\(P\)-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].

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].

Reviewer: Audrey A. Terras (La Jolla)

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |