Teoría de Lie

Complex structures on affine groups by Barberis, M. L. This is a joint with I. Dotti.
On any hermitian manifold there is a distinguished connection (the Bismut connection), which arises, for instance, in N=2 supersymmetry under the presence of the Wess-Zumino term. Physicists call such a connection a KT-connection (KT stands for Kaehler with torsion). In case the manifold is hyperhermitian there is an analogous connection, called HKT-connection, which may or may not exist, but in case it exists it is unique. There are two types of HKT-structures on manifolds, stron and weak, depending on whether the torsion of the corresponding connection is closed or not.
In this work we consider complex structures on a Lie group G acting holomorphically on a complex vector space V. In this case we show that the corresponding affine group carries a natural left invariant complex structure. A particular case of this construction yields a procedure for obtaining weak HKT structures on homogeneous manifolds.

On conformal algebras by Boyallián, Carina This is a joint work with V. G. Kac and J. Liberati.
The associative conformal algebra Cend$_N$ and the corresponding general Lie conformal algebra $gc_N$ are the most important examples of simple conformal algebras which are not finite.. In this work we classify all associative conformal subalgebras of $\hbox{Cend}_1$. We also describe a new family of Lie conformal subalgebras of $ gc_1$ and propose a conjectural list of all of them. In the second part we classify all irreducible conformal representations of the conformal algebras $ gc_1$ and $ oc_1$ of finite growth.

A new proof of the $J^2$-condition for real rank one simple Lie algebras and their classification by Ciatti, Paolo In this work a new purely algebraic proof of the $J^2$-condition for the nilpotent Iwasawa algebras in real rank one simple Lie algebras is presented, yielding the classification of real rank one Lie algebras.

Local integrability of characters and D-modules by Galina, Esther Harish-Chandra had proved that the character of an irreducible unitary representation of a semisimple Lie group is a G-invariant eighendistribution. Moreover, he proved that, in the sense of distributions, all the G-invariant eighendistribution are locally integrable functions on G. Hotta and Kashiwara had proved that a G-invariant eighendistribution is a solution of a regular holonomic D-module, where D is the ring of differential operators given by the Lie algebra of G. We present an algebraic formulation of the theorem of Harish-Chandra and the principal steps of a proof based on the theory of D-modules.

Restriction principle and orthogonal polynomials by Olafson, Gestur Let T=G/K be a Hermitian symmetric space of tube type. Then T has also a realization as a bounded symmetric domain D=G_b/K_b. D and T are isomorphic via a Caley transform and the same holds for the groups G and G_b. Let H be a symmetric subgroup of G such that H/L, L=K\cap H, is a totally real submanifold of T (and D). We discuss representations of H that are gotten by restricting a holomorphic representation of G to H. We use this to construct a family of orthgonal polynomials that generalise the Laguerra polynomials. We derive a differential equation and a recursion formula for those polynomials by using the holomorphic representation.

Matrix valued spherical functions associated to the complex projective plane by Pacharoni, M. Inés This is joint work with F. Grünbaum and J. Tirao.
The main purpose is to determine all irreducible spherical functions on G=SU(3) of arbitrary type, that are associated to any irreducible representation of K=S(U(2)\times U(1)). This is accomplished by associating to a spherical function on G a matrix valued function H on the complex projective plane P_2(C)=G/K.
By taking full advantage of the K-orbit structure of G/K the problem is reduced to determining a function H in one real variable that is an eigenfunction of two second order differential operator with matrix coefficients. The solutions to this pair of systems can be exhibited explicitely in terms of a special class of generalized hypergeometric functions {}_{p+1}F_p.
By a proper combination of several spherical functions corresponding to each irreducible representation of K we get a family of matrix valued function H(t,w) with 0<t<1 and w=0,1,..., such that the functions H(t,w) satisfy a matrix valued differential equation in the parameter t and a three term recursion relation in the parameter w.

Harmonic analysis on the Heisenberg groups associated to the action of U(p,q) by Saal, Linda This is a joint work with T. Godoy.
Here,we develop the harmonic analysis on the 2n+1 dimensional Heisenberg group H,associated to the action of U(p,q)by automorphisms.We determine that the algebra of left invariant differential operators, which are also U(p,q)- invariant, has two generators L and T, where T denotes the central element of the heisenberg algebra. We compute the U(p,q)-invariant eigendistribution and obtain an spectral decomposition on L(H). Also we study the decomposition into irreducible representations of the action of U(p,q)xH on L(H) where H acts by left traslations. Finally we determine relative fundamental solutions of the operators L+iaT, a is a complex number. Most of these topics were developed by several authors when we replace U(p,q) by a compact subgroups of the symplectic group, being an important case the orthogonal group.

On unitary representations and cohomological induction by Salamanca, Susana I will discuss a program to reduce the classification Problem of the Unitary dual of a real Lie group, via lowest K type theory and cohomological induction, to a special Class of unitary representations.

Free Akivis algebras, primitive elements, and hyperalgebras by Shestakov Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved. The conjecture of K. H. Hofmann and K.Strambach on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed. Finally, it is proved that every algebra $B$ can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on $B$ are interpreted by certain primitive elements.

The classifying ring of a rank one semisimple Lie group by Tirao, Juan Let G_o be a connected, non compact real semisimple Lie group with finite center, and let K_o denote a maximal compact subgroup of G_o. We denote with g_o and k_o the Lie algebras of G_o and K_o and g and k will denote the respective complexified Lie algebras. Let U(g) be the universal enveloping algebra of g and let U(g)^K denote the centralizer of K_o in U(g).
By the fundamental work of Harish-Chandra it is known that many deep questions concerning the infinite dimensional representation theory of G_o reduce to questions about the structure and finite dimensional representation theory of the algebra U(g)^K, called the classifying ring of G_o.
Let a be the complex abelian Lie algebra associated to an Iwasawa decomposition G_o=K_o A_o N_o of G_o adapted to K_o. To study the algebra U(g)^K Kostant suggested to consider the projection map P from U(g) onto U(k) tensor U(a), corresponding to the direct sum U(k) tensor U(a) + U(g)n associated to the Iwasawa decomposition. It is known that the map P restricted to U(g)^K becomes an injective antihomomorphism into U(k)^M tensor U(a), where U(k)^M denotes the centralizer of M_o in U(k), M_o being the centralizer of A_o in K_o and U(k)^M tensor U(a) is given the tensor product algebra structure. A very interesting problem it is to determine the image of P(U(g)^K). Towards this end we introduced a subalgebra B of U(k)^M tensor U(a) defined by a set of equations derived from certain imbeddings among Verma modules.
In this talk we shall present the main ideas towards the proof of the the following theorem:
If the split rank of G_o is one, then P(U(g)^K) =B^{W_\rho} when rank(G_o) is not equal to rank(K_o) and P(U(g)^K) =B when rank(G_o)=rank(K_o). B^{W_\rho} denotes the algebra of all elements in B that are invariant under the tensor product action of W on U(k)^M and the translated Weyl group action on U(a).

Restriction of square integrable representations by Vargas, Jorge Let G be a connected semisimple noncompact Lie group and p a square irreducible representation of G. Let H be a semisimple subgroup of G. We choose a maximal compact subgroup K of G so that its intersection K_1 with H is a maximal compact subgroup of K. The lowest K-type of p resctricted to K_1 decomposes as a sum of irreducible representations, assume that one of this factors is the lowest K_1 type of a discrete series representations Z for H. We have
Theorem: Z is containded in the restriction of p to H.
When G=Spin(2n,1) and H=Spin(2n-1,1) inmersed in the usual way and p integrable, we have
Theorem: the restriction of p to H is a direct integral of unitary principal series induced from a minimal parabolic MAN by p_1, ..., p_k. The restriction of p_j to M is obtained as follows: we restrict the lowest K-type of p to K_1, this give us irreducible representations q_1,...q_k. Since rank of M is equal to rank of K_1, for each q_j we fix an irreducible representation p_j of M having the same highest weight.
Finally, we will talk of joint work with B. Orsted on the L^2-continuity of the restriction map r. That is, we realize p as the L^2-kernel of an elliptic operator on an homogeneous vector bundle over the symmetric space G/K. Then H/K_1 is a submanifold of X, since we may restrict smooth sections of a vector bundle over X to sections of the pull back of such a bundle to H/K_1, the map r is well defined on the kernel on an elliptic operator. We have: Theorem: if p is integrable then r is L^2-continuous.
We also have examples where p is not L^2-continuos.

The Meromorphic continuation of the resolvent of the Laplacian on line bundles over CH(n) by Will, Cynthia Let G=SU(n,1), K=S(U(n)x U(1)), and for an integer l, let T_l be a one-dimensional K-type and let E_l the line bundle over G/K associated to T_l. We prove that the resolvent of the Laplacian, acting on compactly suported smooth sections of E_l is given by convolution with a kernel which has a meromorphic continuation to C. We also prove that this extension has only simple poles and we identify the images of the corresponding residues with (g,K)-submodules of the principal series representations. We show that for certain values of the parameters these modules are holomorphic (or antiholomorphic) discrete series.

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