The eigencone and saturation for Spin(8). (Q1022831)

Let \(G\) be a connected complex semisimple adjoint group, with maximal compact subgroup \(K\) and associated symmetric space \(X:=G/K\). Let \(\Delta\) be the associated Weyl chamber. The space \(X\) can be given a so-called \(\Delta\)-distance, and a fundamental question is to determine which triples of elements in \(\Delta\) can occur as side-lengths of a geodesic triangle in \(X\). This collection turns out to be a convex homogeneous polyhedral cone \(\mathcal C\) which is also the solution to the generalized eigenvalues of a sum problem. The cone \(\mathcal C\) (which depends only on the root system) was first described by a set of (generally) redundant `triangle inequalities', and later refined to a smaller set of `restricted triangle inequalities' by \textit{P. Belkale} and \textit{S. Kumar} [Invent. Math. 166, No. 1, 185-228 (2006; Zbl 1106.14037)]. For a type \(A\) root system, it is known that the two sets of inequalities coincide and are in fact irredundant. The first main result of this paper is an explicit computation of the restricted triangle inequalities for a root system of type \(D_4\) (i.e., \(G=\text{PSO}(8)\)). This is obtained via some explicit computations in the cohomology of \(G/P\) for a maximal parabolic subgroup \(P\). Further, with the aid of a computer program, it is verified that the inequalities are irredundant. A related problem involves the Langlands dual group \(G^\vee\). Consider a triple of dominant weights and the tensor product of the three irreducible modules having corresponding highest weights. The problem is to determine those triples such that the \(G^\vee\)-invariants of this tensor product is non-zero. Denote this set by \(\mathcal R\). It is conjectured for simply-laced root systems that for a triple of weights \((\lambda,\mu,\nu)\) (whose sum lies in the root lattice) and a positive integer \(N\), if \((N\lambda,N\mu,N\nu)\) lies in \(\mathcal R\), then \((\lambda,\mu,\nu)\) necessarily lies in \(\mathcal R\). (The conjecture can equivalently be stated in terms of relationship between \(\mathcal R\) and the cone \(\mathcal C\) discussed above.) The conjecture is known to be true in type \(A\) by work of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055-1090 (1999; Zbl 0944.05097)]. (It is also known that the conjecture can fail in non-simply laced cases.) The second main result is a verification via the aid of computer computations that the conjecture holds for a root system of type \(D_4\) and \(G^\vee=\text{Spin}(8)\). A final application is given to structure constants of the spherical Hecke algebra associated to \(G=\text{PSO}(8)\).