1) What is the proper term for a closed subgroup H of an algebraic group G such that every linear representation of H arises as the restriction of a representation of G?

2) Where can I read about this?

  • $\begingroup$ For reductive connected groups, the only examples I can think of are $G = H \times H'$. Is this the sort of thing you are interested in? $\endgroup$ – David E Speyer Jun 22 '12 at 16:44
  • $\begingroup$ This is no real answer (so I post it as a comment) but as far as 2) may be answered quite generically, if the context is the one of algebraic groups, I found very interesting, though not conclusive for the specific question, the book Algebraic homogeneous spaces and invariant theory by Frank Grosshans, Lect Note Math 1673. $\endgroup$ – Nicola Ciccoli Jun 23 '12 at 9:24
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    $\begingroup$ May I suggest to change the question title for something more explicit? $\endgroup$ – Julien Puydt Jun 24 '12 at 13:12
  • $\begingroup$ @Mike: As my added comments indicate, your question would be better phrased by substituting "arises in" for "arises as". $\endgroup$ – Jim Humphreys Jun 24 '12 at 15:06

The most likely answer to 1) is that no "proper term" exists. In any case, it would be helpful to clarify the framework of the question a little more: are you dealing with linear algebraic groups just in charactristic 0 or more generally? are the (presumably rational) representations in question assumed to be finite dimensional? are there nontrivial examples you have in mind of such behavior?

In characteristic 0, for example, any finite dimensional rational representation of a reductive subgroup of maximal rank in a semisimple group (such as a Levi subgroup in some parabolic) will be a direct sum of irreducible representations with a highest weight along with possibly 1-dimensional representations. These irreducibles rarely extend to the full group; this is obvious for the latter types, since a connected semisimple group has no nontrivial characters. On the other hand, it's usually impossible to extend an arbitrary representation of a solvable or non-connected subgroup, etc. In prime characteristic, such things become far more complicated as representations proliferate.

Concerning 2), you might get some perspective by looking at the work of Cline-Parshall-Scott which often has cohomological motivations. For example, in a two-part paper published in Invent. Math. and J. London Math. Soc. (1978-79) they looked explicitly at what conditions are required for a representation of a Borel subgroup to extend to a representation of an ambient semisimple group.

ADDED: The original wording of the question threw me off, and I didn't stop long enough to figure out what you were really asking about. The word you want is observable: not especially illuminating, but hard to find a short substitute for. This comes out of late 1950s work by Hochschild and Mostow on extending representations of subgroups of Lie groups. I think they introduced the term "observable subgroup" in the algebraic groups context in their 1963 paper (online via JSTOR):

Białynicki-Birula, A.; Hochschild, G.; Mostow, G.D., Extensions of representations of algebraic linear groups. Amer. J. Math. 85 1963 131–144

One is given a linear algebraic group $G$ (over an algebraically closed field, sometimes of characteristic 0 in the literature), together with a closed subgroup $H$. In the context of finite dimensional rational representations, one says $H$ is "observable in $G$" if each such representation of $H$ occurs as a subrepresentation of a representation of $G$ (meaning as an $H$-submodule). So in this sense representations of $H$ "extend" to representations of $G$. The difficulty is to identify which closed subgroups of $G$ are in fact observable. One equivalent condition is that $G/H$ should be open in an affine variety.

The question merges into invariant theory in the work of Frank Grosshans, for example in his paper: Observable groups and Hilbert’s fourteenth problem, Amer. J. Math. 95 (1973), 229–253. Cline-Parshall-Scott extended Richardson's work on affine quotients $G/H$ with their notion of "strongly observable". Most of the literature in these directions is scattered and probably not definitive.

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  • $\begingroup$ Thank you for the reply Dr. Humphreys. I could have sworn that I saw the definition somewhere on Wikipedia at some point, but I couldn’t find it your book, Borel’s book, Jantzen’s book, etc., so I’m obviously mistaken. Let me give the question a great deal more context. The overgroups I’m concerned with are the unipotent upper triangular groups of any size, call them $U_n$. The subgroup I’m concerned with is an arbitrary unipotent group, call it $H$. A general fact is that $H$ can be embedded as a closed subgroup of $U_n$ for some $n$. $\endgroup$ – Mike Crumley Jun 24 '12 at 0:16
  • $\begingroup$ I have a result in hand concerning the characteristic $p>0$ representation theory of $U_n$, $n$ arbitrary, which essentially says that, so long as $p >> \text{ dimension of a representation }$, representations of $U_n$ in characteristic $p$ are functorially identical to representations of $U_n^\infty$ (countable direct product) in characteristic zero. In short, Lie theory works just fine for $U_n$ in characteristic $p>0$ just as it does for $U_n^\infty$ in characteristic zero, so long as $p$ is sufficiently large when compared to the dimension of a representation. $\endgroup$ – Mike Crumley Jun 24 '12 at 0:16
  • $\begingroup$ If I can pass representations of a unipotent group $H$ to its overgroup $U_n$ for some $n$, then the proof goes through for $H$ just as it does for $U_n$. Of course, I have no reason to believe this is true, and no, I don’t have any non-trivial examples in mind for such behavior; it’s a shot in the dark, and would just be a wonderful shortcut for proving my theorem for arbitrary unipotent groups, which was my primary motivation for proving the theorem for the groups $U_n$. $\endgroup$ – Mike Crumley Jun 24 '12 at 0:16
  • $\begingroup$ It’s no accident that you mention Cline-Parshall-Scott; this was my advisor’s original motivation for this stuff, but I’ve since seemed to have gone off on a tangent. Any advice? $\endgroup$ – Mike Crumley Jun 24 '12 at 0:17
  • $\begingroup$ Ok, so my comment on Grosshans'book was not too far away from the target, since the whole book is about "observable subgroups". A lot of equivalent formulations of observability, there, so you may possibly find one that applies to your situation. $\endgroup$ – Nicola Ciccoli Jun 25 '12 at 6:24

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