1. To persist for a long time, as of a significant issue or problem. The problems between them go deep, so I doubt they'll be reconciling any time soon.
2. sports slang In American football, for a receiver to run down the field in order to catch a long pass. Often used as as imperative. Jackson, run a slant. Williams, you go deep.
To be well established and strongly integrated into the nature or character of something. In this part of the country, college basketball rivalries run deep. Among that demographic, loyalty to the party runs deep, so I don't think you'll find many undecided voters.
COMMON If something such as a feeling, emotion, or problem runs deep or goes deep, it is very serious or strong, often because it has existed for a long time. My allegiance to Kendall and his company ran deep. Hatred of the army runs deep, and most here have come to side with the rebels. His anger and anguish clearly went deep.