The truncated symbol of a differential symmetry breaking operator

TL;DR

This paper introduces a generalized symbol map for differential symmetry breaking operators, enabling classification and construction of such operators on flag varieties, with surprising connections to Cayley continuants and Krawtchouk polynomials.

Contribution

It generalizes the symbol map to non-abelian nilpotent radicals and applies the inverse to classify and construct differential operators and homomorphisms, revealing new algebraic structures.

Findings

Classification of differential intertwining operators on $SL(3,\mathbb{R})/B$

Discovery of Cayley continuants in operator coefficients

Factorization identities for operators and homomorphisms

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.