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.
RESEARCH PAPERS
- Pantano, A., Paul, A., Salamanca Riba, S., `Unitary principal series of split orthogonal groups’, Pacific Journal of Mathematics, 271 (2014), 479-510 [PPS4]
- Pantano, A., Paul, A., Salamanca Riba, S., `The genuine omega-regular unitary dual of the
metaplectic group’, Canadian Journal of Mathematics 64 (2012), 669-704 [PPS3] - Pantano, A., Paul, A., Salamanca Riba, S., `Unitary Genuine Principal Series of the Metaplectic Group’, AMS Representation Theory, 14 (2010), 201-248 [PPS2]
- Pantano, A., Paul, A., Salamanca Riba, S., `The Omega-Regular Unitary Dual of the Metaplectic Group of Rank 2’, AMS Contemporary Mathematics, 467 (2008), 1-48 [PPS1]
- Pantano, A., Barbasch, D., Ciubotaru, D., `Unitarizable Minimal Principal Series of Reductive Groups’, AMS Contemporary Mathematics, 472 (2008), 63-136 [BCP]
SELECTED RESEARCH TALKS
- “Nilpotent orbits in complex classical semisimple Lie algebras”, Advanced Course. XV Latin
American School of Mathematics. Córdoba, Argentina. (2011) - “On the unitary dual of classical and exceptional real split groups”, MIT Lie Group Seminar,
Massachusetts Institute of Technology, Cambridge, MA. (March 2010). - “Complementary series of split real groups”, Symposium on Representation Theory of Real Reductive Groups, University of Utah, Salt Lake City, Utah. (July 2009).
- “Unitary genuine principal series of the metaplectic group”, Representation Theory XI, Interuniversity Centre, Dubrovnik, Croatia. (June 2009).
- “Weyl group representations and unitarity of spherical representations”, University of Windsor Colloquium Series, University of Windsor, Windsor, ON, Canada. (October 2008).
- “Subquotients of minimal principal series”, Atlas of Lie Groups AIM workshop, American Institute of Mathematics (AIM), Palo Alto, CA. (July 2008).
- “Unitary genuine principal series of the metaplectic group”, Lie Theory and Geometry: The
Mathematical Legacy of Bertram Kostant, University of British Columbia, Vancouver, BC, Canada. (May 2008). - “A step towards the determination of the unitary dual”, Symposium on Structure, Geometry and Symbolic Computation of Algebraic Groups and Symmetric Spaces, AMS National Meetings, San Diego, CA. (January 2008).
- “Unitary representations of real split groups”, Representation Theory Seminar, University of Utah, Salt Lake City, UT. (December 2007).
- “Excursions in unitary representation theory”, Algebra and Analysis Seminar, New Mexico State University, Las Cruces, NM. (November 2007).
- “Complementary series for real reductive groups”, Queen’s Algebraic Geometry Seminar, Queen’s University, Kingston, ON, Canada. (April 2007).
- “Multiplicity of K-types in principal series”, Atlas of Lie Groups Winter Meeting, Massachusetts Institute of Technology, Cambridge, MA. (March 2007).
- “Methods for studying complementary series of split groups”, Symposium on Representation Theory and Theta Correspondence, AMS National Meetings, New Orleans, LA. (January 2007).
- “On the double cover of split F4 and its petite K-types”, Atlas of Lie Groups AIM workshop,
American Institute of Mathematics (AIM), Palo Alto, CA. (July 2006). - “On the complementary series of the double cover of F4”, Cornell Lie Groups Seminar, Cornell University, Ithaca, NY. (March 2006).
- “Petite and relevant K-types for exceptional groups”, Atlas of Lie Groups AIM workshop, American Institute of Mathematics (AIM), Palo Alto, CA. (July 2005).
- “Unitarity of non-spherical principal series”, Atlas of Lie Groups AIM workshop, American Institute of Mathematics (AIM), Palo Alto, CA. (July 2005).
- “The geometry of conjugacy classes of nilpotent matrices”, Oliver Club Talk, Cornell University, Ithaca, NY. (April 2005).
- “The non-spherical unitary dual of real split groups”, Cornell Lie Groups Seminar, Cornell University, Ithaca, NY. (September 2004).
- “On the construction of petite K-types”, Atlas of Lie Groups AIM workshop, American Institute of Mathematics (AIM), Palo Alto, CA. (July 2004).
- “Weyl group representations and signatures of intertwining operators”, Representation Theory Seminar, University of Massachusetts at Amherst, Amherst, MA. (April 2004).
- “The classification of nilpotent orbits by means of weighted Dynkin diagrams”, MIT Student Seminar in Representation Theory, Massachusetts Institute of Technology, Cambridge, MA. (October 2003).