In this note we explicitly construct top-dimensional components of certain cyclic convolution varieties. These correspond (via the geometric Satake equivalence) to irreducible summands V(λ+μ−Nβ)⊂V(λ)⊗V(μ) for SLn+1(C), where N≥1 and β is a positive root. Furthermore, deduce from these constructions nontrivial lower bound on multiplicity subrepresentations when not simple Finally, demonstrate th...

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ Cartan decomposition of its algebra. $K$ maximal compact subgroup algebra k$ $\tau$ an irreducible representation on complex vector space $V$. $h$ Hermitian scalar product $V$ such that $\tau(G)$ is compatible respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $\mu_{\mathfrak p}:\m...