The truncated symbol of a differential symmetry breaking operator

TL;DR

This work introduces the truncated symbol $ ext{Symb}_0$ to extend the F-method to differential symmetry breaking operators with non-abelian nilpotent radicals, enabling a unified treatment of DSBOs and Verma-module homomorphisms. For $G=SL(3, ext{R})$ with a Borel subgroup $B$, it classifies and constructs differential intertwining operators and $(rak g,B)$-homomorphisms, revealing five operator families whose coefficients involve Cayley continuants and Jacobi-polynomials, with binary Krawtchouk polynomials appearing in the proofs. The paper also develops a comprehensive framework linking $ ext{Symb}_0$ to $F_c^{-1}$, provides uniform polynomial expressions, and establishes factorization identities that decompose complex operators into compositions of simpler components. These results illuminate deep connections between representation theory (Verma modules, infinitesimal characters) and classical combinatorial objects, offering explicit, computationally tractable formulas for DSBOs in non-abelian parabolic settings.

Abstract

In this paper, we introduce the truncated symbol $\mathrm{Symb}_0(\mathbb{D})$ of a differential symmetry breaking operator $\mathbb{D}$ between parabolically induced representations. This generalizes the symbol map $\mathrm{Symb}$, which is defined for the case of abelian nilpotent radicals, to the non-abelian setting. The inverse $\mathrm{Symb}_0^{-1}$ of the truncated symbol map $\mathrm{Symb}_0$ enables one to apply a recipe of the F-method for any nilpotent radical. As an application, we classify and construct differential intertwining operators $\mathcal{D}$ on the full flag variety $SL(3,\mathbb{R})/B$ and homomorphisms $\varphi$ between Verma modules. It turned out that, surprisingly, Cayley continuants $\mathrm{Cay}_m(x;y)$ appeared in the coefficients of one of the five families of operators that we constructed. At the end, the factorization identities of the differential operators $\mathcal{D}$ and homomorphisms $\varphi$ are also classified. Binary Krawtchouk polynomials $K_m(x;y)$ play a key role in the proof.

Figures (1)

• Figure 1: Lattices of $(k,\ell)$ with $k, \ell \in \mathbb{Z}_{\geq 0}$

Theorems & Definitions (86)

• Theorem 2.3: Duality theorem
• Theorem 2.19: F-method, KP1
• Lemma 2.21
• proof
• Remark 2.26
• Theorem 3.2: cf. Hel00
• Definition 3.9
• Example 3.10
• Remark 3.11
• Lemma 3.17