A solution of Warner's 3rd problem for representations of holomorphic type
Author:
Floyd L. Williams
Journal:
Trans. Amer. Math. Soc. 293 (1986), 605-612
MSC:
Primary 22E46; Secondary 11F70, 32M15
DOI:
https://doi.org/10.1090/S0002-9947-1986-0816313-2
MathSciNet review:
816313
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Abstract | References | Similar Articles | Additional Information
Abstract: In response to one of ten problems posed by G. Warner, we assign (to the extent that it is possible) a geometric or cohomological interpretation-- in the sense of Langlands--to the multiplicty in of an irreducible unitary representation
of a semisimple Lie group
, where
is a discrete subgroup of
, in the case when
has a highest weight.
- [1] Hans R. Fischer and Floyd L. Williams, Borel-LePotier diagrams—calculus of their cohomology bundles, Tohoku Math. J. (2) 36 (1984), no. 2, 233–251. MR 742597, https://doi.org/10.2748/tmj/1178228850
- [2] Phillip Griffiths and Wilfried Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253–302. MR 259958, https://doi.org/10.1007/BF02392390
- [3] R. P. Langlands, Dimension of spaces of automorphic forms, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 253–257. MR 0212135
- [4] R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. Sect. A Math. Sci. 89 (1980), no. 1, 1–24. MR 573381
- [5] Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1–42. MR 286942, https://doi.org/10.2307/1970751
- [6] Garth Warner, Selberg’s trace formula for nonuniform lattices: the 𝑅-rank one case, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 1–142. MR 535763
- [7] Floyd L. Williams, Vanishing theorems for type (0,𝑞) cohomology of locally symmetric spaces, Osaka Math. J. 18 (1981), no. 1, 147–160. MR 609983
- [8]
-, An alternating sum formula for the multiplicities of unitary highest weight modules in
, unpublished manuscript.
- [9] Floyd L. Williams, Discrete series multiplicities in 𝐿²(Γ\𝐺). II. Proof of Langlands’ conjecture, Amer. J. Math. 107 (1985), no. 2, 367–376. MR 784287, https://doi.org/10.2307/2374418
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1986-0816313-2
Article copyright:
© Copyright 1986
American Mathematical Society