Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866909159983677440 |
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| author | Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo |
| author_facet | Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo |
| contents | In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the multiplicity-free case. We then develop a suitable theory of spherical functions that, in the case of induction of the trivial representation of the subgroup, reduces to the classical theory of spherical functions for finite Gelfand pairs. \par We also examine in detail the case when we induce from a normal subgroup, showing that the corresponding harmonic analysis can be reduced to that on a suitable Abelian group. \par The second part of the work is devoted to a comprehensive study of two examples constructed by means of the general linear group GL$(2,\mathbb{F}_q)$, where $\mathbb{F}_q$ is the finite field on $q$ elements. In the first example we induce an indecomposable character of the Cartan subgroup. In the second example we induce to GL$(2,\mathbb{F}_{\!q^2})$ a cuspidal representation of GL$(2,\mathbb{F}_q)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1811_09526 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups Ceccherini-Silberstein, Tullio Scarabotti, Fabio Tolli, Filippo Representation Theory Functional Analysis Group Theory 20C15, 20C08, 20G05, 20G40, 43A65, 43A90, 43A35 In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the multiplicity-free case. We then develop a suitable theory of spherical functions that, in the case of induction of the trivial representation of the subgroup, reduces to the classical theory of spherical functions for finite Gelfand pairs. \par We also examine in detail the case when we induce from a normal subgroup, showing that the corresponding harmonic analysis can be reduced to that on a suitable Abelian group. \par The second part of the work is devoted to a comprehensive study of two examples constructed by means of the general linear group GL$(2,\mathbb{F}_q)$, where $\mathbb{F}_q$ is the finite field on $q$ elements. In the first example we induce an indecomposable character of the Cartan subgroup. In the second example we induce to GL$(2,\mathbb{F}_{\!q^2})$ a cuspidal representation of GL$(2,\mathbb{F}_q)$. |
| title | Harmonic analysis and spherical functions for multiplicity-free induced representations of finite groups |
| topic | Representation Theory Functional Analysis Group Theory 20C15, 20C08, 20G05, 20G40, 43A65, 43A90, 43A35 |
| url | https://arxiv.org/abs/1811.09526 |