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I'm interested in learning more about the representation theory of the group schemes $SL_2$, $GL_2$ and $\mathbb{G}_m$ (as group schemes over the rationals, say) - specifically, the structure of all irreducible representations.

There is a lot of literature out there about the irrepucible representations of specific points of these groups e.g. $SL_2(\mathbb{R})$, $GL_2 (\mathbb{C})$ using the the theory of Lie groups/differential geometry etc. I would like to know:

1) What are some especially good references (books/articles/notes) about the representation theory of these algebraic groups and the structure of their irreducible reps?

2) In particular, are there any sources which deal with the representation theory of these groups from a more 'algebraic' (i.e. as algebraic groups/group schemes as opposed to Lie groups) viewpoint? Is it even necessary to make such a distinction between the reps of the group scheme and the representations of its group values on certain $\mathbb{Q}$-algebras?

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    $\begingroup$ I mean, you really just want to know the theory of dominant weights for reductive algebraic groups. You should see Milne's notes. There are a veritable cornucopia of resources to study representations of algebraic groups. As to your second question the answer is--it depends. If you allow for all $\mathbb{Q}$-algebras then the answer is obviously that the two notions coincidide. In fact, thinking about the value of the representation on the coordinate ring gives you the comodule which determines the representation. For something like just $\mathbb{Q}$ the answer is no. Although it's worth $\endgroup$ – Alex Youcis Nov 17 '16 at 13:29
  • $\begingroup$ mentioning that over something like $\mathbb{C}$, holomorphic representations of the analytifications (i.e. $\mathbb{C}$-points) coincide with algebraic representations (this is non-obvious). $\endgroup$ – Alex Youcis Nov 17 '16 at 13:30
  • $\begingroup$ PS, there are simple answers if you want to know just the groups you mentioned. E.g. for $\mathbf{G}_m$, it's just characters, for $\text{SL}_2$ every irreducible is isomorphic to $\text{SL}_2$ acting on some degree $m$ homogenous polynomials, etc. $\endgroup$ – Alex Youcis Nov 17 '16 at 13:32
  • $\begingroup$ A more advanced reference is Jantzen's book, though that might be overkill. $\endgroup$ – Tobias Kildetoft Nov 17 '16 at 13:47
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$\newcommand \G{\mathbf{G}}$ $\newcommand \GL{\mathrm{GL}}$ $\newcommand \SL{\mathrm{SL}}$ $\newcommand \Z{\mathbb{Z}}$ $\newcommand \Q{\mathbb{Q}}$ Just to get this out of the unanswered queue, let me give the following answer:

1) Yes! There are actually tons. Any book on 'linear algebraic groups' will cover what you want (although Waterhouse's book is strange it sidesteps a lot of the theory). Specifically though, I would recommend these notes of Milne--I think they are about as good as one could possibly hope for in terms of completeness. They have the downside of being somewhat cagey about using modern algebraic geometry, but only in language. Namely, Milne does talk about nilpotents (which are pivotal in the characteristic $p$ theory since such simple maps like $\text{SL}_p\to\text{PGL}_p$ have non-reduced kernel) but he insists of talking about $\text{MaxSpec}(A)$ instead of $\text{Spec}(A)$ for some unbeknownst reason--it doesn't make a difference, but it's worth noting.

I am also always eager to rep Brian Conrad's notes (for most things). Specifically, there are these from a first course on linear algebraic groups. They have the strong plusses of being written from a 'modern algebro-geometric viewpoint' (e.g. quotients are defined as quotient fppf sheaves, which is what they should be defined as), but falls prey to the neuroticism that befalls all live-TeXed notes--they're fairly all over the place. Unfortunately, that doesn't really cover what you're asking for because it doesn't cover the structure theory of reductive groups (and their representations). For that you'll have to look at notes from his follow-up course. Again, these are probably my favorite notes for the topic, but are somewhat hard to quickly look things up in (something Milne's notes excel in).

Finally, if you want a 'quick fix' you can look at the first (~50 page) section of these notes of Brian Conrad--I think the same material is roughly contained in the appendix to his book (with Prasad and Gabber) Pseudo-reductive Groups.

Let me just note that if you're really only interested in the groups you've mentioned (over $\mathbb{Q}$), then everything is easily describable.

For $\mathbf{G}_m$ (everything in here will be over $\mathbb{Q}$) representations $\mathbf{G}_m\to\GL(V)$ correspond to $\Z$-gradings $\displaystyle V=\bigoplus_{i\in\Z}V_i$ where $\G_m$ acts on $V_i$ by the character $z\mapsto z^i$. In particular, every representation is semisimple and the simple representations are just characters (which are precisely the maps $z\mapsto z^i$).

For $\SL_2$ things are ever so slightly more complicated. Namely, $\SL_2$ is reductive (in fact, semisimple) and thus every algebraic representation of $\SL_2$ is semisimple. Thus, we really only need to describe the simple representations of $\SL_2$. To do this, note that $\SL_2$ naturally acts on homogenous polynomials of degree $m$ in the variables $x,y$ (by $(a_{ij})f(x,y):=f(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$, call this representation $V_m$. Then, every simple representation of $\SL_2$ is isomorphic to precisely one of these $V_m$.

(Note that it's not a coincidence that $V_m\cong \mathcal{O}(m)(\mathbb{P}^1)$ since $\mathbb{P}^1$ is the flag variety $\SL_2/B_2$ associated to $\SL_2$ and the Borel-Weil theorem describes a precise relation between representations of $\SL_2$ and (certain) line bundles on the flag variety--this works more genenerally for a semisimple group).

Finally, $\GL_2$ is also reductive (but not semisimple), thus to describe its representation theory we need only describe its simple representations. These come in two flavors. Namely, there is the irreducible tautological representation of $\GL_n$ (acting on $k^n$ in the usual way) and there are the representations $\wedge^i(k^n)$ for $i\leqslant n$. That's all of them.

Of course, to see where these came from--the real answer is the theory of dominant weights. That said, it's always easier to think about the analogy you know from the representation theory of a finite group. Namely, if $G$ is a finite group and $\text{Reg}(G)$ denotes the left regular representation of $G$, in other words $G$ acting on the group algebra $\mathbb{C}[G]$, then there is a decomposition

$$\text{Reg}(G)\cong\bigoplus_{\rho\in\text{Irr}(G)}\rho^{\dim\rho}$$

where $\text{Irr}(G)$ is the set of isomorphism classes of irreducible algebraic representations of $G$. A similar thing happens for a reductive group $G$. For example, for $\GL_n$ one has that the coordinate ring of $\GL_n$ (as a variety) decomposes as a $\GL_n\times\GL_n$-rep (via the multiplication map) as a direct sum of $V\boxtimes V^\ast$ where $V$ runs over the irreducible representations of $\GL_n$. So, see if you can use this to sort out what happens at least for $\G_m$.

2) As I mentioned in the comment, this really depends on what $\Q$-algebras you pick. For example, you know from the Yoneda philosophy that any representation $\rho:G\to\GL_n$ of an algebraic group $G$ is determined by the group maps $G(R)\to\GL_n(R)$ as $R$ varies over all $\Q$-algebras. In, fact it suffices to think about it for $R=\mathcal{O}_G(G)$ which gives you the notion of a comodule.

Of course, things are going to be bad if you want to consider just the values of the representation on some random $R$, even for examples $R=\Q$ I think. I don't have an example off-hand to be honest (there's probably not a hard one) but if you look anything other than $\Q$, say $\Q(i)$, you'll get lots of representations of $\GL_2(\Q(i))$ (say) that won't be algebraic--think about the one that is induced from the non-trivial automorphism of $\Q(i)$.

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  • $\begingroup$ A note of caution: The jump from the group being semisimple to the reps being so is nontrivial (and fails in positive characteristic). Positive characteristic is also a good place to go for examples like at the end of the answer even if this starts to deviate quite a bit from the original problem (Frobenius kernels are very nice ). $\endgroup$ – Tobias Kildetoft Nov 20 '16 at 13:56
  • $\begingroup$ @TobiasKildetoft Sure--linearly reductive is not the same thing as reductive in positive characteristic. That's why I was explicit about working over $\mathbb{Q}$--I didn't want to complicate things. Thanks for pointing it out though! $\endgroup$ – Alex Youcis Nov 20 '16 at 14:04
  • $\begingroup$ Dear Alex - sorry for taking a while to get back to this, but once again thank you for a very comprehensive answer! Your references are extremely helpful, and the comments you've made about my three cases are great! (These cases are currently all I'm concerned with, although in the future I will need to know more about the whole theory) One question: in the group scheme case is it still true that irreducible representations of commutative groups are one-dimensional? (e.g. so I can at least settle your initial comments about the $\mathbb{G}_m$ case in my mind) $\endgroup$ – Alex Saad Nov 22 '16 at 10:28
  • $\begingroup$ @AlexSaad Yes, this is true. Schur's lemma still works--let's work in char 0 for simplicity. Suppose that $\rho:G\to\GL(V)$ is a representation with $V$ simple (note that this is not the same thing as irreducible if $G$ is not reductive!). To show that $\dim V=1$ it suffices to base change to $\overline{k}$, so we can assume that $k$ is algebraically closed. Let $\alpha\in GL(V)(k)$ be such that $\rho(g)\alpha=\alpha\rho(g)$ for all $g\in G(k)$. Note then that if $a_0$ is an eigenvalue of $\alpha$ that $\alpha-a_0I$ commutes with all $\rho(g)$ for $g\in G(k)$. This says that $\endgroup$ – Alex Youcis Nov 22 '16 at 10:53
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    $\begingroup$ again by smoothness $\rho(G)\subseteq Z(\GL(V))$. This evidently implies by simplicity that $\dim V=1$. So, the summary of the argument is: it's normal Schur's lemma and noting that having $\rho$ factor through $\GL(W)$ for $W\subseteq V$ is equivalent to having the $k$-points have such a factorization if $k$ is algebraically closed and $G$ is smooth (which is automatic in char $0$). $\endgroup$ – Alex Youcis Nov 22 '16 at 10:56

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