Noncommutative Geometry and Calogero-Moser space

Victor Ginzburg

Abstract: In this talk (and the one at UC Berkeley following it) we discuss new unexpected relations among three apparently different objects: 1) right ideals in the algebra C[z, d/dz] of polynomial differential operators in one variable 2) Calogero-Moser phase space 3) 'Adelic Grassmannian' introduced by G. Wilson (Invent. Math. 1998). Calogero-Moser space is, according to Kazhdan-Kostant-Sternberg (1978), the set of GL_N-conjugacy classes of pairs (X,Y) of trace zero matrices such that the matrix XY-YX -1 is conjugate to the rank 1 diagonal matrix: diag(0,...,0,-n). It turns out that the Calogero-Moser space should be thought of as a non-commutative deformation of the Hilbert scheme of N points in the 2-plane, in the same sense as the algebra C[x, d/dx] should be thought of as a non-commutative deformation a of the polynomial algebra C[x,y]. We will show how recent results of Berest-Wilson can be explained naturally by replacing some familiar concepts of Symplectic Geometry by those of `Non-commutative Symplectic Geometry', introduced by Kontsevich (1994) for totally different reasons. Connections with Cyclic homology and `Representation' Functor, and Koszul operads will be discussed.

created Fri Mar 17 16:38:49 PST 2000