In the 1950s, Borel, Weil, and Bott showed that the
irreducible representations of a complex reductive group can be
realized in the cohomology of line bundles on flag varieties. In the
1970s, Deligne and Lusztig constructed a family of subvarieties of
flag varieties whose cohomology realizes the irreducible
representations of reductive groups over finite fields. I will survey
these stories, explain recent progress towards finding geometric
constructions of representations of $p$-adic groups, and discuss
interactions with the Langlands program.