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Toshihisa Kubo

Publications and source records attributed to Toshihisa Kubo.

16 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.

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arXiv2026

Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces

In this paper we classify and construct differential symmetry breaking operators $\mathbb{D}$ from a line bundle over the real projective space $\mathbb{R}\mathbb{P}^n$ to a vector bundle over $\mathbb{R}\mathbb{P}^{n-1}$. We further determine the factorization identities of $\mathbb{D}$ and the branching laws of the corresponding generalized Verma modules of $\mathfrak{sl}(n+1,\mathbb{C})$. By utilizing the factorization identities, the $SL(n,\mathbb{R})$-representations realized on the image $\text{Im}(\mathbb{D})$ are also investigated.

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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.

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arXiv2025

On the intertwining differential operators between vector bundles over the real projective space of dimension two

The main objective of this paper is twofold. One is to classify and construct $SL(3,\mathbb{R})$-intertwining differential operators between vector bundles over the real projective space $\mathbb{RP}^2$. It turns out that two kinds of operators appear. We call them Cartan operators and PRV operators. The second objective is then to study the representations realized on the kernel of those operators both in the smooth and holomorphic setting. A key machinery is the BGG resolution. In particular, by exploiting some results of Davidson-Enright-Stanke and Enright-Joseph, the irreducible unitary highest weight modules of $SU(1,2)$ at the (first) reduction points are classified by the image of Cartan operators and kernel of PRV operators.

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arXiv2024

On the intertwining differential operators from a line bundle to a vector bundle over the real projective space

We classify and construct $SL(n,\mathbb{R})$-intertwining differential operators $\mathcal{D}$ from a line bundle to a vector bundle over the real projective space $\mathbb{RP}^{n-1}$ by the F-method. This generalizes a classical result of Bol for $SL(2,\mathbb{R})$. Further, we classify the $K$-type formulas for the kernel $\text{Ker}(\mathcal{D})$ and image $\text{Im}(\mathcal{D})$ of $\mathcal{D}$. The standardness of the homomorphisms $φ$ corresponding to the differential operators $\mathcal{D}$ between generalized Verma modules are also discussed.

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arXiv2021

Classification of $K$-type formulas for the Heisenberg ultrahyperbolic operator $\square_s$ for $\widetilde{SL}(3,\mathbb{R})$ and tridiagonal determinants for local Heun functions

The $K$-type formulas of the space of $K$-finite solutions to the Heisenberg ultrahyperbolic equation $\square_sf=0$ for the non-linear group $\widetilde{SL}(3,\mathbb{R})$ are classified. This completes a previous study of Kable for the linear group $SL(m,\mathbb{R})$ in the case of $m=3$, as well as generalizes our earlier results on a certain second order differential operator. As a by-product we also show several properties of certain sequences $\{P_j(x;y)\}_{j=0}^\infty$ and $\{Q_j(x;y)\}_{j=0}^\infty$ of tridiagonal determinants, whose generating functions are given by local Heun functions. In particular, it is shown that these sequences satisfy a certain arithmetic-combinatorial property, which we refer to as a palindromic property. We further show that classical sequences of Cayley continuants $\{\mathrm{Cay}_j(x;y)\}_{j=0}^\infty$ and Krawtchouk polynomials $\{\mathcal{K}_j(x;y)\}_{j=0}^\infty$ also admit this property. In the end a new proof of Sylvester's formula for certain tridiagonal determinant $\mathrm{Sylv}(x;n)$ is provided from a representation theory point of view.

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arXiv2018

On the space of $K$-finite solutions to intertwining differential operators

In this paper we give Peter-Weyl type formulas for the space of $K$-finite solutions to intertwining differential operators between degenerate principal series representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of $\widetilde{SL}(3,\mathbb{R})$ attached to the minimal nilpotent orbit.

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arXiv2016

Conformal symmetry breaking operators for anti-de Sitter spaces

For a pseudo-Riemannian manifold $X$ and a totally geodesic hypersurface $Y$, we consider the problem of constructing and classifying all linear differential operators $\mathcal{E}^i(X) \to \mathcal{E}^j(Y)$ between the spaces of differential forms that intertwine multiplier representations of the Lie algebra of conformal vector fields. Extending the recent results in the Riemannian setting by Kobayashi-Kubo-Pevzner [Lecture Notes in Math.~2170, (2016)], we construct such differential operators and give a classification of them in the pseudo-Riemannian setting where both $X$ and $Y$ are of constant sectional curvature, illustrated by the examples of anti-de Sitter spaces and hyperbolic spaces.

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arXiv2016

Conformal symmetry breaking operators for differential forms on spheres

We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formulæ for these new matrix-valued operators in the flat coordinates in terms of basic operators in differential geometry and classical orthogonal polynomials. We also establish matrix-valued factorization identities among all possible combinations of conformally covariant differential operators. The main machinery of the proof is the "F-method" based on the "algebraic Fourier transform of Verma modules" (Kobayashi-Pevzner [Selecta Math. 2016]) and its extension to matrix-valued case developed here. A short summary of the main results was announced in [C. R. Acad. Sci. Paris, 2016].

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arXiv2016

Classification of differential symmetry breaking operators for differential forms

We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of principal series representations of the Lie group $O(n+1,1)$. Further, we provide explicit formulæ for these matrix-valued operators in the flat coordinates and find factorization identities for them.

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arXiv2014

Vector-valued covariant differential operators for the Möbius transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb C), SL(2,\mathbb R))$ that arise from conformal geometry.

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arXiv2014

Systems of Differential Operators and Generalized Verma Modules

In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.

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arXiv2013

On the homomorphisms between the generalized Verma modules arising from conformally invariant systems

It is shown by Barchini, Kable, and Zierau that conformally invariant systems of differential operators yield explicit homomorphisms between certain generalized Verma modules. In this paper we determine whether or not the homomorphisms arising from such systems of first and second order differential operators associated to maximal parabolic subalgebras of quasi-Heisenberg type are standard.

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arXiv2011

A System of Third-Order Differential Operators Conformally Invariant under $\mathfrak{so}(8,\mathbb{C})$

In earlier work, Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to Heisenberg parabolic subalgebras in simple Lie algebras. The construction was systematic, but the existence of such a system was left open in several anomalous cases. Here, a conformally invariant system is shown to exist in the most interesting of these remaining cases. The construction may also be interpreted as giving an explicit homomorphism between generalized Verma modules for the Lie algebra of type $D_4$.

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arXiv2011

A System of Third-Order Differential Operators Conformally Invariant under $\mathfrak{sl}(3,\mathbb{C})$ and $\mathfrak{so}(8,\mathbb{C})$

In earlier work, Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to Heisenberg parabolic subalgebras in simple Lie algebras. The construction was systematic, but the existence of such a system was left open in two cases, namely, the $Ω_3$ system for type $A_2$ and type $D_4$. Here, such a system is shown to exist for both cases. The construction of the system may also be interpreted as giving an explicit homomorphism between generalized Verma modules.

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