_0)$ is one-dimensional, and we classify the set of all conical distributions on $Ξ_0$ when $G/K$ has rank one.

Motivation & Objective

• To classify conical distributions on the space $\Xi_0$ of flat horocycles in the Cartan motion group $G_0 = K \ltimes \mathfrak{p}$.
• To determine the structure of the space of conical distributions in each generic eigenspace of the algebra $\mathbb{D}(\Xi_0)$ of $G_0$-invariant differential operators.
• To provide a complete characterization of conical distributions when $G/K$ has rank one, particularly in terms of $M'$-invariant distributions on $K/M$.
• To establish that the space of conical distributions in each generic eigenspace is one-dimensional, extending known results from the non-flat case.

Proposed method

• The paper uses the Cartan motion group $G_0 = K \ltimes \mathfrak{p}$, where $G_0$ acts on $\mathfrak{p}$ by $ (k,X) \cdot Y = X + k \cdot Y $, and defines flat horocycles as $G_0$-translates of $\mathfrak{q} = \mathfrak{p} \ominus \mathfrak{a}$.
• It identifies the space $\Xi_0$ of flat horocycles with $K/M \times \mathfrak{a}$ via the diffeomorphism $ (kM, H) \mapsto H + k \cdot \mathfrak{q} $, and studies $G_0$-invariant differential operators on $\Xi_0$.
• The algebra $\mathbb{D}(\Xi_0)$ of $G_0$-invariant differential operators is isomorphic to $S(\mathfrak{a})$, and joint eigendistributions are parametrized by $\lambda \in \mathfrak{a}_c^*$.
• Conical distributions are defined as joint eigendistributions of $\mathbb{D}(\Xi_0)$ that are invariant under the isotropy subgroup $M'$ of $G_0$ fixing $\mathfrak{q}$, and are constructed via integration against $M'$-invariant distributions on $K/M$.
• The proof relies on analyzing the action of $M^*$ on the root space decomposition of $\mathfrak{g}$, particularly using the antipodal map and averaging over $M'$ to show that $M'$-invariance implies $\mathfrak{q}$-invariance.
• For rank one, the classification is completed by showing that any conical distribution in $\mathcal{D}'_0(\Xi_0)$ is of the form $\Phi(\varphi) = \int_{K/M} \int_{-\infty}^\infty \widetilde{\varphi}(kM, tH) \, dt \, dT_0(kM) + \int_{K/M} \int_{-\infty}^\infty \widetilde{\varphi}(kM, tH) \, t \, dt \, dT_1(kM) $, with $T_0, T_1$ $M'$-invariant.

Experimental results

Research questions

• RQ1 What is the dimension of the space of conical distributions in each generic eigenspace of $\mathbb{D}(\Xi_0)$ for the Cartan motion group $G_0$?
• RQ2 How can conical distributions on the space of flat horocycles be characterized in terms of invariant distributions on $K/M$?
• RQ3 What is the structure of the space of conical distributions when $G/K$ has rank one?
• RQ4 Under what conditions is an $M'$-invariant distribution on $K/M$ sufficient to generate a conical distribution on $\Xi_0$?
• RQ5 Why does $M'$-invariance of a distribution in $\mathcal{D}'_0(\Xi_0)$ imply its $\mathfrak{q}$-invariance?

Key findings

• The space of conical distributions in each generic eigenspace of $\mathbb{D}(\Xi_0)$ is one-dimensional, establishing a sharp classification result.
• For rank-one symmetric spaces, all conical distributions in $\mathcal{D}'_0(\Xi_0)$ are given by $\Phi(\varphi) = \int_{K/M} \int_{-\infty}^\infty \widetilde{\varphi}(kM, tH) \, dt \, dT_0(kM) + \int_{K/M} \int_{-\infty}^\infty \widetilde{\varphi}(kM, tH) \, t \, dt \, dT_1(kM) $, where $T_0$ and $T_1$ are $M'$-invariant distributions on $K/M$.
• The $M'$-invariance of $T_0$ and $T_1$ is both necessary and sufficient for the associated distribution $\Phi$ to be conical.
• The proof shows that $M'$-invariance implies $\mathfrak{q}$-invariance, a nontrivial fact tied to the antipodal symmetry and averaging over $M'$.
• For $\lambda = 0$, the conical distributions are fully characterized by the $M'$-invariance of the spectral measures $T_0$ and $T_1$ on $K/M$, and the structure is independent of the choice of $H \in \mathfrak{a}^+$.
• The classification is complete and explicit in the rank-one case, extending previous results on conical distributions in the non-flat setting.