In his famous article [1] Klein constructs a representation of $G=PSL_2(\mathbb{F}_7)$ in $\mathbb{C}^3$ (of which the first invariant polynomial of three variables gives rise to the famous *Klein's quartic*).

All other irreducible representations of $G$ are very simple or natural to obtain: the action on $\mathbb{P}^1_{\mathbb{F}_7}$ produces a representation of dimension $7$. The isomorphism $G\simeq GL_3(\mathbb{F}_2) = PGL_3(\mathbb{F}_2)$ and the action on the Fano plane $\mathbb{P}^2_{\mathbb{F}_2}$ produces a representation of dimension $6$. There's the trivial representation, one of dimension $8$ that is simply induced from the normalizer of a $7$-Sylow subgroup (which can be also thought geometrically as these subgroups are point-stabilizers of the above mentioned actions). The two missing are Klein's representation and its conjugate. However, to construct those we appeal to explicit generators and relations by writing explicit $3\times 3$ matrices that correspond to the generators of $G$ of orders $7,3$ and $2$ (this last involution is the *hard* one to find).

Is there a geometric intuition behind this representation? and if not (as it seems to be) is there an inherent reason for this representation being "hard" to obtain?

In the fantastic article Elkies explains this last involution as the discreet Fourier transform on the space of functions $\mathbb{F}_7 \rightarrow \mathbb{C}$. To my mind this appearance of a Fourier transform is indication of some deeper explanation of this action involving some more elaborate geometric tricks as Mukai transforms more than an explanation itself. Many things that you may want to know about $G$ or Klein's quartic are beautifully explained in that article of Elkies.

Finally, it is worth noticing that this phenomenon occurs for other groups of the form $PSL_2(\mathbb{F}_q)$ where most irreducible representations are geometric in nature, except the few ones that you need to write by hand the generators.

[1]: F. Klein, “Ueber die Transformationen siebenter Ordnung der elliptischen Funktionen”, Math. Annalen 14 (1879), 428–471.

Weil representationof $\operatorname{SL}(2,q)$ ($q$ odd). The Weil rep. decomposes into two irreps of dims $(q\pm 1)/2$, where one irrep has $-I$ in the kernel and can thus be viewed as irrep of $\operatorname{PSL}(2,q)$. And the Weil rep. sends elements of order $4$ in $\operatorname{SL}(2,q)$ to the Fourier transform. See arxiv.org/abs/0903.1486 for some elementary proofs about the Weil rep in the finite case. $\endgroup$3more comments