By Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner
This paintings is the 1st systematic research of all attainable conformally covariant differential operators reworking differential varieties on a Riemannian manifold X into these on a submanifold Y with concentrate on the version house (X, Y) = (Sn, Sn-1).
The authors supply a whole type of all such conformally covariant differential operators, and locate their particular formulæ within the flat coordinates when it comes to uncomplicated operators in differential geometry and classical hypergeometric polynomials. ensuing households of operators are average generalizations of the Rankin–Cohen brackets for modular kinds and Juhl's operators from conformal holography. The matrix-valued factorization identities between all attainable mixtures of conformally covariant differential operators also are established.
The major equipment of the evidence depends on the "F-method" lately brought and built via the authors. it's a normal option to build intertwining operators among C∞-induced representations or to discover singular vectors of Verma modules within the context of branching ideas, as suggestions to differential equations at the Fourier rework facet. The e-book provides a brand new extension of the F-method to the matrix-valued case within the basic atmosphere, that could be utilized to different difficulties as well.
This ebook bargains a self-contained advent to the research of symmetry breaking operators for infinite-dimensional representations of reductive Lie teams. this selection may be important for lively scientists and obtainable to graduate scholars and younger researchers in differential geometry, illustration concept, and theoretical physics.
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Extra info for Conformal Symmetry Breaking Operators for Differential Forms on Spheres
Example text
Let (σλ ,V ) and (τν ,W ) be finite-dimensional representations of P and P with trivial actions of N+ and N+ , respectively. (1) (duality) There is a natural isomorphism: ∼ DX→Y : Homg ,P (indgp (W ∨ ), indgp (V ∨ )) −→ DiffG (VX , WY ). (2) (extension) The restriction WZ |Y WY induces the bijection ∼ RestY : DiffG (VX , WZ ) −→ DiffG (VX , WY ). 4)} . 5) Then there is a natural isomorphism ∼ Homg ,P (indgp (W ∨ ), indgp (V ∨ )) −→ Sol(n+ ; σλ , τν ). 6) (4) Assume that the nilradical n+ is abelian.
Following Weyl ([30, Chap. V, Sect. 17) where [λ ] is the O(N)-irreducible summand of F(U(N), λ ) which contains the highest weight vector. 11 H s (CN ) = [s] (s ∈ N), and (CN ) = [1 ] (0 ≤ ≤ N). Moreover, Types I and II are related by the following O(N)-isomorphism: [(a1 , · · · , ak , 1, · · · , 1, 0, · · · , 0)] = det ⊗[(a1 , · · · , ak , 0, · · · , 0)]. 18) The restriction of the O(N)-module [(a1 , . . , ak , 0, . . , 0)] to the subgroup SO(N) is reducible if and only if N = 2k. In this case we have: [(a1 , · · · , ak , 0, · · · , 0)]|SO(N) = F(SO(N), (a1 , · · · , ak )) ⊕ F(SO(N), (a1 , · · · , ak−1 , −ak )).
Then λ1 is nothing but the maximal column length, denoted by (λ ). It is readily seen that Λ + (N)BD consists of elements of the following two types: Type I: (a1 , · · · , ak , 0, · · · , 0), N−k Type II: (a1 , · · · , ak , 1, · · · , 1, 0, · · · , 0), N−2k k with a1 ≥ a2 ≥ · · · ≥ ak > 0 and 0 ≤ k ≤ N 2 . Following Weyl ([30, Chap. V, Sect. 17) where [λ ] is the O(N)-irreducible summand of F(U(N), λ ) which contains the highest weight vector. 11 H s (CN ) = [s] (s ∈ N), and (CN ) = [1 ] (0 ≤ ≤ N).