Construction of cohomology of discrete groups

Authors:
Y. L. Tong and S. P. Wang

Journal:
Trans. Amer. Math. Soc. **306** (1988), 735-763

MSC:
Primary 32N15; Secondary 11F27, 11F55, 22E40

DOI:
https://doi.org/10.1090/S0002-9947-1988-0933315-8

MathSciNet review:
933315

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 Rallis-Schiffmann 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****[F.1]**Mogens Flensted-Jensen,*Discrete series for semisimple symmetric spaces*, Ann. of Math. (2)**111**(1980), no. 2, 253–311. MR**569073**, https://doi.org/10.2307/1971201**[F.2]**-,*Harmonic analysis on semisimple symmetric spaces*, Lecture Notes in Math., vol. 1077. Springer-Verlag, 1984, pp. 166-209.**[G]**B. Brent Gordon,*Intersections of higher-weight cycles over quaternionic modular surfaces and modular forms of Nebentypus*, Bull. Amer. Math. Soc. (N.S.)**14**(1986), no. 2, 293–298. MR**828829**, https://doi.org/10.1090/S0273-0979-1986-15446-7**[Ho.1]**R. Howe,*𝜃-series and invariant theory*, Automorphic forms, representations and 𝐿-functions (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****[Ho.2]**Roger Howe,*Transcending classical invariant theory*, J. Amer. Math. Soc.**2**(1989), no. 3, 535–552. MR**985172**, https://doi.org/10.1090/S0894-0347-1989-0985172-6**[**Roger Howe and I. I. Piatetski-Shapiro,**H.-PS**.]*Some examples of automorphic forms on 𝑆𝑝₄*, Duke Math. J.**50**(1983), no. 1, 55–106. MR**700131****[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****[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****[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**, https://doi.org/10.1007/BF01457221**[K.V.]**M. Kashiwara and M. Vergne,*On the Segal-Shale-Weil representations and harmonic polynomials*, Invent. Math.**44**(1978), no. 1, 1–47. MR**0463359**, https://doi.org/10.1007/BF01389900**[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****[Ma]**Hans Maass,*Siegel’s modular forms and Dirichlet series*, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR**0344198****[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**, https://doi.org/10.2307/1970348**[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**, https://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**, https://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****[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**, https://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**, https://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****[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**, https://doi.org/10.1215/S0012-7094-85-05234-2**[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****[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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
32N15,
11F27,
11F55,
22E40

Retrieve articles in all journals with MSC: 32N15, 11F27, 11F55, 22E40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0933315-8

Article copyright:
© Copyright 1988
American Mathematical Society