A solution of Warner’s 3rd problem for representations of holomorphic type
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- by Floyd L. Williams PDF
- Trans. Amer. Math. Soc. 293 (1986), 605-612 Request permission
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 ${L^2}(\Gamma \backslash G)$ of an irreducible unitary representation $\pi$ of a semisimple Lie group $G$, where $\Gamma$ is a discrete subgroup of $G$, in the case when $\pi$ has a highest weight.References
- 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, DOI 10.2748/tmj/1178228850
- Phillip Griffiths and Wilfried Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253–302. MR 259958, DOI 10.1007/BF02392390
- 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
- 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
- Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1–42. MR 286942, DOI 10.2307/1970751
- 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
- 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 —, An alternating sum formula for the multiplicities of unitary highest weight modules in ${L^2}(\Gamma \backslash G)$, unpublished manuscript.
- Floyd L. Williams, Discrete series multiplicities in $L^2(\Gamma \backslash G)$. II. Proof of Langlands’ conjecture, Amer. J. Math. 107 (1985), no. 2, 367–376. MR 784287, DOI 10.2307/2374418
Additional Information
- © Copyright 1986 American Mathematical Society
- 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