Irreducible representations of $\text{SL}(2, \mathbb{F}_q)$ which don't exist in decomposition?

Question

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0$. Hence, $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group homomorphism $\chi: \mathbb{F}^\times \to S^1 \subset \mathbb{C}^\times$, in $\mathbb{C}\{X\}$, we define a subspace$$\mathbb{C}\{X\}^\chi := \{f \in \mathbb{C}\{X\} : f(z \cdot x) = \chi(z) \cdot f(x), \text{ for all }z \in \mathbb{F}^\times\}.$$My question is, do there exist irreducible representations of $G$ not occur in the decomposition of $\mathbb{C}\{X\}$? I would preferably like to see a way of attacking this which doesn't use too "high-tech" machinery.

Answer 1

What you have described is known as the "principal series representations" of $G={\rm SL}(2,\Bbb{F}_q)$ (representations induced from a character of the Borel subgroup), which are parametrized by the characters $\chi$ of a split torus of $G$. Roughly half of irreducible representations of $G$ arise in this way, with the other half being associated with the characters of a non-split torus. One way to see how much you are missing is via Burnside's formula asserting that the dimensions squared of irreducible representations add up to the order of the group, $$ \sum d_i^2=|G|. $$ Induced representations corresponding to $\chi$ and $\chi^{-1}$ are isomorphic and for $\chi\ne \chi^{-1}$ (generic case), the corresponding representation is irreducible (otherwise, it splits into 2 irreducible components). Thus from the principal series representations, you get about $(q-1)/2$ irreducible representations of dimension $q+1$ each, or about $q^3/2$ contribution to $\sum d_i^2$, whereas the order of ${\rm SL}(2,\Bbb{F}_q)$ is about $q^3$.

Answer 2

Yes. The character table of SL(2,q) is known. If q is odd, the sum of the dimensions of the irreducible representations of SL(2,q) is q²+q. If q is even, this sum is q². In either case, this is bigger than the dimension of $\mathbb{C}\{X\}$, so there must exist irreps not in $\mathbb{C}\{X\}$.

Answer 3

I'll add an overlong comment to provide more perspective, in community-wiki format. The question is probably a bit misguided, even taken as a purely pedagogical one (the older mathematics involved having been developed over a century ago by Frobenius and Schur).

In the context of finite group representations, the groups $SL(2,q)$ provide an excellent example of how the classical theory works and are treated in a number of textbooks as well as in the notes by Mark Reeder linked by Peter. It's a good way to see how basic ideas in linear algebra, group theory, and field theory combine to produce some nontrivial results. On the other hand, the original computation of the character table (which encapsulates most of the essential information about the irreducible representations) involves an ad hoc step after the study of the principal series. The problem is, as the question recognizes, that it's not immediately clear why you can't get all the desired representations via parabolic induction (here from the Borel subgroup of upper triangular matrices). The original calculations by Frobenius (over prime fields) and then by Schur (over arbitrary finite fields) are clever but don't provide a conceptual solution.

Some "high-tech" machinery may actually be essential here, to understand where the missing representations come from. The character table can be produced in this case by some clever moves involving the orthogonality relations, but beyond rank one progress gets very difficult. (J.A. Green used combinatorial and recursive methods for finite general linear groups, while his student Srinivasan pushed the ideas as far as $Sp(4,q)$.)

Only around 1976 did the much more sophisticated Deligne-Lusztig approach get developed, followed by Lusztig's extensive refinements; this is exposed in the rank one case in a concise text by Cedric Bonnafe (Springer, 2011). Much earlier, my 1975 expository article in the Math Monthly here laid out the Frobenius-Schur approach (working over a prime field, just for convenience), still with their ad hoc flavor.

Ultimately the pedagogy depends on what students actually know and where they intend to go next, but the question raised here about avoiding too much "high-tech" machinery probably has no good answer. The suggestions by Peter and Victor rely heavily on knowing the classical theory (relating representations and characters) as well as the main conclusions about characters of $SL(2,q)$, but they suggest no real explanation of what is going on.