How to understand character sheaves There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247
How to understand these for a person with less than excellent representation theory background?
 A: That's a pretty vague question.  The vague answer is that all the operations (like induction from subgroups) that can be done for representations and characters can be done for sheaves, and doing these results in a category of sheaves on a group (like GL_n over a finite field), which are close enough to the characters of representations to tell us something about them, but which also have more structure, since they are sheaves, not just functions.
Here's a somewhat more precise description: if you have a variety X which a group G acts on, then you can take the action of G on the cohomology of X.  Better yet, you can get a sheaf on the group G, whose stalk over a group element g is the cohomology of the fixed points of g on X.  The function sheaf correspondence sends this sheaf to the character of the representation on the cohomology of X (this follows from Lefschetz).  Deligne and Lusztig defined certain varieties (the set of flags over F_q in a given relative position to their conjugates by Frobenius) on which GL(n,F_q) acts (actually, this works for any split simple algebraic group), and the corresponding sheaves (or rather the simple perverse constitutuents) are called character sheaves, and roughly capture the structure of the corresponding representations.
