Geodesic
Functions associated with group representations
Sbornik. Mathematics, Tome 206 (2015) no. 12, pp. 1722-1730 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Given a linear representation of a group, the usual approach is to consider numerical functions on the group that are associated with the representation. These may be generalized functions, that is, distributions. The simplest functions are matrix elements. They are given by a pair of vectors whose choice is arbitrary and random. However, we are interested in functions which are free of this arbitrariness (and thus are naturally associated with the representation); we refer to them as the modified traces of the representation. The usual trace of the representation (if it exists, possibly as a distribution) and spherical functions generated by a fixed vector of some subgroup give examples of these functions. However, it is possible that neither the trace nor spherical functions can be defined for a given representation. It is still desirable to introduce functions on the group that are naturally associated with the representation. We solve this problem for diffeomorphism groups and some representations of these groups. Bibliography: 4 titles.
Keywords: group representations, modified trace, spherical function, tensor product. 
@article{SM_2015_206_12_a3,
     author = {R. S. Ismagilov},
     title = {Functions associated with group representations},
     journal = {Sbornik. Mathematics},
     pages = {1722--1730},
     year = {2015},
     volume = {206},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2015_206_12_a3/}
}
TY  - JOUR
AU  - R. S. Ismagilov
TI  - Functions associated with group representations
JO  - Sbornik. Mathematics
PY  - 2015
SP  - 1722
EP  - 1730
VL  - 206
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2015_206_12_a3/
LA  - en
ID  - SM_2015_206_12_a3
ER  - 
%0 Journal Article
%A R. S. Ismagilov
%T Functions associated with group representations
%J Sbornik. Mathematics
%D 2015
%P 1722-1730
%V 206
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2015_206_12_a3/
%G en
%F SM_2015_206_12_a3
R. S. Ismagilov. Functions associated with group representations. Sbornik. Mathematics, Tome 206 (2015) no. 12, pp. 1722-1730. http://geodesic.mathdoc.fr/item/SM_2015_206_12_a3/

[1] V. Giiemin, S. Sternberg, Geometricheskie asimptotiki, Mir, M., 1981 ; V. Guillemin, Sh. Sternberg, Geometric asymptotics, Amer. Math. Soc., Providence, RI, 1977 | MR | Zbl | MR | Zbl

[2] R. S. Ismagilov, “Spherical functions on the group of diffeomorphisms preserving the volume”, Funct. Anal. Appl., 25:2 (1991), 150–152 | DOI | MR | Zbl

[3] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 ; I. C. Gohberg, M. G. Kreǐn, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl

[4] A. Gisharde, Kogomologii topologicheskikh grupp i algebr Li, Mir, M., 1984 ; A. Guichardet, Cohomologie des groupes topologiques et des algèbres de Lie, Textes Math., 2, CEDIC, Paris, 1980 | MR | Zbl | MR | Zbl