arXiv subjects
Publications and source records attributed to Toshiyuki Kobayashi.
Let G be a complex semisimple Lie group and H a complex closed connected subgroup. Let g and h be their Lie algebras. We prove that the regular representation of G in $L^2(G/H)$ is tempered if and only if the orthogonal of h in g contains regular elements.
We prove a geometric criterion for the bounded multiplicity property of "small" infinite-dimensional representations of real reductive Lie groupsin both induction and restrictions. Applying the criterion to symmetric pairs, we give a full description of the triples $H \subset G \supset G'$ such that any irreducible admissible representations of $G$ with $H$-distinguished vectors have the bounded multiplicity property when restricted to the subgroup $G'$. This article also completes the proof of the general results announced in the previous paper [Adv. Math. 2021, Section 7].
Let $G$ be a semisimple real Lie group with finite center and $H$ a connected closed subgroup. We establish a geometric criterion which detects whether the representation of $G$ in $L^2(G/H)$ is tempered.
We give a complete description of the discrete spectra in the branching law $Π|_{G'}$ with respect to the pair $(G,G')=(O(p,q), O(p',q') \times O(p'',q''))$ for irreducible unitary representations $Π$ of $G$ that are "geometric quantization" of minimal elliptic coadjoint orbits. We also construct explicitly all holographic operators and prove a closed Parseval-type formula.
Let G be a real semisimple algebraic Lie group and H a real reductive algebraic subgroup. We describe the pairs (G,H) for which the representation of G in $L^2(G/H)$ is tempered. When G and H are complex Lie groups, the temperedness condition is characterized by the fact that the stabilizer in H of a generic point on G/H is virtually abelian.
Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect representation theoretic properties on function spaces, with focus on multiplicities and spectrum.
We consider branching laws for the restriction of some irreducible unitary representations $Π$ of $G=O(p,q)$ to its subgroup $H=O(p-1,q)$. In Kobayashi (arXiv:1907.07994), the irreducible subrepresentations of $O(p-1,q)$ in the restriction of the unitary $Π|_{O(p-1,q) }$ are determined. By considering the restriction of packets of irreducible representations we obtain another very simple branching law, which was conjectured in Orsted-Speh (arXiv:1907.07544).
Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $π$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types occurring in $π$ based on symplectic techniques. This leads us to a simple proof of the criterion for discrete decomposability of the restriction of unitary representations with respect to noncompact subgroups (the author, Ann. Math. 1998), and also provides a proof of a reverse statement which was announced in [Proc.ICM 2002, Thm.D]. A number of examples are presented in connection with Kostant's convexity theorem and also with non-Riemannian locally symmetric spaces.
We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces $Γ\backslash G/H$, beyond the classical setting where $H$ is compact (e.g. theory of automorphic forms for arithmetic $Γ$) or $Γ$ is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces).
Given irreducible representations $Π$ and $π$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $θ$-stable parameter necessary and sufficient conditions so that \[ \operatorname{Hom}_{G'}(Π|_{G'}, π)\not = \{0\}. \] In the special case that both $Π$ and $π$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1) \times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both $Π$ and $ π$ have the trivial infinitesimal character $ρ$ then we use a theorem that the periods are nonzero on the minimal $K$-type to obtain a nontrivial bilinear form on the $({\mathfrak g},K)$-cohomology of the representations.
The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models, whereas holographic operators increase it. Various expansions in classical analysis can be interpreted as particular occurrences of these transforms. From this perspective, we investigate two remarkable families of differential operators: the Rankin-Cohen operators and the holomorphic Juhl conformally covariant operators. Then we establish for the corresponding symmetry breaking transforms the Parseval-Plancherel type theorems and find explicit inversion formulæ with integral expression of holographic operators. The proof uses the F-method which provides a duality between symmetry breaking operators in the holomorphic model and holographic operators in the $L^2$-model, leading us to deep links between special orthogonal polynomials and branching laws for infinite-dimensional representations of real reductive Lie groups.
We investigate the structure of the ring ${\mathbb D}_G(X)$ of $G$-invariant differential operators on a reductive spherical homogeneous space $X=G/H$ with an overgroup $\widetilde{G}$. We consider three natural subalgebras of ${\mathbb D}_G(X)$ which are polynomial algebras with explicit generators, namely the subalgebra ${\mathbb D}_{\widetilde{G}}(X)$ of $\widetilde{G}$-invariant differential operators on $X$ and two other subalgebras coming from the centers of the enveloping algebras of $\mathfrak g$ and $\mathfrak k$, where $K$ is a maximal proper subgroup of $G$ containing $H$. We show that in most cases ${\mathbb D}_G(X)$ is generated by any two of these three subalgebras, and analyze when this may fail. Moreover, we find explicit relations among the generators for each possible triple $(\widetilde{G},G,H)$, and describe "transfer maps" connecting eigenvalues for ${\mathbb D}_{\widetilde{G}}(X)$ and for the center $Z({\mathfrak g}_{\mathbb C})$ of the enveloping algebra of ${\mathbb g}_{\mathbb C}$.
For a pair $(G,G')=(O(n+1,1), O(n,1))$ of reductive groups, we investigate intertwining operators (symmetry breaking operators) between principal series representations $I_δ(V,λ)$ of $G$, and $J_ε(W,ν)$ of the subgroup $G'$. The representations are parametrized by finite-dimensional representations $V,W$ of $O(n)$ respectively of $O(n-1)$, characters $δ$, $\varepsilon$ of O(1), and $λ, ν\in C$. The multiplicty [V:W] of W occurring in the restriction $V|_{O(n-1)}$ is either 0 or 1. If $[V:W] \ne 0$ then we construct a holomorphic family of symmetry breaking operators and prove that dim $Hom_{G'}(I_δ(V, λ)|_{G'}, J_ε(W, ν))$ is nonzero for all the parameters $λ$, $ν$ and $δ$, $ε$, whereas if [V:W] = 0 there may exist sporadic differential symmetry breaking operators. We propose a "classification scheme" to find all matrix-valued symmetry breaking operators explicitly,and carry out this program completely when V and W are exterior tensor representations. In conformal geometry, our results yield the complete classification of conformal covariant operators from differential forms on a Riemannian manifold X to those on a submanifold Y in the model space $(X, Y) = (S^n, S^{n-1})$. We use these results to determine symmetry breaking operators for any pair of irreducible representations of G and the subgroup $G'$ with trivial infinitesimal character. Furthermore we prove the multiplicity conjecture by Gross and Prasad for tempered principal series representations of $(SO(n+1,1),SO(n,1))$ and also for 3 tempered representations $Π, π, \varpi$ of $SO(2m+2,1)$, $SO(2m+1,1)$ and $SO(2m,1)$ with trivial infinitesimal character. In connection to automorphic form theory, we apply our main results to find "periods" of irreducible representations of the Lorentz group having nonzero (g, K)-cohomologies.
Consider the restriction of an irreducible unitary representation $π$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ν$ occurring in the restriction $π|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $π$ and $ν$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $\sharp(O^G \cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as "classical limits" of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).
Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A-C for branching problems (T.Kobayashi [Progr.Math.2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on differential forms on the model space $(X,Y)=(S^n, S^{n-1})$, which generalizes the scalar case (Kobayashi--Speh [Memoirs of Amer.Math.Soc. 2015]) and the case of local operators (Kobayashi--Kubo--Pevzner [Lecture Notes in Math. 2016]). Some applications to automorphic form theory, motivations from conformal geometry, and the methods of proof are also discussed.
We prove an explicit residue formula for a meromorphic continuation of conformally covariant integral operators between differential forms on ${\bf R}^n$ and on its hyperplane. The results provide a simple and new construction of the conformally covariant differential symmetry breaking operators between differential forms on the sphere and those on its totally geodesic hypersurface that were introduced in [Kobayashi-Kubo-Pevzner, Lect. Notes Math. (2016)]. Moreover, we determine the zeros of the matrix-valued regular symmetry breaking operators between principal series representations of $O(n+1,1)$ and $O(n,1)$.
We consider irreducible unitary representations $A_i$ of G=SO(n+1,1) with the same infinitesimal character as the trivial representation and representations $B_j$ of H=SO(n,1) with the same properties and discuss H-equivariant homomorphisms Hom_H($A_i,B_j$). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.
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.