A notion of rank for unitary representations of general linear groups

Author:
Roberto Scaramuzzi

Journal:
Trans. Amer. Math. Soc. **319** (1990), 349-379

MSC:
Primary 22E50; Secondary 22E46

DOI:
https://doi.org/10.1090/S0002-9947-1990-0958900-8

MathSciNet review:
958900

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Abstract: A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.

**[B]**J. Bernstein,*-invariant distributions of**and the classification of unitary representations of**(non-Archimedean case)*, Lie Group Representations. II, Proceedings, University of Maryland, Springer, 1983, pp. 50-102. MR**748505 (86b:22028)****[BZ]**I. N. Bernstein and A. V. Zelevinsky,*Induced representations of reductive**-adic groups*. I, Ann. Sci. École Norm. Sup. (4)**10**(1977), 441-472. MR**0579172 (58:28310)****[F1]**J. Fell,*The dual spaces of**-algebras*, Trans. Amer. Math. Soc.**94**(1960), 364-403. MR**0146681 (26:4201)****[F2]**-,*Weak containment and induced representations of groups*, Canad. J. Math.**14**(1962), 237-268. MR**0150241 (27:242)****[HC]**Harish-Chandra,*Discrete series for semisimple Lie groups*. II, Acta Math.**116**(1966), 1-111. MR**0219666 (36:2745)****[H1]**R. Howe,*On a notion of rank of unitary representations of the classical groups*, Harmonic Analysis and Group Representations (Proc. C.I.M.E. II circlo 1980) (A. Figa-Talamanca, ed.), Liguori, 1980, pp. 223-332. MR**777342 (86j:22016)****[H2]**-,*-series and invariant theory*, Proc. Sympos. Pure Math., vol. 33, part I, Amer. Math. Soc., Providence, R.I., 1979, pp. 275-286. MR**546602 (81f:22034)****[H3]**-,*Small unitary representations of classical groups*, preprint.**[H4]**-,*The Fourier transform and germs of characters (case of**over a**-adic field)*, Math. Ann.**208**(1974), 305-322. MR**0342645 (49:7391)****[HM]**R. Howe and C. Moore,*Asymptotic properties of unitary representations*, J. Funct. Anal.**32**(1979), 72-96. MR**533220 (80g:22017)****[K]**D. Kazhdan,*Connection of dual space of a group with the structure of its closed subgroups*, Functional Anal. Appl.**1**(1967), 63-65. MR**0209390 (35:288)****[M1]**G. Mackey,*Unitary representations of group extensions*. I, Acta Math.**99**(1958), 265-301. MR**0098328 (20:4789)****[M2]**-,*Induced representations of locally compact groups*. I, Ann. of Math. (2)**55**(1952), 101-139. MR**0044536 (13:434a)****[R]**M. Rieffel,*Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner*, Studies in Analysis (G. C. Rota, ed.), Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York, pp. 43-82. MR**546802 (81h:22004)****[Sa1]**S. Sahi,*Spherical unitary representations of general linear groups over local fields*, doctoral dissertation, Yale University, 1985.**[Sa2]**-,*On Kirillov's conjecture for Archimedean fields*, preprint.**[S]**R. Scaramuzzi,*Unitary representations of small rank of general linear groups*, doctoral dissertation, Yale University, 1985.**[Si]**A. Silberger,*Introduction to harmonic analysis on reductive**-adic groups*, Princeton Univ. Press, Princeton, N.J., 1979. MR**544991 (81m:22025)****[St]**E. Stein,*Analysis in matrix space and some new representations of*, Ann. of Math. (2)**86**(1967), 461-490. MR**0219670 (36:2749)****[T]**M. Tadić,*Solution of the uniterizability problem for general linear group (non-Archimedean case)*, preprint.**[V]**D. Vogan,*The unitary dual of**over an Archimedean field*, preprint. MR**827363 (87i:22042)****[VN]**J. von Neumann,*Die Eindeutgkeit der Schràderschen Operatores*, Math. Ann.**104**(1931), 570-578.**[Wg]**S. P. Wang,*The dual space of semi-simple Lie groups*, Amer. J. Math.**91**(1969), 921-937. MR**0259023 (41:3665)****[W]**G. Warner,*Harmonic analysis on semi-simple Lie groups*, Springer, 1972.**[Z]**G. J. Zuckerman,*Continuous cohomology and unitary representations of real reductive groups*, Ann. of Math. (2)**107**(1978), 495-516. MR**496844 (81c:22025)**

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DOI:
https://doi.org/10.1090/S0002-9947-1990-0958900-8

Article copyright:
© Copyright 1990
American Mathematical Society