\(K\)-types of \(\text{SU}(1,n)\) representations and restriction of cohomology (Q1568920)

Let \(G\) be the group \(SU(1,n)\) \((n > 1)\) and \(K \cong U(n)\) a maximal compact subgroup of \(G\); let \(T\) be the Cartan subgroup of \(K\) of all diagonal matrices in \(G\) and \(\mathfrak{g}_0\), \(\mathfrak{k}_0\), \(\mathfrak{t}_0\), respectively, the Lie algebras of \(G\), \(K\), \(T\) with complexifications \(\mathfrak{g}\), \(\mathfrak{k}\), \(\mathfrak{t}\). Let \(\widehat{T}\) be the set of analytically integral weights \(\mu = (\mu_0,\mu_1,\dots,\mu_n)\) on \(\mathfrak{t}\) and \(\widehat{T}^+\) the subset of the positive elements of \(\widehat{T}\), i.e., of the \(\mu \in \widehat{T}\) with \(\mu_1 \geq \mu_2 \geq\cdots\geq \mu_n\). If \(X\) is a \((\mathfrak{g},K)\)-module, \(\theta\) a Cartan involution of \(\mathfrak{g}_0\), and \(\mathfrak{q} = \mathfrak{l} + \mathfrak{u}\) a \(\theta\)-stable parabolic subalgebra of \(\mathfrak{g}\), where \(\mathfrak{l}\) is the Levi factor and \(\mathfrak{u}\) the radical of a Levi decomposition of \(\mathfrak{q}\), then the restriction mappings \(\text{Hom}(\bigwedge^n\mathfrak{u},X) \mapsto \text{Hom}(\bigwedge^n (\mathfrak{u}\cap\mathfrak{k}) ,X)\) \((n \in \mathbb{N})\) induce a mapping \[ \tau: H^*(\mathfrak{u},X) \mapsto H^*(\mathfrak{u}\cap\mathfrak{k},X) \] of cohomology vector spaces. Let \(L\) be the normalizer of \(\mathfrak{q}\) in \(G\). If \(\nu\) (identified with an \(L \cap K\)-type of \(X\)) is the highest weight of a representation of \(L \cap K\) appearing in \(H^n(\mathfrak{u}\cap\mathfrak{k},X)\), then \(\nu_K = w(\nu + \rho_K) - \rho_K\) is a \(K\)-type of \(X\) called the associated \(K\)-type to \(\nu\), when \(\rho_K\) is the half sum of the positive roots of \(\mathfrak{k}\) corresponding to the choice of \(\widehat{T}^+\) and \(w\) is an element of the Weyl group of \(K\) such that \(w(\nu + \rho_K)\) is positive. If \(\mathcal{C}\) is the closed convex hull in \(i\mathfrak{k}_0^*\) of the set of highest weights of the \(K\)-types appearing in \(X\), then a \(K\)-type \(\mu\) of \(X\) is said to lie on the geometric edge of the set of \(K\)-types of \(X\) if \(\mu \in \widehat{T}^+\) and it lies on the boundary of \(\mathcal{C}\). It is shown in this paper that if the \((\mathfrak{g},K)\)-module \(X\) is carrying an irreducible representation of \(G\), then any \(K\)-type of \(X\) lying on a geometric edge is the associated \(K\)-type to an \(L \cap K\)-type of \(X\) lying in the image of \(\tau\) for some \(\theta\)-stable parabolic subalgebra \(\mathfrak{q} = \mathfrak{l} + \mathfrak{u}\) and, actually, only two parabolic subalgebras are needed. In addition, all \(K\)-types of \(X\) may be recovered from the closed convex hull \(\mathcal{C}\) by intersecting it with a translate of the root lattice of \(K\).