I will talk about the following results I proved with Bernhard
Leeb and Misha Kapovich in a series of papers - see for example AMS
Memoirs 896. The main point was to generalize
the saturation theorem of Knutson and Tao to any simple reductive group
(except one needs the saturation factor k_G , see below).
In a rank one symmetric space of noncompact type X=G/K the only invariant
of pairs of points x,y is the distance between them. In a symmetric space
of rank k the
distance gets replaced by a point in the Weyl chamber \Delta in a Cartan
subspace. Using this we can define the \Delta valued distance
d_{\Delta}(x,y). Given three points
\alpha,\beta,\gamma \in \Delta one may ask for conditions on \alpha,
\beta,\gamma that are necessary and sufficient in order that one can draw a
geodesic triangle in X
with side-lengths \alpha,\beta,\gamma, precisely find three points x,y,z in
Z so that d_{\Delta}(x,y) = \alpha, d_{\Delta}(y,z) = \beta, d_{\Delta}(z,x)
= \gamma.
The answer is that there is a system of homogeneous linear inequalities
determined by the Schubert calculus in the Grassmanians G/P P ({ a maximal
parabolic subgroup of G)
that give these necesssary and sufficient conditions.
Now suppose that G = G(R) is the group of real points of a reductive
algebraic group defined over Z. Then we have G(Q_p) and the Langlands’
dual G^{\vee}. Suppose further that
\alpha,\beta,\gamma are integral in the sense that they are cocharacters
of a maximal torus T defined over Z. Such cocharacters \lambda parametrize
basis elements f_{\lambda}
for the spherical Hecke algebra of G(Q_p_) and basis elements ch_{\lambda}
of the representation ring of G^{\vee}. Hence we may define triple
structure constants m and n
of the two rings parametrized by triples of dominant cocharacters by
f_{\alpha}. f_{\beta} . f_{\gamma} = m(\alpha,\beta,\gamma} 1 + …
and
ch_{\alpha}. ch_{\beta} .ch_{\gamma} = n(\alpha, \beta,\gamma) 1 + ….
Let k_G be the LCM of the coefficients of the highest root when expressed
in terms of the simple roots. Assume \alpha + \beta + \gamma is in the
coroot lattice. Then we have
the following
Theorem
I. n(\alpha,\beta,\gamma) is nonzero implies m(\alpha,\beta,\gamma) is
nonzero implies \alpha, \beta,\gamma satisfy the triangle inequalities.
2. \alpha, \beta,\gamma satisfy the triangle inequalities implies m( k_G
alpha, k_G \beta, k_G \gamma) is nonzero implies n( k_G ^2 \alpha,\ k_G^2
beta, k_G^2 \gamma) is nonzero.
For GL(n) we have k_G = 1 so the result includes the saturation theorem of
Knutson and Tao. For the other classical groups k_G = 2.