Classification of Irreducible (𝔤, 𝔨)-Modules Associated to the Ideals of Minimal Nilpotent Orbits for Simple Lie Groups of TypeA

Abstract

We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent coadjoint orbit for $\mathfrak {g}=\mathfrak {s}\mathfrak {l}(n,\mathbb {C})$ and classify irreducible $(\mathfrak {g},\mathfrak {k})$-modules, which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak {k}$-modules.

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

“…Kable [20] and the authors [47] recently showed a Peter-Weyl type theorem for the kernel Ker(D) of an intertwining differential operator D. The theorem allows us to compute the K-type formula of Ker(D) explicitly by solving the hypergeometric/Huen differential equation ( [46,47]). Tamori [57] independently used a similar idea to determine the K-type formula of Ker(D) for his study of minimal representations.…”