Flag varieties and representations of p-adic groups
Geometry has had a remarkable influence on representation theory over the last century. In the 1950s, Borel, Weil, and Bott showed that the irreducible representations of complex semisimple Lie groups 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 finite groups of Lie type. 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.