I am currently reading Shalika's article "Representation of the two by two unimodular group over local fields" and various other related articles, which deal with the classification of complex representation of reductive groups over local rings. It is cumbersome, that some authors consider only local rings with residue fields of characteric $p \neq 2$.

Why is $p=2$ special here?

Perhaps some words about the strategy, which one should use - at least from my perspective: Consider a finite extension $K$ of $\mathbb{Q}_p$. Let $\mathfrak{o}$ be the ring of integers in $K$ and $\mathscr{p}$ its maximal ideal. We want to classify all representation of $\mathrm{GL}_2( \mathfrak{o})$. Since we deal with a pro-$p$-group, the representations live on $\mathrm{GL}_2( \mathfrak{o}/\mathfrak{p}^n)$ for some $n>0$. We proceed by induction over $n$:

1) Classify all representation of $\mathrm{GL}_2(\mathbb{F}_q)$, where $\mathbb{F}_q$ is the residue field.

2) Use Mackey's formalism for the group extension (non split) $$ 0 \rightarrow M_{2\times2}(\mathbb{F}_q) \rightarrow \mathrm{GL}_2( \mathfrak{o}/\mathfrak{p}^n) \rightarrow \mathrm{GL}_2( \mathfrak{o}/\mathfrak{p}^{n-1}) \rightarrow 0.$$

Apparently the difficulties do already arise in step 1, since Piatesko-Shapiro in his lecture "Complex representations of $\mathrm{GL}_2$ over a finite field" only considers characteristic $\neq 2$.

Representation of the two by two unimodular group over local fields, Contributions to automorphic forms, geometry, and number theory, 1–38, Johns Hopkins Univ. Press, Baltimore, MD, 2004. Ilya Piatetski-Shapiro,Complex representations of GL(2, K) for finite fields K, Contemporary Mathematics, 16. American Mathematical Society, Providence, R.I., 1983. (The MR review of the latter makes clear his interest in ideas that carry over to local fields. That probably motivates`$p > 2$`

.) $\endgroup$ – Jim Humphreys Mar 23 '11 at 21:31