arXiv subjects

Víctor Pérez-Valdés

Publications and source records attributed to Víctor Pérez-Valdés.

5 recordsLinked to original sources

arXiv2026

The ordered F-system for the F-method and differential symmetry breaking operators for $(GL(3,\mathbb{R}), GL(2,\mathbb{R}))$

In this paper, we introduce a new aspect of the F-system for the F-method arising in the case where the nilpotent radical is not necessarily abelian. For this, we define the symmetrization operator on the space of polynomial functions on the dual $\mathfrak{g}^\vee$ of a finite-dimensional Lie algebra $\mathfrak{g}$. In the context of the F-method, the conjugation by this operator of the F-system yields a new system, which we call the ordered F-system. These two techniques allow one to consider the symmetrized form (with symmetrization) and ordered form (without symmetrization) of differential symmetry breaking operators (DSBOs) uniformly. As an application of this theory, we classify and construct all the DSBOs between principal series representations for $(GL(3,\mathbb{R}), GL(2,\mathbb{R}))$ in both symmetrized and ordered forms. Here, we consider all embeddings $GL(2,\mathbb{R}) \hookrightarrow GL(3,\mathbb{R})$ corresponding to the positive roots of $\mathfrak{gl}(3,\mathbb{R})$. Furthermore, we utilize the DSBOs for the above pair to investigate differential intertwining operators (DIOs) for $GL(3,\mathbb{R})$ and DSBOs for $(SL(3,\mathbb{R}), SL(2,\mathbb{R}))$. The branching laws of the corresponding Verma modules are also studied to support our results of DSBOs.

math.RT↗

arXiv2026

On sporadic symmetry breaking operators for principal series representations of the de Sitter and Lorentz groups

In this paper, we construct and classify all differential symmetry breaking operators between certain principal series representations of the pair $SO_0(4,1) \supset SO_0(3,1)$. In this case, we also prove a localness theorem, namely, all symmetry breaking operators between the principal series representations in concern are necessarily differential operators. In addition, we show that all these symmetry breaking operators are sporadic in the sense of T. Kobayashi, that is, they cannot be obtained by residue formulas of meromorphic families of symmetry breaking operators.

math.RT↗

arXiv2026

Construction and classification of differential symmetry breaking operators for principal series representations of the pair $(SO_0(4,1), SO_0(3,1))$ for special parameters

We construct and give a complete classification of all the differential symmetry breaking operators $\mathbb{D}_{λ, ν}^{N,m}: C^\infty(S^3, \mathcal{V}_λ^{2N+1}) \rightarrow C^\infty(S^2, \mathcal{L}_{m, ν})$, between the spaces of smooth sections of a vector bundle of rank $2N+1$ over the $3$-sphere $\mathcal{V}_λ^{2N+1} \rightarrow S^3$, and a line bundle over the $2$-sphere $\mathcal{L}_{m, ν} \rightarrow S^2$ in the special case $|m| = N$.

math.RT↗

arXiv2025

The truncated symbol of a differential symmetry breaking operator

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 $φ$ 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 $φ$ are also classified. Binary Krawtchouk polynomials $K_m(x;y)$ play a key role in the proof.

math.RT↗

arXiv2023

Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 over S^3

We construct and give a complete classification of all the differential symmetry breaking operators D_{λ,ν}^m : C^\infty(S^3, V^3_λ) \rightarrow C^\infty(S^2,L_{m,ν}), between the spaces of smooth sections of a vector bundle of rank 3 over the 3-sphere V^3_λ \rightarrow S^3, and a line bundle over the 2-sphere L_{m,ν} \rightarrow S^2. In particular, we give necessary and sufficient conditions on the tuple of parameters (λ, ν, m) for which these operators exist.

math.DG↗