Description du sujet de thèse

Analytic and algebraic aspects of infinite dimensional branching laws.
In representation theory, a branching problem describes the behavior of an irreducible representation π of a group G when restricted to a given subgroup G'. Although a classical topic, it is often highly comlex and multifaceted, depending strongly on the underlying structures. Combinatorial rules such as the Littlewood–
Richardson rule and fusion rules (i.e., the decomposition of the tensor product of two irreducible representations) are special cases, typically involving pairs (G, G') of the form (G₁ × G₁, diag(G₁)). When (G, G') is a pair of reductive groups and π is an infinite-dimensional representation of G, branching problems include important examples such as the theta correspondence, Plancherel formulas, the Gan–
Gross–Prasad conjecture, etc.
The effective construction of operators realizing an abstract branching law in an explicit geometric model is a rapidly developing research area. Rankin–Cohen brackets, Yamabe and Paneitz operators, and higher Fefferman–Graham Laplacians are examples of such operators, with multiple applications in number
theory, conformal geometry, and theoretical physics (notably via the AdS/CFT correspondence).
Recently, we have developed a new method for constructing and analyzing such symmetry breaking operators, their inverses (holographic operators), and their 'generating series' (generating operators), for a broad class of infinite-dimensional representations of reductive Lie groups corresponding to six different
parabolic geometries. This has revealed deep connections with the theory of orthogonal polynomials in one and several variables [2,3,4].
Moreover, the algebraic study of symmetry breaking operators promotes multiplicity spaces as representations of algebraic structures related to orthogonal polynomials and mathematical physics. The Racah algebra is a fundamental example for the branching laws of tensor products of SL(2) representations [1,5] and motivates further generalizations to other branching scenarios.
The aim of this PhD project is to explore the global structure of families of symmetry breaking operators and their associated holographic transforms, by simultaneously developing both approaches through the complementary expertise of the supervisor and his collaborators within the CNRS IRL2025 FJ-LMI.
The research results will be disseminated through publications in scientific journals and presentations at conferences.
References
1. N. Crampé, L. Poulain d'Andecy, L. Vinet, A Calabi-Yau algebra with E6 symmetry and the Clebsch-Gordan series of sl(3). J. Lie Theory, 31 (2021), 1085–1112.
2. T. Kobayashi, M. Pevzner, Differential symmetry breaking operators. I. General theory and F-method, Selecta Math., 22, (2016), pp. 801–845.
3. T. Kobayashi, M. Pevzner, Differential symmetry breaking operators. II. Rankin–Cohen operators for symmetric pairs, Selecta Math., 22, (2016), pp. 847–911.
4. T. Kobayashi, M. Pevzner, A generating operator for Rankin–Cohen brackets, To appear in J. Funct. Anal., (2025).
5. Q. Labriet, L. Poulain d'Andecy, Realizations of Racah algebras using Jacobi operators and convolution identities. Adv. in Appl. Math. 153 (2024), Paper No. 102620, 31 pp.

Contexte de travail

Reims Math Laborarory counts 35 faculty members in pure and applied mathematics. The Ph.D. student will join the research group « Groups and Quantization ».

Contraintes et risques

The thesis will be affiliated with the MPSNI doctoral school of the University of Reims Champagne-Ardenne. Stays at the partner IRL in Tokyo are expected annually.