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In the present paper we study the remaing nontrivial case, that of a negative central charge-N.
      
The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections.
      
We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants.
      
We express them in terms of generatorsEij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities.
      
In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties.
      
Whenever the action of a maximal torus on the coneCλ* has some nice properties, we obtain simple closed formulas for all weight multiplicities and theirq-analogs in the representationsVnλ,n∈?.
      
In the interesting case when the group is of Coxeter typeDn (n≥4) we use higher polarization operators introduced by Wallach.
      
We give complete proofs of the K-theoretic construction of the quantized enveloping algebra of affine gl(n) sketched in [GV].
      
Lichtenstein in the caseu =(n, ?) or(n?), we prove that ?(q).ζ1/2 is non zero for all harmonic polynomialsq ∈S() \ {0}.
      
In this paper we compute the cohomology with trivial coefficients for the Lie superalgebraspsl(n, n), p (n) andq(2n); we show that the cohomology ring ofq(2n+1) is of Krull dimension 1 and we calculate the ring forq(3) andq(5).
      
As a corollary we obtain af·g·p·d·f subgroup of SLn(?) (n ≧ 3.
      
LetMm be a closed smooth manifold with an involution having fixed set of the form (point)?Fn, 0>amp;lt;n>amp;lt;m.
      
Let ρ:G?Gl(n,) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra.
      
We derive two consequences: the first is a new proof of Lusztig's description of the intersection cohomology of nilpotent orbit closures for GLn, and the second is an analogous description for GL2n/Sp2n.
      
In this paper we establish two results concerning algebraic (?,+)-actions on ?n.
      
Secon, we are interested in dominant polynomial mapsF:?n→?n-1 whose connected components of their generic fibers are contractible.
      
For such maps, we prove the existence of an algebraic (?,+)-action φ on ?n for whichF is invariant.
      
A conjectural generalization of the n! result to arbitrary groups
      
Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch-Okounkov, and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions.
      
We establish an explicit formula for the n-point correlation functions in the sense
      
 

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