Lees, Paul (1976) Steinberg representations for the general linear groups over the integers modulo a prime power. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Representations of Lie groups, Modular representations of groups, Finite simple groups, Lie algebras
Official Date:	1976
Dates:	Date Event 1976 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Lusztig, George, 1946-
Sponsors:	Science Research Council (Great Britain)
Extent:	vi, 141 leaves
Language:	eng

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