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Conformally Invariant Differential Operators on Heisenberg Groups and Minimal Representations
 
J. Frahm Aarhus University, Aarhus, Denmark
A publication of the Société Mathématique de France
Softcover ISBN:  978-2-85629-986-9
Product Code:  SMFMEM/180
List Price: $71.00
AMS Member Price: $56.80
Not yet published - Preorder Now!
Expected availability date: July 26, 2024
Please note AMS points can not be used for this product
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Conformally Invariant Differential Operators on Heisenberg Groups and Minimal Representations
J. Frahm Aarhus University, Aarhus, Denmark
A publication of the Société Mathématique de France
Softcover ISBN:  978-2-85629-986-9
Product Code:  SMFMEM/180
List Price: $71.00
AMS Member Price: $56.80
Not yet published - Preorder Now!
Expected availability date: July 26, 2024
Please note AMS points can not be used for this product
  • Book Details
     
     
    Mémoires de la Société Mathématique de France
    Volume: 1802024; 148 pp
    MSC: Primary 22; 35; 43

    For a simple real Lie group \(G\) with Heisenberg parabolic subgroup \(P\), the author studies 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, the author takes the Heisenberg group Fourier transform in the non-compact picture and shows that it yields a new realization of the minimal representation on a space of \(L^2\)-functions. The Lie algebra action is given by differential operators of order \(\leq3\) and the author finds explicit formulas for the functions constituting the lowest \(K\)-type.

    These \(L^2\)-models were previously known for the groups \(\mathrm{SO}(n,n)\), \(E_{6(6)}, \), \(E_{7(7)}\) and \(E_{8(8)}\) by Kazhdan and Savin, for the group \(G_{2(2)}\) by Gelfand, and for the group \(\widetilde{\mathrm{SL}}(3,\mathbb{R}))\) by Torasso, using different methods. This new approach provides a uniform and systematic treatment of these cases and also constructs new \(L_{2}\)-models for \(E_{6(2)}, E_{7(-5)}\) and \(E_{8(-24)}\) for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups \( \widetilde{\mathrm{SO}}(p,q)\) with either \(p\geq q=3\) or \(p,q\geq4\) and \(p+q\) even.

    As a byproduct of our construction, the author finds 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 \(G\).

    A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

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Volume: 1802024; 148 pp
MSC: Primary 22; 35; 43

For a simple real Lie group \(G\) with Heisenberg parabolic subgroup \(P\), the author studies 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, the author takes the Heisenberg group Fourier transform in the non-compact picture and shows that it yields a new realization of the minimal representation on a space of \(L^2\)-functions. The Lie algebra action is given by differential operators of order \(\leq3\) and the author finds explicit formulas for the functions constituting the lowest \(K\)-type.

These \(L^2\)-models were previously known for the groups \(\mathrm{SO}(n,n)\), \(E_{6(6)}, \), \(E_{7(7)}\) and \(E_{8(8)}\) by Kazhdan and Savin, for the group \(G_{2(2)}\) by Gelfand, and for the group \(\widetilde{\mathrm{SL}}(3,\mathbb{R}))\) by Torasso, using different methods. This new approach provides a uniform and systematic treatment of these cases and also constructs new \(L_{2}\)-models for \(E_{6(2)}, E_{7(-5)}\) and \(E_{8(-24)}\) for which the minimal representation is a continuation of the quaternionic discrete series, and for the groups \( \widetilde{\mathrm{SO}}(p,q)\) with either \(p\geq q=3\) or \(p,q\geq4\) and \(p+q\) even.

As a byproduct of our construction, the author finds 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 \(G\).

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.