Systems of Differential Operators and Generalized Verma Modules

Abstract

Abstract. In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.

Special values for conformally invariant systems associated to maximal parabolics of quasi-Heisenberg type

“…We close these cases in [22] together with the case when q is of type D n (n − 2). When q is of type B n (n − 1), the special constituent V (μ + nγ ) is of type 1b, and when q is of type C n (i), the special constituent V (μ + γ ) is of type 3.…”

“…. The systematic study of conformally invariant systems started with the work of Barchini-Kable-Zierau in [1] and [2] and is continued in [11][12][13][14][15], and [21][22][23]. , D n of linear differential operators on sections of V is called a conformally invariant system if, for each X ∈ g, there are smooth functions C X ij (m) on M so that, for all 1 ≤ i ≤ n, and sections f of V, we have The notion of conformally invariant systems generalizes that of quasi-invariant differential operators introduced by Kostant in [19] and is related to work of Huang ([9]).…”

“…The case that q is of type D n (n − 2) is handled in[22].Licensed to University of Sussex. Prepared on Sun Feb 8 02:39:29 EST 2015 for download from IP 139.184.14.159.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use…”

“…Also see some comments in the introductions of[22] and[23] on the definition of special constituents.…”

On the space of 𝐾-finite solutions to intertwining differential operators

“…Higher order case is still in development. For some recent studies, see, for instance, Barchini-Kable-Zierau [2,3], Kable [21,22,23,24,25, 26], Kobayashi-Ørsted-Somberg-Souček [36], and the first author [43,44].…”