# A question related to supercuspidal representations of $\operatorname{GL}_2$ over local fields

I was learning about the representation of $$\operatorname{GL}_2$$ over local fields and came to know something like: if the residual characteristic of the local field is an odd prime, then every supercuspidal representation is a dihedral supercuspidal representation. I know what is a dihedral supercuspidal representation (I read it in Bump's Automorphic forms book).

But I could not find the previous statement when residual characteristic is odd. Please suggest some good references for this.

From the first two lines it is pretty clear that if the residual characteristic of the local field is even, then there are non-dihedral supercuspidal representations. I also want to know about them. Please suggest some references for this too.

• What is a dihedral supercuspidal representation? Where are you learning about supercuspidals of $\operatorname{GL}_2$? (The second question just to know what might be useful further references.) – LSpice Feb 16 '20 at 2:14
• Let $E/F$ be a quadratic extension of non-Archimedean locan fields and let $\zeta$ be a quasicharacter of $E^*$ that does not factor through the norm map $N: E^* \to F^*$. Using this yoy will be able to construct irrducible admissible representation of $GL(2,F)$_+ and induce that to $GL(2,F)$. The induced representation is irreducible and supercuspidal which is called digedral supercuspidal representation. For more details, you can look page no 541 from Bump's automorphic form and representation book. – Kiddo Feb 16 '20 at 14:52

## 2 Answers

When the residue characteristic is odd, all supercuspidal representations of GL(2,F) arise from admissible pairs, and via the LLC they correspond two dimensional irreducible continuous representations of Weil group of F.

So I think the word dihedral here should mean that the image of the corresponding Galois representation (in PGL(2, C)) is dihedral.

For details you may check Bushnell—Henniart’ book on GL_2.

• Hi, are you talking about the book The Local Langlands Conjecture for GL(2)?? – Kiddo Feb 16 '20 at 14:55
• @Kiddo yes, that is the book mentioned in my comment above. – Peng Feb 16 '20 at 17:23

I take your word that 'dihedral' is Bump's terminology (I don't have the book to hand), but I think that the modern terminology for such supercuspidals is 'tame'. I don't know the first place where it is proven that all supercuspidals of $$\operatorname{GL}_2$$ are tame in odd residual characteristic; certainly it's proven in Bushnell and Henniart - The local Langlands conjecture for $$\operatorname{GL}(2)$$ (MSN), as @Peng mentions, but that's not the earliest place—although it is probably the best modern reference if this exact fact is what you want to learn. The result you want in the odd-residual-characteristic case is Theorem 20.2; and the non-dihedral (I think usually called exceptional) case is discussed in §45 ff.

The idea of constructing supercuspidals by induction goes back to Mautner in the '60s, although I don't know a reference (Mautner - Spherical functions over $$\mathfrak p$$-adic fields (MSN) is related, but seems to be about the spherical case); and I think an exhaustion result was first proven in Gel'fand and Graev - Irreducible unitary representations of the group of second-order unimodular matrices with elements in a locally compact field (MSN) and by Shalika - Representations of the two by two unimodular group over local fields (MSN), but for $$\operatorname{SL}_2$$ (which is harder). (I for one am grateful that authors are no longer afraid of using math mode in titles.) I believe that the current state of the art in such results is Kim - Supercuspidal representations: An exhaustion theorem (MSN), where one should also mention the work of Fintzen, specifically Fintzen - Types for tame $$p$$-adic groups, which improves on Kim's hypotheses. I think one of the first references for what you call the non-dihedral case is Kutzko - The exceptional representations of $$\operatorname{Gl}_2$$ (MSN).