Multiplet Classification of Reducible Verma Modules over the G ₂ Algebra

Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra G 2(2) which is split real form of G2. We give the classification of reducible Verma modules G2. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over G 2(2) .Dedicated to I.E. Segal in commemoration of the centenary of his birth. The author remembers with great pleasure the talk he gave at Segal's seminar at MIT in 1975.

Heisenberg Parabolic Subgroups of Exceptional Non-Compact G2(2) and Invariant Differential Operators

“…(α 2 , α 1 ) = −3. (Note that in [4] we have chosen α 1 as the long root, and α 2 as the short root).…”

“…Below we give the general preliminaries necessary for our approach. In Section 2 we introduce the Lie algebra G 2 (following [4]), its real from G 2(2) and, shortly, the corresponding Lie group. In Section 3 we consider the representations induced from the minimal parabolic subalgebra of G 2 (2) .…”

Heisenberg Parabolic Subgroups of Exceptional Noncompact $G_{2(2)}$ and Invariant Differential Operators

“…Remark: In [4] were considered also the following multiplets for G C which are not interesting for the real form G. Fix k = 1, . .…”

“…In the next section we give the general preliminaries necessary for our approach. In Section 3 we introduce the Lie algebra G 2 (following [4]), its real from G 2 (2) , and shortly the corresponding Lie group. In Section 4 we consider the representations induced from the minimal parabolic subalgebra of G 2 (2) .…”

An Exceptional Symmetry Algebra for the 3D Dirac–Dunkl Operator

“…The latter is primarily known from the classification of simple Lie algebras. The associated Lie group and algebra continue to spark interest, see for instance the recent paper of Dobrev [4] and references therein. Our purpose is related instead to the action of the Weyl group associated with G 2 on a (two-dimensional subspace of a) three-dimensional space.…”