### A Comparison Theory for the Structure of Induced Representations II.

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We study certain $\mathrm{\U0001d530\U0001d529}(2,\u2102)$-actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $(\U0001d524,\U0001d52d)$, $({\U0001d524}^{\text{'}},{\U0001d52d}^{\text{'}})$ of Lie algebras and their parabolic subalgebras.

Let ${G}_{\mathbb{R}}$ be a real form of a complex semisimple Lie group $G$. Recall that Rossmann defined a Weyl group action on Lagrangian cycles supported on the conormal bundle of the flag variety of $G$. We compute the signed average of the Weyl group action on the characteristic cycle of the standard sheaf associated to an open ${G}_{\mathbb{R}}$-orbit on the flag variety. This result is applied to find the value of the constant term in Harish-Chandra’s limit formula for the delta function at zero.

This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of $d$-orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when $d=1$. The underlying algebraic framework allowed a systematic derivation...