The truncated symbol of a differential symmetry breaking operator

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.