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Title: Steinberg representations for the general linear groups over the integers modulo a prime power
Author: Lees, Paul
ISNI:       0000 0001 3607 6830
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1976
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In this work we construct two representations, Sg and StG, for the general linear group G = GLn (Ϟ/ph Ϟ),over the integers modulo a prime power, which, in different ways may be regarded as analogues of the usual Steinberg representation SG1 of G.1= GLn Ϟ /p Ϟ ). Chapter 1 contains technical results which are required for the representation theory of the sequel. We Investigate the parabolic structure of G and discuss 'unramified semi- simple' and 'regular' elements of G. In Chapter 2 we construct SG by an analogic of Curtis' formula; this is inductive on h, requiring prior construction of SM, for M ≡ GLn(Ϟ ph1Ϟ). We also express SG as an alternating sum of permutation representations. SG is not irreducible if h > 2 but its character is 0 or + a power of p at many elements. In Chapter 3 StG appears as The 'largest' irreducible component of 1GB (suitably defined) and is constructible homologically, but it has complicated character values. It is contained in the 'Gelfand-Graev representation' XGU with multiplicity 1; this is proved by considering a related 'affine Steinberg representation' StG, which is isomorphic to "XHU. We also show that "XGU is multiplicity-free in certain cases. In Chapter 4 we give geometric interpretations of SG and StG using a Bruhat-TIts building; these enable us to show that Sg is a subrepresentation of 1GB and contains StG (except possibly if p=2). Finally, in Chapter 5 we give some examples and counter­examples; in particular we show that the character of isnot always 0 or + a power of p and we compute the character of StG at split semisimple elements for n=2 and 3» giving a conjecture for the general case.
Supervisor: Not available Sponsor: Science Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics