Algebra Seminar: Romanov -- An orbit model for the spectra of nilpotent Gelfand pairs

Speaker: Anna Romanov (University of Sydney) 

Date: Friday 23 November 

Time: 12-13 

Venue: Carslaw 375 

Title: An orbit model for the spectra of nilpotent Gelfand pairs 

Abstract: Let N be a connected and simply connected nilpotent Lie group, and let K be a
subgroup of the automorphism group of N.  We say that the pair (K,N) is a nilpotent
Gelfand pair if the set of integrable K-invariant functions on N forms an abelian
algebra under convolution.  In 2008, Benson and Ratcliff established a one-to-one
correspondence between the set of bounded K-spherical functions for such a Gelfand pair
and a set of K-orbits in the dual of the Lie algebra of N.  The sets on either side of
this bijection can be given the structure of topological spaces, and Benson and Ratcliff
conjectured that this set-theoretic correspondence is actually topological.  In this
talk, we will describe Benson-Ratcliff’s construction and provide a proof of their
conjecture for a certain class of nilpotent Gelfand pairs, establishing a geometric
model for the spectrum of such pairs.  This is joint work with H.  Friedlander, W.
Grodzicki, W.  Johnson, G.  Ratcliff, B.  Strasser, and B.  Wessel.