Spherical unitary dual of general linear group over non-Archimidean local field

Tadic, Marko

Annales de l'Institut Fourier, Tome 36 (1986) no. 2, pp. 47-55.

Soit $F$ un corps local non-archimédien. Dans cet article, nous commençons par décrire toutes les représentations irréductibles sphériques de $GL(n,F)$. En particulier, nous montrons que de telles représentations sont paraboliquement induites par des caractères non-ramifiés. Le résultat de Bernstein sur l’irréductibilité de la représentation de $GL(n,F)$ paraboliquement induite par une représentation unitaire irréductible, et la construction de Olshansky des séries complémentaires donnent directement le dual unitaire sphérique de $GL(n,F)$.

Let $F$ be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of $GL(n,F)$ is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of $GL(n,F)$ by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all equivalence classes of irreducible unitary spherical representations of $GL(n,F)$.

[1] I. N. Bernstein, P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-archimedean case), in Lie Group Representations II, Proceedings, University of Maryland 1982-1983, Lecture Notes in Math., vol. 1041, Springer-Verlag, Berlin, (1983), 50-102. | Zbl

[2] P. Cartier, Representations of p-adic groups: a survey, in Proc. Sympos. Pure Math. Vol. XXXIII, part 1, Amer. Math. Soc., Providence, R.I., 1979, 111-155. | MR | Zbl

[3] W. Casselman, The unramified principal series of p-adic groups I, The spherical functions, Comp. Math., vol. 41 (1980), 387-406. | Numdam | MR | Zbl

[4] J. Dieudonné, Treatise on analysis, vol. VI, Academic Press, New York, 1978. | MR | Zbl

[5] I. M. Gelfand, M. I. Graev, Representations of a group of the second order with elements from a locally compact field, Russian Math. Surveys, 18 (1963), 29-100. | MR | Zbl

[6] I. G. Macdonald, Spherical functions on a group of p-adic type, Rammanjan Institute, Univ. of Madras Publ. (1971). | Zbl

[7] F. I. Mautner, Spherical functions over p-adic fields I, II, Amer. J. Math., vol. 80 (1958), 441-457 and vol. 86 (1964), 171-200. | Zbl

[8] G. I. Olshansky, Intertwining operators and complementary series in the class of representations of the general group of matrices over a locally compact division algebra, induceded from parabolic subgroups, Math. Sb., vol. 93, n°. 2 (1974), 218-253.

[9] I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Inst. Hautes Études Sci. Publ. Math., 18 (1963), 1-69. | Numdam | MR | Zbl

[10] M. Tadic, Classification of unitary representations in irreducible representations of general linear group (non-archimedean case), to appear in Ann. Scient. École Norm. Sup. | Numdam | Zbl

[11] A. V. Zelevinsky, Induced representations of reductive p-adic groups II, Ann. Scient. École Norm. Sup., 13 (1980), 165-210. | Numdam | MR | Zbl