We propose a generalization of the theory of spherical harmonics to the context of symmetric subgroups of reductive groups acting on flag manifolds. We give some sample results for the case of the orthogonal group acting on Grassmann manifolds, especially the case of 2-planes.
We consider a construction of the fundamental spin representations of the simple Lie algebras in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a -graded associative algebra (rather than the usual -filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some . Additionally...
Let be a Hermitian symmetric space of the noncompact type and let be a discrete series representation of holomorphically induced from a unitary character of . Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple by a suitable modification of the Berezin calculus on . We extend the corresponding Berezin transform to a class of functions on which contains the Berezin symbol of for in the Lie algebra of . This allows...
Let be a semisimple algebraic Lie group and a reductive subgroup. We find geometrically the best even integer for which the representation of in is almost . As an application, we give a criterion which detects whether this representation is tempered.
For the dual pair considered, the Cauchy Harish-Chandra Integral, as a distribution on the Lie algebra, is the limit of the holomorphic extension of the reciprocal of the determinant. We compute that limit explicitly in terms of the Harish-Chandra orbital integrals.