TITLE: Why is p=2 special, if we want to classify cplx. representation of GL2(Zp)?
QUESTION [8 upvotes]: 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 $\neq 2$.

REPLY [4 votes]: A few more reasons/repetitions of reasons above in addition to the answers above :

For $p = 2$, two different quadratic extensions can give rise to the same representation of $GL_2$ (this can happen for p not equal to 2, but only in the case of a Weil representation associated to a non-trivial character that has order 2 when restricted to the norm 1 elements of the extension). I don't know how bad this makes things, but criteria for this happening involves the discriminant (which can be pretty weird when $p = 2$).
For p = 2, the characters are hard to compute, even for the "Weil representations" Mander referred to above. This is because Weil representations are not so useful for character computation as with applying Frobenius formula to induction with a relatively simple representation of a subgroup of $GL_2(F)$ of the form $T B_n$, where T is an elliptic torus and B_n is something like a congruence subgroup. Such a representation is often a just a (quasi)character, and hence easier to handle. You see - the approach is not even to compute the character of a representation of $GL_2$ of the integers or an Iwahori but to start from a smaller subgroup times torus. The characters are computed on a torus-by-torus basis, and the presence of other tori in $T B_n$ can create havoc, when $p \neq 2$. Or so I hear. That the characters for $p \neq 2$ were computed enabled Sally and Shalika to compute Fourier transforms of orbital integrals and deduce Plancherel formula (for $SL_2$) from them.