So now assume you have $T:U\to V$ and $S:V\to W$ and bases for $U,V,W$ such that the matrices for $T$ and $S$ in these bases are both in SNF. Then, evidentally, you have found a basis for $U$ and $W$ putting $ST$ in SNF (i.e. the SNF of the product is the product of the SNF).

Let's say you have a linear transformation $T:V\to W$ where $V$ and $W$ are modules over a PID (for simplicity, lets say of the same rank). Finding the SNF of this transformation is the same as finding a basis $v_1,\ldots,v_n$ for $V$ and $w_1,\ldots,w_n$ for $W$ such that $Tv_i=d_iw_i$, where the $d_i$ are the diagonal entries in the SNF.

I'm having trouble with your example. The subfield lattice of $\mathbb{F}_{p^n}$ is a chain (corresponding under the Galois correspondence to the subgroup lattice of $\mathbb{Z}/p^n\mathbb{Z}$). So in this case, uniqueness is obvious. I expect uniqueness also holds in the general case.

Could you spell out exactly what you mean by $M(\lambda)^\vee$? How does the Lie algebra act on it? If you write these things out carefully, your question may answer itself.

You can reformulate the question in terms of $J$, $A_1$, $B_1$ and $C$ only, and the input of $A$ seems irrelevant. Let $J_1=CJC^{-1}$, then $B_1=J_1A_1J_1$. So the possibilities for $B_1$ are related to the adjoint orbit of $J$ under the action of an appropriate general linear group. Maybe there is a nice description of this? Similarly, your second question is to find $[A_1,B_1]$, which again is determined up to conjugation by something in the adjoint orbit of $J$.

(Cont.) finite dimensional representations, and this is a very interesting subject. For this, I would recommend you take a look on Mathscinet at some of the papers by Chari and Pressely.

Clinton, I am sorry to press you further, but I still do not understand your question. For any simple finite dimensional Lie algebra $\mathfrak{g}$, and $q$ NOT a root of unity, the finite dimensional representation theory of $\mathfrak{g}$ and $U_q(\mathfrak{g})$ are essentially the "same" in a precise sense. On the other hand, (unless I am making a huge mental blunder) there is no finite dimensional representation theory of $U_q(\hat{\mathfrak{g}})$ for the $\hat{\mathfrak{g}}$ you have written down. If you remove the scaling element, then there is are . . .

I understand now. When $\lambda=0$, each conjugacy class in $W$ gives one class sum, so you don't get a full $|W|$ elements...

Peter- I am completely baffled by your argument. If I take $\lambda=0$ in your definition of $x$, then it seem you get $|W|$ linearly independent elements in $Z(\mathbb{C}[W])$. In fact, unless I am misunderstanding the algebra structure (in which case, I would appreciate some clarification), then your construction gives precisely elements of the form $z\sum_{w\in W}e^{w\lambda}$.

I think its the same. The braid group of type $B_n$ is generated by elements $b_0,\ldots,b_{n-1}$ subject to the relation $b_ib_j=b_jb_i$ for $|i-j|>1$, $b_ib_{i+1}b_i=b_{i+1}b_ib_{i+1}$ for $i>0$, and $b_0b_1b_0b_1=b_1b_0b_1b_0$. The map $b_0\mapsto X_1$, $b_i\mapsto T_i$ is a surjective homomorphism onto the affine Hecke algebra.

Nice answer (it was almost the one I wanted to give). The only thing I would add is that the De Concini, Kac, Kazhdan paper "Boson-Fermion Correspondence over $\mathbb{Z}$" describes a generating series for the $h_i^{\perp}$.