Let p be a prime and let $\mathbb Z_p$ denote the p-adic integers.

If n<m, then what are the embeddings $SL_n(\mathbb Z_p)\rightarrow SL_m(\mathbb Z_p)$? I am particularly interested in those which carry $SL_n(\mathbb Z)$ into $SL_m(\mathbb Z)$.

There are obvious "block" embeddings, e.g., carrying a matrix to the upper-left hand corner of a larger matrix. There are also certain conjugates of these. In general, the embeddings should come from representations of $SL_n(\mathbb Z_p)$, but I do not know where they are catalogued or what exactly to do with the catalog.

deletethem, so you can edit them by cut-and-pasting, editing, and then deleting the original. Regarding mathematics: SL_2(Z_p) maps onto SL_2(Z/pZ) which will I guess have a faithful representation into some SL_N(Z). Now tensor up with the identity map SL_2(Z_p)-->SL_2(Z_p) and don't I have a counterexample to your claim that all embeddings are algebraic (or did I miss something)? My guess is that the Lie algebra argument only proves they're locally algebraic. – Kevin Buzzard Nov 30 '09 at 19:47