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In this post I want to briefly record the following nice recipe for predicting the dimension of a Galois representation you expect to extract from a Shimura variety (and the local factors you expect to see). More precisely, let’s suppose I have a cohomological cuspidal representation $\pi$ of a reductive group $G/F$. Then Langlands would conjecture the existence of a map

$Gal(\bar{F}/F) \rightarrow ^LG(\bar{\mathbb{Q}_l})$.

However, it’s not clear the L-group necessarily has a preferred representation, and ultimately this is what we will construct in the cohomology of our Shimura variety. To see where this extra information comes from, recall that the Shimura variety is defined by both the group $G$ and a map $h:\mathbb{S} \rightarrow G_\mathbb{R}$. This latter datum can be made into a Hodge cocharacter $\mu: \mathbb{G}_m \rightarrow G$. But a cocharacter of $G$ is a character of its Langlands dual, i.e. a weight, and it is possible to take the irreducible algebraic representation with this as highest weight, and extend it to the whole L-group.

Composing the map predicted by Langlands reciprocity with this representation gives the Galois representation we might hope to find in the cohomology of our Shimura variety. As a nice bonus, the cocharacters floating around are miniscule, so computing the dimensions of these irreducible reps is really easy (the weight spaces are multiplicity free and just Weyl-translates of the highest weight).