# Symmetry Breaking for Orthogonal Groups and a Conjecture by B. Gross and D. Prasad

## Abstract

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 \(\operatorname {Hom}_H(A_i,B_j)\). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.

## Keywords

Special orthogonal group Tempered representations Branching laws Gross-Prasad conjectures## Notes

### Acknowledgements

Many of the results were obtained while the authors were supported by the Research in Pairs program at the Mathematisches Forschungsinstitut in Oberwolfach (MFO), Germany.

The research by T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science.

The research by B. Speh was partially supported by NSF grant DMS-1500644. Part of this research was conducted during a visit of the second author at the Graduate School of Mathematics of the University of Tokyo, Komaba. She would like to thank it for its support and hospitality during her stay.

The authors thank an anonymous referee for careful reading of the manuscript and comments.

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