The following question seems pretty natural, but searching online
and looking in some obvious places didn't turn up much, so maybe
I can ask it here. (Disclaimer: I'm a newcomer to this topic, so
apologies if the question is obviously misguided.)


Suppose that $V\_1 = X\_1/\Gamma\_1$ and $V\_2= X\_2/\Gamma\_2$ are
arithmetic quotients of Hermitian symmetric domains. Let $V\_1^\ast$
and $V\_2^\ast$ be their respective Baily--Borel compactifications. Now suppose
we have an analytic map $f: V\_1 \rightarrow V\_2$. Does it extend
to a morphism $f^\ast: V\_1^\ast \rightarrow V\_2^\ast$?


If the answer in general is no, are there any nontrivial cases in which it is yes? (For instance, the simplest example that comes to mind is the case where $X\_1=X\_2$, $\Gamma\_1 \subsetneq \Gamma\_2$, and $f: V\_1 \rightarrow V\_2$ is the quotient.)



Is the Baily--Borel compactification functorial?
A useful reference might be the article "Satake Compactification and extension of Holomorphic Mappings", Inv.Math. 16, 237-248, 1972, by Kiernan and Kobayashi. They show that if the map $V\_1 \to V\_2$ is induced from a map $X\_1 \to X\_2$ then it extends. In particular, the answer is positive for your "simplest example", though that presumably follows from the construction of the compactification.


