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$$P$$-invariant distributions on $$\text{GL}(n)$$ and the classification of unitary representations of $$\text{GL}(n)$$ (non-Archimedean case). (English) Zbl 0541.22009
Lie group representations II, Proc. Spec. Year, Univ. Md., College Park 1982-83, Lect. Notes Math. 1041, 50-102 (1984).
Here $$G=\text{GL}(n,F)$$ the general linear group over a local field $$F$$ and $$P$$ is the parabolic subgroup of matrices in $$G$$ with last row equal to $$(0,\ldots,0,1)$$. Various results are obtained concerning restrictions to $$P$$ of representations of $$G$$. A corollary is:
Theorem: Each irreducible unitary representation of $$G$$ remains irreducible when restricted to $$P$$.
And it is proved that any nondegenerate irreducible representation $$(\pi,E)$$ of $$G$$ is generic; i.e., the scalar product on $$E$$ can be written as a standard integral in the Kirillov model of $$\pi$$. This gives an alternate proof of the uniqueness and injectivity of the Kirillov model. An algorithm is given for the classification of the irreducible unitary representations of $$G$$. A. V. Zelevinsky’s classification is used. This involves a study of derivatives of representations of $$G$$. Zelevinsky conjectures that the multiplicities of representations can be expressed via Kazhdan-Lusztig polynomials for symmetric groups. Such a conjecture would make the classification algorithm more precise. There is a problem in that the Kazhdan-Lusztig polynomials are defined recursively and not by an explicit formula.
References for the paper include A. A. Kirillov [Sov. Math., Dokl. 3, 652–655 (1962); translation from Dokl. Akad. Nauk SSSR 144, 37–39 (1962; Zbl 0119.26804)], I. N. Bernstein and A. V. Zelevinskii [Russ. Math. Surv. 31, No. 3, 1–68 (1976); translation from Usp. Mat. Nauk 31, No. 3(189), 5–70 (1976; Zbl 0342.43017)], and A. V. Zelevinskii [Funct. Anal. Appl. 15, 83–92 (1981); translation from Funkts. Anal. Prilozh. 15, No. 2, 9–21 (1981; Zbl 0463.22013)].
[For the entire collection see Zbl 0521.00012].

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields