# Mathematics Colloquia and Seminars

In last year's algebra seminar I discussed joint work with Misha Kapovich and Bernhard Leeb explaining the connection between the problem of decomposing tensor products of triple of irreducible finite dimensional representations of a simple complex Lie group $G^{\vee}$ defined over the integers Z and the geometric problem of constructing triangles in the symmetric space $X = G/K$ with given "side-lengths" $a,b,c$. Here $G$ is Langlands' dual to $G^{\vee}$ and the "side-lengths" are elements in the positive Weyl chamber of in the Lie algebra of a maximal split torus $A$ in $G$. I also related these two problems to the problem of determining the structure constants of the spherical Hecke algebra of $G(F)$ where $F$ is a p-adic field. There is a chain of implications between the various problem that can be reversed only up to multiplying the highest weights $a,b,c$ by a "saturation factor $k_G$".
In this year's seminar I will discuss joint work with Tom Haines and Misha Kapovich. I will replace the representation theory problem of decomposing tensor products with the the problem of "branching to Levi's" i.e. the problem of finding formulas for the restriction of an irreducible representation of $G^{\vee}$ to $M^{\vee}$ where $M^ {\vee}$ is a Levi subgroup of a standard parabolic subgroup $P^{\vee}$ in $G^ {\vee}$. Once again there is an associated geometry problem in the symmetric space $X = G/K$. The standard parabolic subgroup $P^{\vee}$ corresponds to a standard parabolic subgroup $P = MU$ of $G$. The new geometry problem is the problem of determining the projection map (along the $U$ orbits) from a "sphere of radius $\lambda$" in $X$ (i.e. the $K$-orbit of $exp(\lambda)K$ in $X$) to the symmetric subspace $Y = M/K_M$ where $K_M$ is a maximal compact subgroup of $M$. In the case that $P$ is the Borel $B$ so $M = T$ = the maximal torus and $T/K_T = A$ and this is the problem of describing the "Iwasawa $A$-parts" for elements in G with the fixed Cartan $A$- part $exp(\lambda)$ i.e. comparing the decompositions $G =KAK$ and $G= UAK$. This is the situation of Kostant's "nonlinear convexity theorem/ problem". Again there is a Hecke algebra problem (there is a "constant term" map from the spherical Hecke algebra of $G(F)$ to that of $M(F)$). There is again a chain of implications between the various problems that can be reversed only up to saturation. It is a remarkable fact that the same saturation factor comes into the branching problem as for the decomposing tensor products problem. If an $M$-highest weight $\mu$ occurs in the projection of the sphere of radius $\lambda$ (where $\lambda$ is a $G$-highest weight) then the restriction of the irreducible representation of $M^{\vee}$ with highest weight $k_G \mu$ occurs in the restriction to $M^\vee$ of the representation of $G^\vee$ with highest weight $k_G \lambda$. There are examples that show that the factor $k_G$ is sometimes necessary.