A proof of Kirillov's conjecture
Let \(K=\mathbb R\) or \(\mathbb C\) and let \(P_n(K)\) be the subgroup of the matrices in GL\(_n(K)\) consisting of those whose last row is \((0,0,\dots,0,1)\). In the paper under review the author proves that the restriction of an irreducible unitary representation of GL\(_n(K)\) to \(P_n(K)\) is irreducible. This claim was called Kirillov's conjecture. The history of this claim is long and interesting. \textit{I. M. Gelfand} and \textit{M. A. Naimark} in their book ``Unitäre Darstellungen der klassischen Gruppen'' [Akademie Verlag, Berlin (1957) (which is the German translation of the Russian publication from 1950; Zbl 0077.03405)] proved that a series of parabolically induced unitary representations of SL\(_n(\mathbb C)\) are irreducible by proving that their restriction to a group similar to \(P_n(\mathbb C)\) is irreducible (recall that GL\(_n(\mathbb C)\) is the product of SL\(_n(\mathbb C)\) and the center of GL\(_n(\mathbb C)\)). I. M. Gelfand and M. A. Naimark claimed that their representations exhaust all the irreducible unitary representations of SL\(_n(\mathbb C)\), what would imply that Kirillov's conjecture holds for GL\(_n(\mathbb C)\). In his paper [Ann. Math. (2) 86, 461--490 (1967; Zbl 0188.45303)], \textit{E. M. Stein} has shown that I. M. Gelfand and M. A. Naimark's list of representations is incomplete. Nevertheless, in their book they showed how Kirillov's conjecture implies the irreducibility of unitary parabolic induction. In his paper [Dokl. Akad. Nauk SSSR 114, 37--39 (1962; Zbl 0119.26804)] \textit{A. A. Kirillov} proposed a proof of the claim about the irreducibility of the restriction based on a nice geometric idea. He claimed that each \(P_n(K)\)-invariant distribution on GL\(_n(K)\) is invariant for the whole group. He observed that this claim implies the irreducibility of the restrictions to \(P_n(K)\) of the irreducible unitary representations of GL\(_n(K)\). To show this, for an irreducible unitary representation \(\pi\) of GL\(_n(K)\) and \(T\in \text{End}_{P_n(K)}(\pi)\), he considered the \(P_n(K)\)-invariant distributions \[ \varphi\mapsto \text{Trace} (T\pi(\varphi)). \] If one knows the GL\(_n(K)\)-invariance of this distribution, this easily implies the claim that \(T\) is a scalar operator, and further Schur's lemma implies the claim about the irreducibility of the restriction \(\pi| P_n(K)\). Kirillov noticed that in GL\(_n(K)\)-conjugacy classes there exist dense \(P_n(K)\)-conjugacy classes. This implies that each continuous function on GL\(_n(K)\), which is \(P_n(K)\)-invariant, is GL\(_n(K)\)-invariant. This implies that an analogous claim holds for the distributions represented by (integration against) continuous functions. Kirillov failed to give an argument of this claim for general distributions. M. E. Baruch's strategy of the proof of Kirillov's conjecture is to show that the above distribution is represented by a function, which is nice enough (a distribution represented by a continuous function on a set of the full measure), and then to apply Kirillov's argument. To prove that the above distribution is nice enough, he uses the fact that the above distribution is an eigendistribution, and that \(P_n(K)\)-conjugacy classes are big in GL\(_n(K)\)-conjugacy classes. The last fact enables him to use the ideas of the proof of the Harish-Chandra regularity theorem (which claims that each invariant eigendistribution on a reductive group is represented by a locally integrable function, which is analytic on the set of regular semisimple elements) in proving that the above distribution is represented by a nice enough function. Kirillov's conjecture has several important consequences. Let us mention only that this conjecture and several standard and well-known facts from the representation theory of reductive groups can be used to give a complete description of the irreducible unitary representations of the groups GL\(_n(\mathbb R)\) and GL\(_n(\mathbb C)\) [see for example the reviewer's paper, Bull. Am. Math. Soc., New Ser. 28, 215--252 (1993; Zbl 0799.22010)]. Let us recall that \textit{D. A. Vogan} classified the unitary duals of GL\(_n(\mathbb R)\) and GL\(_n(\mathbb C)\) in his paper [Invent. Math. 83, 449--505 (1986; Zbl 0598.22008)] along completely different ideas.
general linear groups over real and complex fields
restrictions of irreducible unitary representations
scientific article; zbMATH DE number 1997276