Construction of cohomology of discrete groups
Authors:
Y. L. Tong and S. P. Wang
Journal:
Trans. Amer. Math. Soc. 306 (1988), 735763
MSC:
Primary 32N15; Secondary 11F27, 11F55, 22E40
MathSciNet review:
933315
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Abstract: A correspondence between Hermitian modular forms and vector valued harmonic forms in locally symmetric spaces associated to is constructed and also shown in general to be nonzero. The construction utilizes RallisSchiffmann type theta functions and simplified arguments to circumvent differential geometric calculations used previously in related problems.
 [B.W.]
Armand
Borel and Nolan
R. Wallach, Continuous cohomology, discrete subgroups, and
representations of reductive groups, Annals of Mathematics Studies,
vol. 94, Princeton University Press, Princeton, N.J.; University of
Tokyo Press, Tokyo, 1980. MR 554917
(83c:22018)
 [F.1]
Mogens
FlenstedJensen, Discrete series for semisimple symmetric
spaces, Ann. of Math. (2) 111 (1980), no. 2,
253–311. MR
569073 (81h:22015), http://dx.doi.org/10.2307/1971201
 [F.2]
, Harmonic analysis on semisimple symmetric spaces, Lecture Notes in Math., vol. 1077. SpringerVerlag, 1984, pp. 166209.
 [G]
B.
Brent Gordon, Intersections of higherweight cycles
over quaternionic modular surfaces and modular forms of
Nebentypus, Bull. Amer. Math. Soc. (N.S.)
14 (1986), no. 2,
293–298. MR
828829 (87e:11068), http://dx.doi.org/10.1090/S027309791986154467
 [Ho.1]
R.
Howe, 𝜃series and invariant theory, Automorphic
forms, representations and 𝐿functions (Proc. Sympos. Pure Math.,
Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math.,
XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285.
MR 546602
(81f:22034)
 [Ho.2]
Roger
Howe, Transcending classical invariant
theory, J. Amer. Math. Soc.
2 (1989), no. 3,
535–552. MR
985172 (90k:22016), http://dx.doi.org/10.1090/S08940347198909851726
 [H.PS.]
Roger
Howe and I.
I. PiatetskiShapiro, Some examples of automorphic forms on
𝑆𝑝₄, Duke Math. J. 50 (1983),
no. 1, 55–106. MR 700131
(84m:10019)
 [H]
L.
K. Hua, Harmonic analysis of functions of several complex variables
in the classical domains, Translated from the Russian by Leo Ebner and
Adam Korányi, American Mathematical Society, Providence, R.I., 1963.
MR
0171936 (30 #2162)
 [K.M.1]
Stephen
S. Kudla and John
J. Millson, Geodesic cyclics and the Weil representation. I.
Quotients of hyperbolic space and Siegel modular forms, Compositio
Math. 45 (1982), no. 2, 207–271. MR 651982
(83m:10037)
 [K.M.2]
Stephen
S. Kudla and John
J. Millson, The theta correspondence and harmonic forms. I,
Math. Ann. 274 (1986), no. 3, 353–378. MR 842618
(88b:11023), http://dx.doi.org/10.1007/BF01457221
 [K.V.]
M.
Kashiwara and M.
Vergne, On the SegalShaleWeil representations and harmonic
polynomials, Invent. Math. 44 (1978), no. 1,
1–47. MR
0463359 (57 #3311)
 [L.V.]
Gérard
Lion and Michèle
Vergne, The Weil representation, Maslov index and theta
series, Progress in Mathematics, vol. 6, Birkhäuser, Boston,
Mass., 1980. MR
573448 (81j:58075)
 [Ma]
Hans
Maass, Siegel’s modular forms and Dirichlet series,
Lecture Notes in Mathematics, Vol. 216, SpringerVerlag, BerlinNew York,
1971. Dedicated to the last great representative of a passing epoch. Carl
Ludwig Siegel on the occasion of his seventyfifth birthday. MR 0344198
(49 #8938)
 [M.M.]
Yozô
Matsushima and Shingo
Murakami, On vector bundle valued harmonic forms and automorphic
forms on symmetric riemannian manifolds, Ann. of Math. (2)
78 (1963), 365–416. MR 0153028
(27 #2997)
 [R.S.1]
Stephen
Rallis and Gérard
Schiffmann, Weil representation. I. Intertwining distributions and
discrete spectrum, Mem. Amer. Math. Soc. 25 (1980),
no. 231, iii+203. MR 567800
(81j:22007), http://dx.doi.org/10.1090/memo/0231
 [R.S.2]
S.
Rallis and G.
Schiffmann, Automorphic forms constructed from the Weil
representation: holomorphic case, Amer. J. Math. 100
(1978), no. 5, 1049–1122. MR 517145
(80d:10038), http://dx.doi.org/10.2307/2373962
 [T]
Yue
Lin Lawrence Tong, Weighted intersection numbers on Hilbert modular
surfaces, Compositio Math. 38 (1979), no. 3,
299–310. MR
535073 (80f:10034)
 [T.W.1]
Y.
L. Tong and S.
P. Wang, Harmonic forms dual to geodesic cycles in quotients of
𝑆𝑈(𝑝,1), Math. Ann. 258
(1981/82), no. 3, 289–318. MR 649201
(84m:32046), http://dx.doi.org/10.1007/BF01450684
 [T.W.2]
Y.
L. Tong and S.
P. Wang, Theta functions defined by geodesic cycles in quotients of
𝑆𝑈(𝑝,1), Invent. Math. 71
(1983), no. 3, 467–499. MR 695901
(85c:11046), http://dx.doi.org/10.1007/BF02095988
 [T.W.3]
Y.
L. Tong and S.
P. Wang, Correspondence of Hermitian modular forms to cycles
associated to 𝑆𝑈(𝑝,2), J. Differential Geom.
18 (1983), no. 1, 163–207. MR 697988
(85d:11047)
 [T.W.4]
Y.
L. Tong and S.
P. Wang, Period integrals in noncompact quotients of
𝑆𝑈(𝑝,1), Duke Math. J. 52
(1985), no. 3, 649–688. MR 808097
(87c:32038), http://dx.doi.org/10.1215/S0012709485052342
 [T.W.5]
, Some nonzero cohomology of discrete groups (preprint).
 [W.1]
S.
P. Wang, Correspondence of modular forms to cycles associated to
𝑂(𝑝,𝑞), J. Differential Geom.
22 (1985), no. 2, 151–213. MR 834276
(88a:32040)
 [W.2]
, Correspondence of modular forms to cycles associated to (preprint).
 [Z]
D.
P. Želobenko, Compact Lie groups and their
representations, American Mathematical Society, Providence, R.I.,
1973. Translated from the Russian by Israel Program for Scientific
Translations; Translations of Mathematical Monographs, Vol. 40. MR 0473098
(57 #12776b)
 [B.W.]
 A. Borel and N. Wallach, Continuous cohomology discrete subgroups, and representations of reductive groups, Princeton Univ. Press, Princeton, N.J., 1980. MR 554917 (83c:22018)
 [F.1]
 M. FlenstedJensen, Discrete series for semisimple symmetric spaces, Ann. of Math. (2) 111 (1980), 253311. MR 569073 (81h:22015)
 [F.2]
 , Harmonic analysis on semisimple symmetric spaces, Lecture Notes in Math., vol. 1077. SpringerVerlag, 1984, pp. 166209.
 [G]
 B. Gordon, Intersections of higher weight cycles over quaternionic modular surfaces and modular forms of nebentypus, Bull. Amer. Math. Soc. 14 (1986), 293298. MR 828829 (87e:11068)
 [Ho.1]
 R. Howe, series and invariant theory, Proc. Sympos. Pure Math., vol. 33, Amer. Math. Soc., Providence, R.I., 1979, pp. 275285. MR 546602 (81f:22034)
 [Ho.2]
 , Transcending classical invariant theory (preprint). MR 985172 (90k:22016)
 [H.PS.]
 R. Howe and I. PiatetskiShapiro, Some examples of automorphic forms on , Duke Math. J. 50 (1983), 55106. MR 700131 (84m:10019)
 [H]
 L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Transl. Math. Monographs, vol. 6, Amer. Math. Soc., Providence, R.I., 1963. MR 0171936 (30:2162)
 [K.M.1]
 S. Kudla and J. Millson, Geodesic cycles and the Weil representation. I, Compositio Math. 45 (1982), 207271. MR 651982 (83m:10037)
 [K.M.2]
 , The theta correspondence and harmonic forms. I, Math. Ann. 274 (1986), 353378. MR 842618 (88b:11023)
 [K.V.]
 M. Kashiwara and M. Vergne, On the SegalShaleWeil representations and pluriharmonic polynomials, Invent. Math. 44 (1978), 147. MR 0463359 (57:3311)
 [L.V.]
 G. Lions and M. Vergne, The Weil representation, Maslov index and theta series, Birkhäuser, Boston, Mass., 1980. MR 573448 (81j:58075)
 [Ma]
 H. Maass, Siegel's modular forms and Dirichlet's series, Lecture Notes in Math., vol. 216, SpringerVerlag, New York, 1971. MR 0344198 (49:8938)
 [M.M.]
 Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric spaces, Ann. of Math. (2) 78 (1963), 365416. MR 0153028 (27:2997)
 [R.S.1]
 S. Rallis and G. Schiffmann, Weil representation. I: Intertwining distributions and discrete spectrum, Mem. Amer. Math. Soc. 231 (1982). MR 567800 (81j:22007)
 [R.S.2]
 , Automorphic forms constructed from the Weil representation: Holomorphic case, Amer. J. Math. 100 (1978), 10491122. MR 517145 (80d:10038)
 [T]
 Y. L. Tong, Weighted intersection numbers on Hilbert modular surfaces, Compositio Math. 38 (1979), 299310. MR 535073 (80f:10034)
 [T.W.1]
 Y. L. Tong and S. P. Wang, Harmonic forms dual to geodesic cycles in quotients of , Math. Ann. 258 (1982), 289318. MR 649201 (84m:32046)
 [T.W.2]
 , Theta functions defined by geodesic cycles in quotients of , Invent. Math. 71 (1983), 467499. MR 695901 (85c:11046)
 [T.W.3]
 , Correspondence of Hermitian modular forms to cycles associated to , J. Differential Geom. 18 (1983), 163207. MR 697988 (85d:11047)
 [T.W.4]
 , Period integrals in noncompact quotients of , Duke Math. J. 52 (1985), 649688. MR 808097 (87c:32038)
 [T.W.5]
 , Some nonzero cohomology of discrete groups (preprint).
 [W.1]
 S. P. Wang, Correspondence of modular forms to cycles associated to , J. Differential Geom. 22 (1985), 151213. MR 834276 (88a:32040)
 [W.2]
 , Correspondence of modular forms to cycles associated to (preprint).
 [Z]
 D. P. Želobenko, Compact Lie groups and their representations, Transl. Math. Monographs, vol. 40, Amer. Math. Soc., Providence, R.I., 1973. MR 0473098 (57:12776b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198809333158
PII:
S 00029947(1988)09333158
Article copyright:
© Copyright 1988
American Mathematical Society
