In Sweet Smell of Success (1957), Steve speaks to Susan on phone:
Susan: Where are you?
Steve: We're on our way to the Elysian to dicker with Van Cleve.
Why is first letter capatalize in the word "Elysian"?
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I would need more context to understand what they are talking about.
In English, the names of places, persons, nationalities, languages, etc. are capitalized because they are proper nouns.
Elysian refers to a place and hence is also capitalized. Absent any context, in your example this could be paradise, a Parisian avenue, a pub, etc.