Conformally invariant differential operators on Heisenberg groups and minimal representations / Jan Frahm.

Format:

Book

Author/Creator:

Frahm, Jan, author.

Contributor:

Société mathématique de France, issuing body.

Series:

Mémoire (Société mathématique de France) ; nouv. sér., no 180.

Mémoires de la Société mathématique de France ; numéro 180, nouvelle série

Language:

English

French

Subjects (All):

Differential operators.

Lie algebras.

Physical Description:

vi, 139 pages : illustrations ; 24 cm

Place of Publication:

Paris : Société mathématique de France, 2024.

Language Note:

Text in English; abstract in English and French.

Summary:

For a simple real Lie group G with Heisenberg parabolic subgroup P, we study the corresponding degenerate principal series representations. For a certain induction parameter the kernel of the conformally invariant system of second order differential operators constructed by Barchini, Kable and Zierau is a subrepresentation which turns out to be the minimal representation. To study this subrepresentation, we take the Heisenberg group Fourier transform in the non-compact picture and show that it yields a new realization of the minimal representation on a space of L2-functions. The Lie algebra action is given by differential operators of order ≤3 and we find explicit formulas for the functions constituting the lowest K-type. These L2-models were previously known for the groups SO(n,n), E6(6), E7(7) and E8(8) by Kazhdan and Savin, for the group G2(2) by Gelfand, and for the group SL (3,R) by Torasso, using different methods. Our new approach provides a uniform and systematic treatment of these cases and also constructsnew L2-models for E6(2), E7(-5) and E8(-24) for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups SO (p,q) with either p≥q=3 or p, q≥4 and p+q even. As a byproduct of our construction, we find an explicit formula for the group action of a non-trivial Weyl group element that, together with the simple action of a parabolic subgroup, generates G.

Notes:

Includes bibliographical references and index.

Other Format:

Online version: Frahm, Jan. Conformally invariant differential operators on Heisenberg groups and minimal representations.

ISBN:

9782856299869

2856299865

OCLC:

1437302799