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Matrix problems, small reduction and representations of a class of mixed Lie groups. (English) Zbl 0829.16009
Representations of algebras and related topics, Proc. Tsukuba Int. Conf., Kyoto/Jap. 1990, Lond. Math. Soc. Lect. Note Ser. 168, 225-249 (1992).
[For the entire collection see Zbl 0746.00076.]
In representation theory of Lie groups the cases of reductive and solvable groups are highly elaborated. Much less seems to be known about mixed groups, i.e. those neither reductive nor solvable. Moreover, the simplest examples (cf. §1) show that in a sense the complete description of their representations is a hopeless problem. Nevertheless, in some cases it turns out to be possible to describe “almost all” of them in a rather appropriate way (cf. theorem 1.2), namely, they behave just like representations of a reductive group.
The lecture splits into two parts. The first (§§1-3) contains the formulation of the main theorem (theorem 3.1) with necessary preliminaries and its reduction to a matrix problem. The second part (§§4-6) is devoted to matrix problems which we treat in terms of representations of bocses [cf. A. V. Rojter, Lect. Notes Math. 831, 288-324 (1980; Zbl 0473.18006)] and culminates in §6 with the proof of the main theorem.

16G20 Representations of quivers and partially ordered sets
22E50 Representations of Lie and linear algebraic groups over local fields