Research interests

My research is in the field of representation theory. In particular, I am interested in the study of the unitary dual of real reductive groups. Finding all the unitary irreducible representations of such groups is a really fascinating problem (see BCP). I thank my undergraduate advisor (Prof. Massimo Picardello, from the University of Rome II, Tor Vergata) and my graduate advisor (Prof. David Vogan from MIT) for introducing me to this wonderful field.

Many results are known for unitary spherical representations, but the non-spherical unitary dual of a real reductive group is still a mysterious object (even in the classical case). In collaboration with Dan Barbasch, I have been exploring the relation between non-spherical unitary representations for non-linear coverings of real split exceptional groups and spherical unitary representation for different linear groups. The main idea is to find a match-up of parameters, so that the Hermitian forms coincide. This match-up determines powerful necessary conditions for unitarity; in some cases it also implies an inclusion of a (non-spherical) complementary series of the non-linear group into the spherical unitary dual of a linear group. Some of the results are posted on the “Atlas of Lie groups and representations” webpage.
The same question can of course be posed for coverings of real split classical groups; this was the subject of a joint work with Annegret Paul and Susana Salamanca-Riba (see PPS2 for the metaplectic group and PPS4 for the split orthogonal groups). In collaboration with A. Paul and S. Salamanca-Riba, I have also been working on the problem of finding the genuine omega-regular unitary dual of the metaplectic group, that is, classifying all the genuine unitary representations of this group whose infinitesimal character is real and at least as regular as that of the oscillator representation (see PPS1 and PPS3).

Although the work is very conceptual, some of the algebra-type problems are solved with computer calculations (Mathematica, Gap, Lie). In this sense, my research fits into the general goal of the "Atlas of Lie groups and representations'', a NSF-sponsored grand project aimed at classifying the unitary dual of reductive groups by means of both mathematical and computational techniques. I gratefully acknowledge the NSF for supporting my research in the summers from 2005 to 2009 through the "Atlas of Lie groups'' FRG grant, and the wonderful colleagues in the Atlas research group.




Prof. Info

Alessandra Pantano