A notion of rank for unitary representations of general linear groups

Scaramuzzi, Roberto

doi:10.1090/S0002-9947-1990-0958900-8

A notion of rank for unitary representations of general linear groups
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Trans. Amer. Math. Soc. 319 (1990), 349-379 Request permission

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.

References

Joseph N. Bernstein, $P$-invariant distributions on $\textrm {GL}(N)$ and the classification of unitary representations of $\textrm {GL}(N)$ (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50–102. MR 748505, DOI 10.1007/BFb0073145
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172
J. M. G. Fell, The dual spaces of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 94 (1960), 365–403. MR 146681, DOI 10.1090/S0002-9947-1960-0146681-0
J. M. G. Fell, Weak containment and induced representations of groups, Canadian J. Math. 14 (1962), 237–268. MR 150241, DOI 10.4153/CJM-1962-016-6
Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813
Roger Howe, On a notion of rank for unitary representations of the classical groups, Harmonic analysis and group representations, Liguori, Naples, 1982, pp. 223–331. MR 777342
R. Howe, $\theta$-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

Small unitary representations of classical groups

Roger Howe, The Fourier transform and germs of characters (case of $\textrm {Gl}_{n}$ over a $p$-adic field), Math. Ann. 208 (1974), 305–322. MR 342645, DOI 10.1007/BF01432155
Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979), no. 1, 72–96. MR 533220, DOI 10.1016/0022-1236(79)90078-8
D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71–74 (Russian). MR 0209390
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
Marc A. Rieffel, Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner, Studies in analysis, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, 1979, pp. 43–82. MR 546802

Spherical unitary representations of general linear groups over local fields

On Kirillov’s conjecture for Archimedean fields

Unitary representations of small rank of general linear groups

Allan J. Silberger, Introduction to harmonic analysis on reductive $p$-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991
E. M. Stein, Analysis in matrix spaces and some new representations of $\textrm {SL}(N,\,C)$, Ann. of Math. (2) 86 (1967), 461–490. MR 219670, DOI 10.2307/1970611

Solution of the uniterizability problem for general linear group (non-Archimedean case)

David A. Vogan Jr., The unitary dual of $\textrm {GL}(n)$ over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505. MR 827363, DOI 10.1007/BF01394418

Die Eindeutgkeit der Schràderschen Operatores

104

S. P. Wang, The dual space of semi-simple Lie groups, Amer. J. Math. 91 (1969), 921–937. MR 259023, DOI 10.2307/2373310

Harmonic analysis on semi-simple Lie groups

Gregg J. Zuckerman, Continuous cohomology and unitary representations of real reductive groups, Ann. of Math. (2) 107 (1978), no. 3, 495–516. MR 496844, DOI 10.2307/1971126