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The Theta Correspondence over R

These notes were written for the workshop on the Theta correspondence held at the University of Maryland, May 1994. They have circulated informally for many years. Except for minor changes and some improvements to the references they have not been changed. There have been too many developments in the field since 1994 to try to bring them up to date. This is dedicated to Roger Howe, who introduced me to representation theory in graduate school in 1977.