[Paper Review] The bounded spherical functions for the free two step nilpotent Lie group

Véronique Fischer|arXiv (Cornell University)|Dec 8, 2010

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper provides a complete characterization of bounded spherical functions for free two-step nilpotent Lie groups $N_p$ under the action of orthogonal groups $O(p)$ or $SO(p)$, using representation theory and harmonic analysis. It derives explicit expressions for these functions, computes their eigenvalues under the Kohn sub-Laplacian, and determines the radial Plancherel measure, offering a full spectral decomposition for $K$-invariant $L^2$ functions on $N_p$. The results generalize known cases like the Heisenberg group and extend to higher-rank free nilpotent groups.

ABSTRACT

In this paper, we give the expressions for the bounded spherical functions, or equivalently the spherical functions of positive type, for the free two-step nilpotent Lie groups endowed with the actions of orthogonal groups or their special subgroups. Next we deduce some results about the (Kohn) sub-Laplacian, and we compute the radial Plancherel measure.

Motivation & Objective

• To provide a complete and rigorous derivation of all bounded spherical functions for the free two-step nilpotent Lie group $N_p$ under the action of $O(p)$ or $SO(p)$, extending known results for the Heisenberg group.
• To compute the eigenvalues of the Kohn sub-Laplacian acting on these spherical functions.
• To determine the radial Plancherel measure for $K$-invariant functions on $N_p$, enabling spectral decomposition in the context of Gelfand pairs.
• To establish a connection between spherical functions, Laguerre and Hermite functions, and the representation theory of $N_p$ via induced representations.

Proposed method

• Construct bounded spherical functions via induced representations $\Pi_{r,\Lambda,l}$ from irreducible representations of the stabilizer subgroup in $O(p)$, using the Gelfand pair structure $(N_p, O(p))$.
• Use the canonical realization of $\mathcal{N}_p = \mathcal{V} \oplus \mathcal{Z}$ with $\mathcal{V} \cong \mathbb{R}^p$, $\mathcal{Z} \cong \mathfrak{so}(p)$, and define the Lie bracket via $[X,Y](V) = \langle X,V\rangle Y - \langle Y,V\rangle X$.
• Express spherical functions $\phi^{r,\Lambda,l}$ as matrix coefficients of the representation $\Pi_{r,\Lambda,l}$, using orthonormal bases of Laguerre and Hermite functions.
• Compute the action of the Kohn sub-Laplacian $L = -\sum_{i=1}^p X_i^2$ on these functions via the differential representation $d\Pi(L)$, yielding eigenvalues $\sum_{j=1}^{p_1} \lambda_j(2l_j + m_j) + r^2$.
• Derive the radial Plancherel measure $m^\natural$ as the tensor product of $\eta'$ on $\mathcal{L}$, the counting measure on $\mathbb{N}^{p'}$, and $\tau$ on $\mathbb{R}^+$, with normalization constant $c(p)$.
• Use polar decomposition of antisymmetric matrices and the invariance of the measure $d\eta$ under $O(p)$ to justify the Plancherel formula.

Experimental results

Research questions

• RQ1 What are the complete expressions for the bounded spherical functions on the free two-step nilpotent Lie group $N_p$ under the action of $O(p)$ or $SO(p)$?
• RQ2 How do the spherical functions relate to the eigenfunctions of the Kohn sub-Laplacian on $N_p$?
• RQ3 What is the radial Plancherel measure for $K$-invariant functions on $N_p$ in the Gelfand pair $(N_p, O(p))$?
• RQ4 How can the spectral decomposition of $L^2(N_p)$ be realized via spherical functions and associated measures?

Key findings

• The bounded spherical functions on $N_p$ are explicitly given by $\phi^{r,\Lambda,l}$, constructed as matrix coefficients of the induced representation $\Pi_{r,\Lambda,l}$, with $r \in \mathbb{R}^+$, $\Lambda \in \mathcal{L}$, and $l \in \mathbb{N}^{p'}$.
• The Kohn sub-Laplacian $L$ acts on $\phi^{r,\Lambda,l}$ with eigenvalue $\sum_{j=1}^{p_1} \lambda_j(2l_j + m_j) + r^2$, where $\lambda_j$ are the positive eigenvalues of the antisymmetric matrix $A$ and $m_j$ is the multiplicity of $\lambda_j$.
• The radial Plancherel measure $m^\natural$ is the product of the measure $\eta'$ on $\mathcal{L}$, the counting measure on $\mathbb{N}^{p'}$, and $\tau$ on $\mathbb{R}^+$, normalized by $c(p)$, and satisfies $\|\psi\|^2_{L^2(N)} = \int |\langle \psi, \phi^{r,\Lambda,l} \rangle|^2 dm^\natural(r,\Lambda,l)$ for $K$-invariant $\psi \in L^2(N)$.
• The non-radial Plancherel measure $m$ is the tensor product of the Haar measure $dk$ on $O(p)$, $\eta'$ on $\mathcal{L}$, and $\tau$ on $\mathbb{R}^+$, with normalization constant $c(p)$, and satisfies the full Plancherel formula for $L^2(N)$.
• The measure $d\eta(\Lambda)$ is defined as $c \prod_{j<k} (\lambda_j^2 - \lambda_k^2)^2 d\Lambda$ for $p=2p'$, and $c \prod_i \lambda_i^2 \prod_{j<k} (\lambda_j^2 - \lambda_k^2)^2 d\Lambda$ for $p=2p'+1$, ensuring the polar decomposition of $\mathcal{A}_p$.
• The construction via induced representations and the use of Laguerre and Hermite functions provide a complete and explicit spectral theory for $K$-invariant functions on $N_p$, generalizing the Heisenberg group case.