Why is p=2 special, if we want to classify cplx. representation of GL2(Zp)?

Question

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 $\neq 2$.

Answer 1

Shalika's paper that you mention deals not with $\mathrm{GL}_2(\mathfrak{o})$, but with $\mathrm{SL}_2(\mathfrak{o})$. The representation theory of $\mathrm{GL}_2(\mathfrak{o})$ is uniform in the residue characteristic $p$, in the sense that for any $r,n\in\mathbb{N}$, there exist polynomials $f_{r,n}(X)\in\mathbb{Z}[X]$ such that the number of irreducible representations of $\mathrm{GL}_2(\mathfrak{o}/\mathfrak{p}^r)$ of dimension $n$, is given by $f_{r,n}(q)$, where $q$ is the residue cardinality of $\mathfrak{o}$ (see U. Onn: Representations of automorphism groups of finite $\mathfrak{o}$-modules of rank two). Moreover, Onn gives a construction of the representations of $\mathrm{GL}_2(\mathfrak{o})$ which works uniformly for any residue characteristic $p$.

In contrast, the representation theory of $\mathrm{SL}_2(\mathfrak{o})$ depends dramatically on the parity of the residue characteristic $p$. The case when $p$ is odd has been known for a long time, but the case when $p=2$ has so far only been done when $\mathfrak{o}=\mathbb{Z}_2$ by Nobs and Wohlfart, and is very different. There are several reasons for this difference, ranging from the trivial to the less so. Some of these are:

1. Every element in $\mathbb{F}_{2^m}$ is a square, while this is not the case for $\mathbb{F}_{q}$ when $q$ is odd.
2. Two by two scalar matrices over $\mathbb{F}_{2^m}$ have trace zero,
3. The presence of wild ramification,
4. The fact that Hensel's lemma cannot be applied in the same way as when $p$ is odd to obtain the number of solutions in $\mathfrak{o}/\mathfrak{p}^r$ of equations of the form $x^2+sxy+\Delta y^2 = 1$, where $s,\Delta\in \mathfrak{p}$.

Of course, wild ramification is present also for $\mathrm{GL}_2(\mathfrak{o})$, so one cannot say that this plays a role in itself, but rather it seems that it is the combination of the above factors (and possibly others), which accounts for the dependence on the parity mentioned above.

Answer 2

Shalika constructs a bunch of representations of $p$-adic $SL_2$ using the Weil representation. In order to get genuine (non-projective) representations he needs to work with an even-dimensional vector space in this construction, so one is naturally led to look at quadratic extensions of the $p$-adic field. If the residue characteristic $p$ is not $2$, then these are quite simple to understand (there are three of the them, they're easy to write down in a pretty uniform way independent of the field, etc.). If $p=2$ there can be gobs of quadratic extensions (if I recall correctly the number grows exponentially with the degree over $\mathbb{Q}_2$), so the situation is more complicated.

I don't know if this is really a direct obstruction to the construction of these representations, but I do know that Shalika and Sally did many calculations with the explicit realizations of these representations associated to the three easy-to-write-down quadratic extensions when $p\neq 2$. In the handful of these that I've looked at or carried out, the analysis is greatly simplified by the assumption that $p\neq 2$.

Answer 3

All supercuspidals of GL(2,K) can be realized using the Weil representation along the lines of Shalika if and only if p is odd. The supercuspidals that can be realized are dihedral. In the case p even not all supercuspidals are dihedral. A good introduction to all of this is the book of Bushnell and Henniart.

While quadratic extensions are more subtle when p=2 that is not at all the most important issue.

Answer 4

A few more reasons/repetitions of reasons above in addition to the answers above :

1. 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 $p = 2$).

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.

Answer 5

Maybe I can address the final comment in the original question concerning step 1. I don't see in the classical literature any special role at this point for the prime 2, though Ramsey makes it clear that working over $p$-adic fields or integers gets more subtle.

Study of the ordinary complex representations and characters of the finite groups $GL_2(\mathbb{F}_q)$ goes back to Frobenius and Schur, followed by many others who eventually broadened the study to other finite groups of Lie type. For the classical viewpoint involving characters, a convenient source is Steinberg's 1951 Canad. J. Math. article here based on his Toronto thesis with Brauer in which he recalled this rank 1 case and went on to some higher ranks. (This all preceded J.A. Green's landmark 1955 paper in Trans. AMS treating all finite general linear groups.)