Non-commutative geometry begins with the Gelfand theorem asserting that commutative C* algebras and locally compact Hausdorff topological spaces are the same thing. Another classical theorem states that unital commutative finitely-generated nilpotent-free algebras (over the complex numbers) is the same thing as complex affine algebraic varieties. This can be taken as the starting point for non-commutative algebraic geometry. Based on this point of view, this talk states a conjecture within the representation theory of p-adic groups. The idea of the conjecture is that a simple geometric structure underlies many delicate and inticate results in this representation theory.
The above is joint work with Anne-Marie Aubert and Roger Plymen.