On the Cohomology of Some Complex Hyperbolic Arithmetic 3-Manifolds

Abstract

Given an arithmetic lattice of the unitary group U(3,1) arising from a hermitian form over a CM-field, we show that all unitary representations of U(3,1) with nonzero cohomology contribute to the cohomology of the attached arithmetic complex 3-manifold, at least when we pass to a finite-index subgroup of the given arithmetic lattice.

Keywords

Cohomology / Unitary representation / Theta lift / Complex hyperbolic 3-manifold

References

[1] Anderson G. Theta functions and holomorphic differential forms on compact quotients of bounded symmetric domains. Duke Math. J.. 1983, 50 4 1137-1170

[2] Borel A. Automorphic L-functions. Automorphic Forms, Representations, and L-Functions. 1979 Providence: Am. Math. Soc.. 27-61

[3] Borel A., Casselman W. L ²-cohomology of locally symmetric manifolds of finite volume. Duke Math. J.. 1983, 50 3 625-647

[4] Borel, A., Wallach, N.: Continuous cohomology, Discrete subgroups, and representations of reductive groups. Ann. Math. Stud. 94 (1980)

[5] Clozel L. On the cohomology of Kottwitz’s arithmetic varieties. Duke Math. J.. 1993, 72 757-795

[6] Gelbart S., Jacquet H. A relation between automorphic forms on GL₂ and GL₃. Proc. Natl. Acad. Sci. USA. 1976, 73 3348-3350

[7] Harris M., Kudla S., Sweet J. Theta dichotomy for unitary groups. J. Am. Math. Soc.. 1996, 9 4 941-1004

[8] Harris M., Li J.S., Skinner C. . Cogdell J. . The Rallis inner product formula and p-adic L-functions. Automorphic Representations, L-Functions and Applications: Progress and Prospects. 2005 225-256

[9] Harris M., Li J.-S., Sun B. Funke J., Cogdell J., Rapoprt M., Yang T. Theta correspondences for close unitary groups. Arithmetic Geometry and Automorphic Forms, Volume in Honor of the 60th Birthday of Stephen S. Kudla. 2011 Somerville: International Press and the Higher Education Press of China. 265-308

[10] Howe R. θ series and invariant theory. Automorphic Forms, Representations, and L-Functions. 1979 Providence: Am. Math. Soc.. 275-285

[11] Ichino A. A regularized Siegel–Weil formula for unitary groups. Math. Z.. 2004, 247 2 241-277

[12] Kashiwara M., Vergne M. On the Segal–Shale–Weil representations and harmonic polynomials. Invent. Math.. 1978, 44 1-47

[13] Kazhdan D. Some applications of the Weil representations. J. Anal. Math.. 1977, 32 235-248

[14] Kudla S. Splitting metaplectic covers of dual reductive pairs. Isr. J. Math.. 1994, 87 361-401

[15] Kudla S.S., Sweet W.J. Jr.. Degenerate principal series representations for U(n,n). Isr. J. Math.. 1997, 98 253-306

[16] Langlands R. Problems in the Theory of Automorphic Forms, Lectures in Modern Analysis and Application. 1970 New York: Springer. 18-86

[17] Langlands R. On the Functional Equations Satisfied by Eisenstein Series. 1976 Berlin: Springer

[18] Li J.-S. Singular unitary representations of classical groups. Invent. Math.. 1989, 97 237-255

[19] Li J.-S. Theta liftings for unitary representations with non-zero cohomology. Duke Math. J.. 1990, 61 913-937

[20] Li J.-S. Non-vanishing theorems for the cohomology of certain arithmetic quotients. J. Reine Angew. Math.. 1992, 428 177-217

[21] Luo W., Rudnick Z., Sarnak P. On the generalized Ramanujan conjecture for GL(n), vol. II. Proc. Symp. Pure Math.. 1999, 66 301-311

[22] Rallis S. Langlands functoriality and the Weil representation. Am. J. Math.. 1982, 104 469-515

[23] Rapoport M., Zink T. Über die lokale Zetafunktion von Shimuravarietäten, Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math.. 1982, 68 21-101

[24] Rogawski J. Automorphic Representations of Unitary Groups in Three Variables. 1990 Princeton: Princeton University Press

[25] Savin G. Limit multiplicities of cusp forms. Invent. Math.. 1989, 95 149-159

[26] Sun, B., Zhu, C.-B.: Conservation relations for local theta correspondence. arXiv:1204.2969

[27] Vogan D., Zuckerman G. Unitary representations with non-zero cohomology. Compos. Math.. 1984, 53 51-90

[28] Weil A. Sur certains groupes d’operateurs unitaires. Acta Math.. 1964, 111 143-211

[29] Weil A. Sur la formulae de Siegel dans la theorie des groupes classiques. Acta Math.. 1965, 113 1-87