I would like to create (in Julia) a 2 dimensional array Y storing the spherical harmonics Y_lm(x) evaluated at some fixed x, indexed by an integer l>=0 and -l<=m<=l.
How can I create the array Y such that I may access elements via tuples, e.g, to access Y_20(x) I would call Y[(2,0)]?
More generally does Julia allow arrays indexed by tuples (x1,...xn) if we don't know anything about the possible range of the xi (like a dictionary, but indexed by tuples of integers instead of strings)?
Short answer, that isn't what arrays are "for" in Julia, this is what Dict is for. In Julia (and many languages) what is generally meant by an array is something that is indexed by a series of contiguous integer values. (That said, you can implement your own object implementing the Array interface that might work differently...).
A Dict allows for any arbitrary set of indices, that can be any type you want, not just strings. For example:
Y = Dict()
Y[(2,0)] = "Hello, World"
println(Y[(2,0)])
For your particular problem there may be a more efficient solution, but I don't know enough about spherical harmonics to know what it would be. It would be worth looking at the package mentioned in the comments. It probably has a more idiomatic approach.