We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, Transf. Groups (2012)], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan-Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators [Juhl, Progr. Math. 2009] and its generalizations to spinor bundles. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

Abstract.Working on the four-sphere S4 , a flat four-torus, S x S , or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit formulas for the functional determinants of the conformai Laplacian (Yamabe operator) and the square of the Dirac operator, and discuss qualitative features of the resulting variational problems. Our analysis actually applies in the conformai class of any Riemannian, locally symmetric, Einstein metric on a compact 4-manifold; and to any geometric differential operator which has positive definite leading symbol, and is a positive integral power of a conformally covariant operator.

Let V be a Hermitian symmetric space of the non-compact type, w its Kaehler form. For A a geodesic triangle in £>, we compute explicitly the integral J a;, generalizing previous results (see [D-T]). As a consequence, if X is a manifold which admits V as universal cover, we calculate the Gromov norm of [u>] E /f 2 (X ) R). The formula for f. u is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see [C-0]). Introduction.Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral component of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form UJ. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to mean triangles the sides of which are geodesic segments). When M is of type 1,11 or III (in E. Cartan's classification), the integral / A a; (the symplectic area of the geodesic triangle A) was computed in [D-T]. By using their techniques, we give the result in the general case. It turns out that these quantities have an upper bound, and with the appropriate normalization, the bound depends only on the rank r of M. We extend these computations to ideal triangles, and we prove (new) sharp estimates for the areas. In particular, we determine precisely the triangles for which the upper bound is achieved. This turns out be of great geometric significance, as the summits of such an extremal triangle are contained in the image of a tight holomorphic totally geodesic imbedding of the complex unit disc into M (Theorem 4.7). Generally speaking, our study of the integrals / A u requires the fine structure of Hermitian symmetric spaces : special role played by the tube-type case, behaviour of geodesies at infinity and structure of G-orbits in the boundary, use of partial Cayley transforms.This study is also related to a previous work (see [C-0]) where we extended the notion of Maslov index to the Shilov boundary S of a Hermitian symmetric space of tube-type. The Maslov index is (up to a factor TT) nothing but the symplectic area of ideal triangles with summits in 5, and in some sense the present work can be understood as a continuation of [C-0].The computation of the integrals was used in [D-T] to calculate the Gromov norm of the Kaehler class of a compact Hermitian locally symmetric manifold X = r\M, where M is of type I and F a discrete, torsion-free, co-compact subgroup of the group G. They observed that it has a nice topological corollary. Let S be a Riemann surface of genus g > 1 and / : S -> X a continuous map. Then /.

We present a new method of calculating intertwining operators between principal series representations of semisimple Lie groups G. Working in the compact realization we find the eigenvalues of the operators on the K-types, and give several examples. Among the advantages of our method is its applicability to bundlevalued cases.1996 Academic Press, Inc.

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