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This paper proves that the $\mathbb{U}$-operator attached to a cocharacter isa right root of the corresponding Hecke polynomial. This result is an importantingredient in the proof of (i) the horizontal norm relations in the context ofGross--Gan--Prasad cycles and of (ii) the generalization of Eichler--Shimurarelations.

Seed Relations for Eichler--Shimura congruences and Euler systems