# Classification of mod p Galois Representations for l not equal to p

Let $$l\neq p$$ be primes and let $$\text{G}_l:=\text{G}_{\mathbb{Q}_l}$$. Let $$k$$ be a finite field of characteristic $$p$$ and $$\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$$ a local Galois representation. Diamond gave a classification of all such Galois representations into four families $$P$$, $$S$$, $$V$$ and $$H$$.

I understand that there is no such classification when $$\text{GL}_2$$ is replaced by $$\text{GL}_n$$ for $$n>2$$ or $$\text{GSp}_{2n}$$ for $$n>1$$. Does one expect that there should be a generalization of Diamond's classification which has not been discovered yet, or is there a reason why no such description should exist as in the $$\text{GL}_2$$ case?