UC San Diego

Classification of "small K-types" for the connected, simply connected split real form of simple Lie type other than type C_n is obtained via Clifford algebras which completes the list of all small K-types of dim > 1 for the connected, simply connected split real form of simple Lie types. An analog, P̂[xi], of Kostant's P̂[gamma] matrix is defined for a K-type V_[xi] of principal series admitting a small K-type, and a product formula of the determinant of P̂[xi] over the rank one subgroups corresponding to the reduced restricted roots is proved. The product formula and the relationship between P̂[xi] and intertwining operator between the genuine principal series representations give a method to compute the shift factors of Vogan and Wallach's generalization of Leslie Cohn's determinant formula for the restriction of the intertwining operator to a K-isotypic component given in terms of ratios of classical gamma functions. The determinant of the intertwining operator between the genuine principal series representations of widetilde [SL(n,R)] (n >̲ 3) is obtained as a ratio of classical gamma functions