Unitary genuine principal series of the metaplectic group

Pantano, Alessandra; Paul, Annegret; Salamanca-Riba, Susana

doi:10.1090/S1088-4165-10-00367-5

Unitary genuine principal series of the metaplectic group
HTML articles powered by AMS MathViewer

by Alessandra Pantano, Annegret Paul and Susana A. Salamanca-Riba

Represent. Theory 14 (2010), 201-248

DOI: https://doi.org/10.1090/S1088-4165-10-00367-5

Published electronically: February 15, 2010

PDF | Request permission

Abstract:

This paper presents some recent progress on the classification of the unitary genuine irreducible representations of the metaplectic group $Mp(2n)$. Our focus will be on Langlands quotients of genuine minimal principal series; the main result is an embedding of the set of unitary parameters of such representations into the union of spherical unitary parameters for certain split orthogonal groups. The latter are known from work of D. Barbasch; hence we obtain the non-unitarity of a large (conjecturally complete) set of parameters for Langlands quotients of genuine principal series of $Mp(2n)$.

References

Jeffrey Adams, Nonlinear covers of real groups, Int. Math. Res. Not. 75 (2004), 4031–4047. MR 2112326, DOI 10.1155/S1073792804141329
Jeffrey Adams and Dan Barbasch, Genuine representations of the metaplectic group, Compositio Math. 113 (1998), no. 1, 23–66. MR 1638210, DOI 10.1023/A:1000450504919
J. Adams, D. Barbasch, A. Paul, P. Trapa, and D. A. Vogan Jr., Unitary Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007), no. 3, 701–751. MR 2291917, DOI 10.1090/S0894-0347-06-00530-3
D. Barbasch, Unitary spherical spectrum for split classical groups, preprint, arXiv:math/ 0609828v4 [math.RT].
Dan Barbasch, Relevant and petite $K$-types for split groups, Functional analysis VIII, Various Publ. Ser. (Aarhus), vol. 47, Aarhus Univ., Aarhus, 2004, pp. 35–71. MR 2127163
Dan Barbasch, Dan Ciubotaru, and Alessandra Pantano, Unitarizable minimal principal series of reductive groups, Representation theory of real reductive Lie groups, Contemp. Math., vol. 472, Amer. Math. Soc., Providence, RI, 2008, pp. 63–136. MR 2454332, DOI 10.1090/conm/472/09237
Dan Barbasch and Allen Moy, Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6 (1993), no. 3, 611–635. MR 1186959, DOI 10.1090/S0894-0347-1993-1186959-0
D. Barbasch, A. Pantano, Petite $K$-types for nonspherical minimal principal series, preprint.
Roger Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535–552. MR 985172, DOI 10.1090/S0894-0347-1989-0985172-6
David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples; Reprint of the 1986 original. MR 1880691
A. W. Knapp and Gregg Zuckerman, Classification theorems for representations of semisimple Lie groups, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977, pp. 138–159. MR 0476923
Jian-Shu Li, Singular unitary representations of classical groups, Invent. Math. 97 (1989), no. 2, 237–255. MR 1001840, DOI 10.1007/BF01389041
Alessandra Pantano, Annegret Paul, and Susana A. Salamanca-Riba, The omega-regular unitary dual of the metaplectic group of rank 2, Council for African American Researchers in the Mathematical Sciences. Vol. V, Contemp. Math., vol. 467, Amer. Math. Soc., Providence, RI, 2008, pp. 1–47. MR 2484675, DOI 10.1090/conm/467/09130
—, The genuine omega-regular unitary dual of the metaplectic group, preprint.
Tomasz Przebinda, The duality correspondence of infinitesimal characters, Colloq. Math. 70 (1996), no. 1, 93–102. MR 1373285, DOI 10.4064/cm-70-1-93-102
Susana Salamanca-Riba, On the unitary dual of some classical Lie groups, Compositio Math. 68 (1988), no. 3, 251–303. MR 971329
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
D. J. Simms, Lie groups and quantum mechanics, Lecture Notes in Mathematics, No. 52, Springer-Verlag, Berlin-New York, 1968. MR 0232579, DOI 10.1007/BFb0069914
Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407