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They result in many nontrivial properties of quantum immanants.
      
We prove the following result: LetG be a finite irreducible linear group.
      
The proof is an application of a recent result by W.
      
We prove as the main result thatM is weakly symmetric with respect toG1 and complex conjugation.
      
As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf.
      
A basis is calledmonomial if each of its elements is the result of applying to a (fixed) highest weight vector a monomial in the Chevalley basis elementsYα, α a simple root, in the opposite Borel subalgebra.
      
We prove a more general version of a result announced without proof in [DP], claiming roughly that in a partially integrable highest weight module over a Kac-Moody algebra the integrable directions from a parabolic subalgebra.
      
As our main result, we prove that every coherentK-sheaf onX extends uniquely to a holomorphicG-sheaf onXc.
      
The classification of infinite-dimensional simple linearly compact Lie superalgebras given in [K2] is based on this result.
      
From it, we recover Joseph and Letzter's result by a kind of "quantum duality principle".
      
The last section is devoted to a result about the cohomology of a Lie superalgebra with reductive even part with coefficients in a finite dimensional moduleM.
      
To prove this, we first state and prove a general result which gives a criterion for checking whether a variety of dimensionN≥3 is a (compactification of a) ball quotient.
      
For a smooth oriented surface Σ, denote byM(Σ) the set of all ways to represent Σ as a result of gluing together standard spheres with holes ("the Lego game").
      
The main result of this paper is to establish the upper bound form, for eachn.
      
to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure.
      
Except for the Borel and some special cases a corresponding result is not known for the semi-centre of the enveloping algebra ofp.
      
The result generalizes and implies the classical "branching rules" that describe the restriction of an irreducible representation of the symmetric groupSn toSn-1.
      
We give a generalization of this result for the isotropy representations of symmetric spaces.
      
By a result of Miyanishi, its ring of invariants is isomorphic to ?[t1,t2].
      
We give a new proof of the famous result that any two embeddings of the affine lineA1 inA2 are equivalent by an automorphism ofA2.
      
 

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