## T. Kobayashi and M. Pevzner, *Differential symmetry breaking operators.
II. Rankin-Cohen operators for symmetric pairs*, Selecta Mathematica (N.S.) **22** (2016), no. 2, 847-911, Published OnLine 14 December 2015. 65 pages.
DOI: 10.1007/s00029-015-0208-8.
arXiv:1301.2111.
[old title of the preprint version: Rankin-Cohen operators for symmetric
pairs]..

Rankin-Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of *SL*(2,*R*). We address a general problem to find explicit formulæ for such intertwining operators
in the setting of multiplicity-free branching laws for reductive symmetric pairs.
For this purpose we use a new method (F-method) developed in [KP, Selecta Math. Part 1] and based on the *algebraic
Fourier transform for generalized Verma modules*.
The method characterizes symmetry breaking operators by means of certain systems of partial differential
equations of second order.

We discover explicit formulæ of new differential symmetry breaking
operators for all the six different complex geometries arising from semisimple symmetric
pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear
in these operators (Rankin-Cohen type)
in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the
branching laws of Verma modules may jump up at singular parameters.

[ DOI |
arXiv |
IHES-preprint |
preprint version(pdf) ]

© Toshiyuki Kobayashi