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FONDECYT-REGULAR - 2001 - 1010423
MODELOS DE GELFAND Y CONSTRUCCION DE REPRESENTACIONES DE GRUPOS
  • Nombre : MODELOS DE GELFAND Y CONSTRUCCION DE REPRESENTACIONES DE GRUPOS
  • Número : 1010423
  • Año Concurso : 2001
  • Concurso : FONDECYT-REGULAR
  • Consejo : CIENCIA
  • Duración : 3 años
  • Estado : APROBADO
  • Sector de Aplicación : CONOCIMIENTO GENERAL

SUMMARY
The main problem we address in this proposal is the construction of complex irreducible representations of finite classical groups. More precisely, we intend to develop new methods of construction of representations which are universal (independent of the base field for the given classical group) and geometric in nature.

A first approach to our problem consists in constructing first, in a geometric way, large multiplicity-free representations of our classical group G, which contain most of its irreducible representations. Multipliciy-free representations of classical groups arise very frequently from natural actions of G in suitable geometric spaces. A very important special case of this setup is the case of Gel'fand Models, i.e. representations of G which contain each irreducible representations of G exactly once.

For this reason, we intend first to construct Gel'fand Models by geometric or cohomological methods. Our first main conjecture is that Gel'fand Models for classical finite groups may be always obtained thru suitable cohomological constructions à la Solomon-Tits, as in the case of the Steinberg representation. We have already some partial results supporting this conjecture in the case of symmetry groups of regular polygones and polyhedra, and general linear groups over finite fields.

We also conjecture that the character of a Gelfand Model for G, which we call the Gel'fand character of G may be always expresses as the difference of two permutation characters of G. This is tantamount to saying that our model lies always in the Green ring of G (i. e. the ring generated by the natural representations of G).

We intend to work towards the proof of these conjectures and also to study and fully decompose various relevant multiplicity-free representations of some classical finite groups and to compute the corresponding spherical functions, as in the following cases:
- natural representations associated to orthogonal similarity groups, in even and odd dimension, in any characteristic.
- natural representations associated to symplectic similarity groups GSp(2n,k) , for instance, those associated to the finite analogue of Siegel's Half Space.

Notice that for n=1 the representations we consider are associated to the finite analogue of (the double cover of) Poincaré's Upper Half Plane. The corresponding spherical functions have turned out to be quite useful in a number of contexts (construction of Ramanujan graphs [1,40,31,32], harmonic analysis of Radon Transforms [88], harmonic analysis of random walks [21], etc.)

A second approach to our main problem is to try to extend the construction of the so called Weil representations for the group GL(2,k), k a finite field, to other classical finite groups, for instance, to GL(3,k) to begin with. Recall that these Weil representations, associated to each semi-simple 2-dimensional algebra A over k, provide by decomposition all irreducible representations of our group. In the case of
G = GL(3,k) for instance, it is clear, from the character table of G that the relevant algebras are the "cubic" semi-simple algebras over k, instead of the aforementioned quadratic algebras for GL(2,k). The construction of the wanted higher order analogues of the classical Weil representations has turned out to be a hard, still open problem. Notice that a satisfactory solution to this problem would provide a constructive proof of the Langlands correspondence for our finite group, as it is the case for the Weil construction for GL(2,k).

Among the methods we intend to employ we may mention the exploitation of duality, in the sense of Howe and Juyumaya [4,39], the method of tensor division of representations and the introduction of n-ary analogues of the Heisenberg groups which lie at the heart of the original Weil construction.

NOTE: Numbers in square brackets refer to the bibliography appended to our research proposal IV.1 (p. 4)

ARTICULOS
  • A BRUHAT DECOMPOSITION OF THE GROUP SI* (2,A), JOURNAL OF ALGEBRA, VOL.262:401-412, 2003.
  • Autores Asociados al Proyecto

    • PANTOJA MACARI, JOSE EDUARDO
    • SOTO ANDRADE, JORGE ANTONIO.




  • TWISTED SPHERICAL FUNCTIONS ON THE FINITE POINCARE UPPER HALF-PLANE, JOURNAL OF ALGEBRA, VOL.248:724-746, 2002.
  • Autores Asociados al Proyecto

    • SOTO ANDRADE, JORGE ANTONIO.


    Autores No Asociados al Proyecto

    • VARAS, JORGE.




INFORME FINAL
  • MODELOS DE GELFAND Y CONSTRUCCION DE REPRESENTACIONES DE GRUPOS, 2004. 12 p.
  • Autores Asociados al Proyecto

    • SOTO ANDRADE, JORGE ANTONIO.




MANUSCRITOS
  • A GEOMETRICAL GEL`FAND MODEL FOR GL (3,K), K A FINITE FIELD, 2 p.
  • Autores Asociados al Proyecto



    Autores No Asociados al Proyecto

    • YAÑEZ, FRANCISCA.




  • ANALISIS ARMONICO EN ESPACIOS CUADRATICOS EN DIMENSION IMPAR, CASO ISOTROPO, 4 p.
  • Autores Asociados al Proyecto

    • JIMENEZ BRIONES, DANIEL ALBERTO.




  • HARMONIC ANALYSIS OF RADON FILTRATIONS FORN SN AND GLN (Q), 11 p.
  • Autores Asociados al Proyecto



    Autores No Asociados al Proyecto

    • YAÑEZ, FRANCISCA.




  • HARMONIC ANALYSIS ON EVEN DIMENSIONAL QUADRATIC SPACES, 25 p.
  • Autores Asociados al Proyecto

    • JIMENEZ BRIONES, DANIEL ALBERTO.




  • ON GENERATORS AND REPRESENTATIONS OF THE GROUPS GL* (2,A)AND SL* (2,A), 9 p.
  • Autores Asociados al Proyecto

    • PANTOJA MACARI, JOSE EDUARDO
    • SOTO ANDRADE, JORGE ANTONIO.




  • ON REALIZATIONS OF THE GEL`FAND CHARACTER OF A FINITE GROUP, 1 p.
  • Autores Asociados al Proyecto



    Autores No Asociados al Proyecto

    • YAÑEZ, FRANCISCA.




  • TENSOR PRODUCTS OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP GL (3,F.), 12 p.
  • Autores Asociados al Proyecto

    • PANTOJA MACARI, JOSE EDUARDO
    • SOTO ANDRADE, JORGE ANTONIO.


    Autores No Asociados al Proyecto

    • ABURTO HAGEMAN, LUISA AMADA.




  • TRANSITIVE NEAR NEARFIELD PLANES, 7 p.
  • Autores Asociados al Proyecto



    Autores No Asociados al Proyecto

    • DRAAYER, D
    • JOHSON, N L
    • POMAREDA, R.




  • UNE APPROCHE COHOMOLOGIQUE DU MODELE DE GEL`FAND DE GL (N,Q), 3 p.
  • Autores Asociados al Proyecto

    • SOTO ANDRADE, JORGE ANTONIO.


    Autores No Asociados al Proyecto

    • AUBERT, ANNE MARIE.




CIENCIA, CONOCIMIENTO GENERAL
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