The inverse limit of the spaces U(n) is equipped with 2-parametric family of canonical U(∞) × U(∞)-quasiinvariant measures. It will be given a self-consistent description of this construction and also a nonformal introduction to the Plancherel formula, which was recently obtained by G. Olshanski and A. Borodin.

For each operator Λ → Λ we associate a formal series K(x₁, x₂, …; y₁, y₂, …) (bisymmetric kernel of operator) symmetric with respect to x_j and with respect to y_j. Our representations is realized by operators corresponding to kernels of the form

We also show, that the set of all operators having such kernels is closed with respect to multiplication and describe this semigroup

(1) We give a classification of the homomorphisms between SGVMs associated to maximal parabolic subalgebras.

(2) Using a comparison theorem, we construct a homomorphism between SGVMs associated to (not necessarily maximal) parabolic subalgebra from a homomorphism between SGVMs associated to a maximal parabolic subalgebra.

In the 1990s Vogan has studied an algebraic version of Parthasarathy's Dirac operator. He conjectured that if the Dirac operator has nonzero kernel for a unitary (g,K)-module X, then this kernel (explicitly) determines the infinitesimal character of X. This conjecture was recently proved by J.-S. Huang and myself.

In the meantime, Kostant has defined a more general, ''cubic'' Dirac operator, attached to other subalgebras than k. In the case of a Levi subalgebra, this Dirac operator is closely related to the corresponding u-cohomology. In a recent preprint with J.-S. Huang and D. Renard, we study this relationship in detail in certain special situations.

Let g be a simple Lie algebra of type A, D, E and n a maximal nilpotent subalgebra of g. Moreover, let N be a maximal unipotent subgroup of a simple Lie group with Lie algebra g. Finally, let Π denote the corresponding preprojective algebra.

Lusztig's semicanonical basis S of U(n) is parametrized by irreducible components of the corresponding nilpotent varieties mod(Π,d). The dual S^* is a basis of C[N].

We can show: For two elements of S^* holds b_C • b_D ∈ S^* if for the corresponding irreducible components holds ext_Π¹(C,D)=0.

On the other hand, the dual canonical basis B_q^* of U_q(n) specializes for q=1 to a basis B. We can show, that S and B have many elements in common, but B and S coincide only if Π is representation finite, i.e. in the cases A_2,3,4. This explains the multiplicative properties of the dual canonical basis observed previously in these cases.

On the other hand it gives us a good control over the dual semicanonical basis in the cases A₅ and D₄, i.e. when Π is tame, since we have in this case a precise combinatorial description of the irreducible components of mod(Π,d) in terms of indecomposable components. This is closely related to an elliptic root system of type E₈^(1,1) resp. E₆^(1,1).

Let G be the adjoint group of the Lie algebra of type F₄, E₆, E₇ or E₈ with real rank four. (For F₄, G is replaced with its double cover). There is a dual pair G₂ × H in G, with G₂ the split real group of type G₂ and H compact.

We restrict the minimal representation of G constructed by Gross and Wallach to this dual pair and obtain the explicit Howe correspondences. For an irreducible (finite-dimensional) representation E of H, we first calculate the K-types of Θ(E) using some ''see-saw'' techniques and branching laws. Then we identify Θ(E) in the unitary dual of G₂ as given by Vogan.

Finally, we show that our results can serve as examples of Langlands correspondences.