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.
Categories: math.RT math.DG