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Register a new userExtensions of real bounded symmetric domains
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For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a bounded domain G_h/K_h. We describe the boundary orbits and relate them to the boundary orbits of G_h/K_h. We relate the crown and the split-holomorphic crown of G/K to the crown Xi_h of G_h/K_h. We identify an extension of a representation of K to a larger group L_c and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.
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Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks most needed in an approach to the classification of simple vectorial Lie superalgebras, we list the outer derivations and nontrivial central extensions. When the answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of nonsymmetric, namely, periplectic Lie superalgebras (of any rank) preserving the nondegenerate supersymmetric odd bilinear forms, and of the Lie algebras obtained from periplectic Lie superalgebras by desuperization when the characteristic of the ground field is equal to 2. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results are counterintuitive.
We study the complete K{a}hler-Einstein metric of a Hartogs domain $widetilde {Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $Omega $, using a power $N^{mu}$ of the generic norm of $Omega$. The generating function of the K{a}hler-Einstein metric satisfies a complex Monge-Amp`{e}re equation with boundary condition. The domain $widetilde {Omega}$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $Xinlbrack0,1[$. This allows to reduce the Monge-Amp`{e}re equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $mu_{0}$ of $mu$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.
Gabber and Joseph introduced a ladder diagram between two natural sequences of extensions. Their diagram is used to produce a twisted sequence that is applied to old and new results on extension groups in category $mathcal{O}$.
It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We give a condition under which the bundle and the direct sum of its irreducible constituents are intertwined by an equivariant constant coefficient differential operator. We show that in the case of the unit ball in $mathbb C^2$ this condition is always satisfied. As an application we show that all homogeneous pairs of Cowen-Douglas operators are similar to direct sums of certain basic pairs.
It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous $n$-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in $mathbb C^n$.
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