My work on ``Automorphic forms on covering groups of GL(2)” was completed as I was a 1978-1979 member of the Institute for Advanced Study, Princeton, at the invitation of Robert Langlands. In the years 1979-1981 I was a Ritt Assistant Professor at Columbia University. Yet much of this time I spent at IHES at the invitation of Pierre Deligne, and at Université de Paris VII. From 1981 to 1985 I was an assistant Professor at Princeton University, at the invitation of Goro Shimura. During these years I developed the trace formula in the context of automorphic representations on low rank groups – including regularizing weighted orbital integrals, computing singular terms – and its applications: establishing base change liftings for GL(3), U(2), U(3), the symmetric square lifting from SL(2) to PGL(3), and transfer to GL(n) from its inner forms.
Invited by David Kazhdan to visit Harvard University in 1985-1987, we developed a simple form of the trace formula for automorphic representations with a single cuspidal component, using it in the theory of liftings, establishing (``Langlands”) correspondence of Galois and admissible (locally) or automorphic (with one cuspidal component, globally) representations of GL(n) over a function field, and extending the metaplectic correspondence from GL(2) to GL(n).
From 1987 to 2015 I served as a Professor at the Ohio State University, teaching graduate courses on various topics in representation theory, algebraic geometry, and arithmetic. I developed various topics using the ``relative trace formula”; Galois cohomology – some with my OSU postdoc Sujatha Ramdorai, and C. Scheiderer; orbital integrals; Hecke algebras. I wrote several books uniting my work in various publications. Along with Pierre Deligne we developed a counting technique -- originating with the work of Vladimir Drinfeld when n=2 -- for Galois and automorphic representations for GL(n) over a function field.