Representations of GL2

so this lecture will be a rough survey of some of the representations of group GL2 um so by representations I'm going to mean complex representations and GL2 you recall is just the set of 2x two matrices of determinant um non zero so they're invertible and of course it has to be GL2 for some field F and you want to know what f is well there are some popular choices for f f might be a finite field um so it might be fq for Q a power of some prime or it might be a p addic field so it might be something like the um P addic integers or it might just be the real numbers or it might just be the complex numbers so these are the most popular choices you can also look at other things like it might be a um a power Series ring over a finite field or something um well first of all GL2 is almost a product of the group SL2 time gl1 um so there's a map from SL2 * gl1 so you can just think of gl1 as just being the diagonal matrices and SL2 just means determinant um equals 1 and this obviously maps to GL2 and it's not quite onto in general it's not quite injective in general but you sort of get an exact sequence where this is something small and this is something small so representations of GL2 are almost but not quite the same as representations of SL2 times representations of gl1 so when people first studied representation Theory they started with representations of SL2 but people soon notice that representations of GL2 are actually easier and better behaved than SL2 so normally what you do is you do the representations of GL2 and then just sort of restrict to get SL2 um the the reason why GL2 is better than SL2 seems to be it's um to do with the fact that if you look at the algebraic group GL2 the centralizers of elements are connected and for slightly complicated reasons if you have an algebraic group with this property its representation Theory tends to be a little bit easier than in general there's SL2 that the centralizers not necessarily connected um so uh so the the key theme about representations of GL2 is that representations of GL2 form families um corresponding roughly to representations of cartan subgroups so this is not an exact correspondence it's just sort of a rule of thumb um this is actually a special case of um much more General collection of correspondences called langland's functoriality which tells you very general conditions under which representations of one group give you representations of another group um so I better explain what is a cartan subgroup well um the simplest example of a cartan subgroup is just the diagonal matrices is a00 of go2 and a general C and subgroup is roughly speaking a subgroup that sort of behaves a bit like this um so in particular we notice this subgroup is a bilon and its elements are semi simple and it's sort of maximal and it's connected and so on um so you can define a c and sub grou group to be something with some collection of these properties the precise definition of cart and subgroup actually varies slightly depending on who's defining it so you have to be a little bit careful um but for GL2 it doesn't really matter um so semisimple means you want to exclude a bilon groups like um the group of these elements here so so this does not count as a carton subgroup this is a unipotent subgroup which behaves quite differently from the ones with semi- simple elements so um so an obvious cart subgroup is is the group of diagonal matrices there a slightly less obvious one so let's look at the group GL2 of the reals and we notice the reals is contained in a subset of the complex numbers and the complex numbers is isomorphic to R2 as a real Vector space so so the complex numbers act on R2 by multiplication if if we identify R2 with the complex numbers and this gives us a map from the nonzero complex numbers to GL2 of R it takes um a + b i to a B minus B A and this is another Coten subgroup um you notice billion and semi simple essentially because these numbers here are of course they want a b would not be z z otherwise this wouldn't be invertible um and we can do the same trick for um many other fields so if I take a field K and embed this in a bigger field Big K with um which is um two dimensional over little K then get a map from K star to GL2 of little k um just by identifying K with K squar and letting Big K act on K squ by identifying it with K so if Big K is equal to K of root T which we can assume if the characteristic is not two then what we get is a carton subgroup which looks like elements a b TB a with determinant a^ 2 minus t b^ 2 is not zero here T is some fixed element in in little k um there's a sort of special case if if we take T um equals 1 then we have big K is K ofun one um and this isn't really a field it's it's really a sum of two copies of K and this gives us as our cart and subgroup just um um matrices in the form a b ba a which is actually conjugate to the set of diagonal matrices so um the the obvious cartan subgroup of diagonal matricies corresponds to a sort of degenerate extension of a field K where where where the extension actually splits as a sum of two fields and all the others correspond to n degenerate extensions where you where you take a field um and I said the um representations of cartan subgroups should correspond to representations of the general linear groups so so so um if we take cartan subgroup plus a represent ation this should give you a representation of GL2 except it's a bit more complicated than that because you know this representation might not be irreducible and so on um in the special case when the cartan subgroup is um K Star Plus K star so it's just the diagonal matrices um this correspondence is very easy to describe you just take the induced representation of um this group here so we first um if we've got a representation of this group we can treat it as a representation of this group just by letting these this element act Tri and then we induce from this group called a bra subgroup o to GL2 so the dial C and subgroups this correspondence is fairly easy to describe it's just induced representations these are called principle series representations and they're mostly irreducible but sometimes they're not um the others are rather tricky and the names of them is is a little bit hazy they're sometimes called discrete series um they're sometimes called lots of other things as well um um the the name Reet series comes um because people originally did this Theory over the reals and the discrete series Ur discreetly in L2 of GL2 of the reals or more precisely I should say SL2 of the reals because they don't occur discreetly in GL2 of the reals um now over other things like finite Fields discrete series are not the only represent presentations that occur discreetly um but the so over other fields discret series tends to mean something which is vaguely analogous to discrete series over the reals I mean the the naming of all these representations isn't really completely rational and or systematic um so um now let's take a look at um GL2 of let's just take a finite field of order p so um this is order it's easy to work out it's just Q ^2 -1 * Q ^2 minus Q just count the number of ways of taking a basis of a two-dimensional Vector space and we want to find the cot and subgroups and as we saw what this means is we want to find the two-dimensional extensions of FP that's a field or sum of fields there are two ways of doing this we can take FP plus FP or we can take fp^ s so there's there's a unique degree to extension of FP and this is giving us the diagonal cartan subgroups and this is giving us cartan subgroups that look a little bit more complicated they're sort of a bit like the complex numbers only only more so um um incidentally you can also describe the cartan subgroups of gln of FP in a rather similar way what you do is you just write n is equal to N1 + N2 plus N3 and so on and then um FP has a an extension that looks like FP the N1 * FP to the N2 and so on and um so you get a cartan subgroup which looks like nonzero elements of this field times nonzero elements of this field and so on um you can also see the number of cart and subgroups is going to be something to do with the number of ways of writing n as a sum of smaller integers which is the number of partitions of n anyway if n equals 2 we can write 2 is equal to 2 or 2 = 1 + 1 so that just two partitions and these correspond to our two cart and subgroups um so let's see what we get um so um these are going to give us the principal series and we can see there are Q -1 squared characters of degree one of this which gives us Q -1 2ar representations of GL2 of fq if you induce them up and these all have um Dimension Q + 1 and some of them are reducible so um um Q -1 reduce as 1+ a q dimensional representation and this is a very interesting one called the Steinberg representation and the others um um well there are Q -1 2us Q -1 other onedimensional representations these actually pair off and what we do is we get Q minus1 2 - Q - 1 / 2 representations of Dimension 1 + Q which are irreducible reason we um these representations form pairs is that um Matrix a b00 0 is actually conjugate to B a0 0 um and this sort of symmetry of the C and subgroup means um that the representations sort of pair off and um we only get this number of um irreducible representations um well there are some other representations you get from fp^ s so the other coton subgroup fp^ s um um what this gives us um so this is going to have q^2 minus1 characters so we get um Q ^2 -1 representations of GL2 of fq q and of these Q -1 turn out to be something like Steinberg minus one this minus one is a little bit funny and maybe I need to explain this a little bit and then we get Q ^2 -1 - Q -1/ 2 representations of Dimension Q minus one and the these ones are sometimes called the discrete series so the question is how do you construct the discrete series representations and this is actually a little bit tricky um there are several ways of doing it um none of them are totally trivial um so I'll just mention a couple of popular ways of doing it um one is due to drinfeld and um um what he found is you can get these representations inside the atal chology of a suitable um variety um and um this turns out to work really well for groups other than GL2 in fact delen and lustig sort of took drinfeld's idea and really ran with it and managed to construct similar representations for all other groups and luster use this to work out all irreducible representations of all the simple group finite groups of of Le type um and using italco omology um you you actually get the these representations appearing in a sort of Oiler characteristic which would be a sort of first chology class minus a zero chology class and what you do is you you find one of them is the Steinberg representation and one of the other homology groups gives you the trivial representation and you subtract them to get the all characteristic which is where this minus one comes from um another popular way of writing these is using a v representation um where what you do is you really take um a v representation of a bigger group like it might be the simplec group and um what you find is that uh this contains SL2 times um um fp^ 2 Star and you can take the v v representation of this group and decomposed as representations of SL2 times representations of this and this gives you a correspondence between representations of this group and representations of this group if you do that you find the thing corresponding to the trivial representation here is actually is dimension Q rather than Q minus one as and and as the whole Steinberg representation and so anyway there there are several ways of constructing the discrete series representations um so um let's give a few examples um let's start by looking at GL2 of F2 um so this is order six it's actually isomorphic to the symmetric group on three points and you've probably all seen the character table of this group it looks like 1 1 1 1 - 1 1 2 0 minus one where this here is the dimension of the representation and these are the values on conjugacy classes of GL2 and of these representations this one is the one-dimensional one um this one here is the discrete series and this one here is the Steinberg representation um there's actually no principal series representation if if you look the number of principal series representations is actually zero for F2 so this example is a bit misleading and the the most common sort of representation doesn't actually occur um if you look at GL2 of F3 um then its representations of Dimension one one two two three three and four um of these this is the principal series um here we get Steinberg representation you've heard there's only one Steinberg representation that's for simple groups GL2 isn't quite simple so you can you can get more than one these ones are the discrete series and these are the onedimensional ones um if you look at GL2 of f4 then this is actually well SL2 of f4 is actually isomorphic to A5 of order 60 um but GL2 means you sort also multiply by z over3 z um so what for these we take the representations of A5 which have Dimension 1 3 3 four and five and we multiply all of them by three so we get three copies of one three copies of three another three copies of three three copies of four and three copies of five so these are the dimensions of the representations of GL2 of f4 and as before these are the Steinberg um and these are the principle series I guess and these are the discrete series and so you can go on like this um let me show you a more complicated example um let's do GL2 of um uh 17 and here is uh um here it is in the atlas of finite groups I don't know if you can see it very well um this is the character table in sort of compressed format and if you wrote out the whole of the character table GL2 of 17 it would cover pages and pages and pages so so um they had to compress it a bit and if you look here there's a representation of Dimension 17 that's the Steinberg representation here's the onedimensional representation um here are some 18 dimensional representations which are the principal series and here are some 16 dimensional ones which are the discrete series and here are some nine-dimensional ones well where did those come from um well I never said anything about nine-dimensional representations well what they come from is um that one of the principal series representations of Dimension 18 actually splits into two nine dimensional representations if you restrict it to SL2 so this is really the character table of SL2 over 17 and um this vertical line here means that if you were looking at GL2 you should um sum these two representations together um if you look down here you can see two eight dimensional representations which are one of the discrete series representations splitting when you restrict it to SL2 um so um next I'll say a little bit about um representations over the uh periodic numbers so let's look at GL2 over the periodic numbers and um as I said first of all what we've got to do is to find the cartan subgroup and to find the cartan subgroups we need to look at the degree 2 extensions of QP um well there's one obvious degree to extension which is QP plus QP so this is the degenerate one that isn't a field it's a sum of two fields and then you can also look at QP of root n for n a non residue or QP of root p p n or QP of root P so there are three non-trivial degree to extension so we we we we expect to get three discrete series and one principal series well that's not always true this is for p odd um for p equal 2 it gets more complic at so for p = 2 we get Q2 + Q2 but Q2 the two addict numbers actually has seven quadratic extensions because we can add < tk3 orun 5 orun 7 or um < tk2 or < tk2 * 3 or < tk2 * 5 or < tk2 * 7 so we get seven discrete series um Sorry Seven series of discrete thisr series representations that's not seven discrete series representation seven series of discret series representations if you see what I mean um so the the principal series uh mostly irreducible um um a few or not some of them have a one-dimensional sub representation or quotient representation so this would be the one-dimensional representation and if they have a one-dimensional sub representation or quotient representation then the then the leftover piece is called a special representation so the remainder is called the soal special represent ation and if you compare this with the um representation over finite Fields you see the special representation corresponds to what people doing finite dimens doing finite Fields would call the Steinberg representation um these really ought to be called the same thing but um it seems that the people who um discussed representation theory of finite fields and the people who did pic Fields weren't talking to each other as much as they should have been so they came up with different names for what is essentially the same thing um um as I said uh there are also discret series representation um these are usually constructed by using the V representation um where you take a sort of metaplectic representation of some simp of simplec group for a four-dimensional vector space um now if p is odd this gives you all the representations we get three discrete series and some principal series and some special representations um if p is equal to two this isn't quite true although some earlier books and papers sort of imply that that's all you get for P's to as well however Andre vay noticed there was something fishy going on um so there's some weird weird stuff for p 2 I mean we've already seen some weird stuff we get seven instead of three discrete series of discrete series um but um things get even weirder than that you get things called tetrahedral and octahedral representations and I'll try and explain where these comes from um first of all um according to Lang's philosophy the rep presentations of um GL2 of of a pic field should have something to do with um um representations of the gwa group of um C that means the absolute galwa group um in um GL2 over the complex numbers um and what Andre vay noticed is that there are extra representations of this in current when when p is equal to two and let me explain why why you get these um so if we look at GL2 of C we can quotient out by the center and look at pgl2 of c and pgl2 of C has a well-known collection of finite groups it's finite subgroups are cyclic or dihedral or they can be the tetrahedral group A4 or the octahedral group S4 or the icosahedral group A5 um on the other hand um the um galma group of absolute Gall group of the rationals looks like a p group times a cylic group times a cylic group and where this is something to do with wild ramification and this is um ordinary ramification and so on so what we want to do is to ask when can we have a map from a group that looks like this which is some sort of gwa group to one of these finite groups um well if the image is cyclic or dihedral this turns out to be all the representations we already know about the principal series and the special representation so these are sort of accounted for A5 it can't of image A5 because A5 is not solvable and this group here is solvable so we can cross this out but A4 and S4 are both might be possibilities so let's look at their structure well A4 and S4 both of structure looks like Z over 2 z^ s do Z over 3 Z and then there may be a z over 2 Z on top of that in the case of S4 but not for A4 so this is this is sort of optional um and then we're also taking a central extension of that by GL2 so there might be a z over 2 Z sitting in there so we want to map P Group by cylic B cylic to this sequence here now we can see how to do this so we can map this cylic group to this C over 3 Z and we can map this citate group to this bit here and we can map this P group to this group here if p is equal to two so if p is odd we can't do this but if p is equal to two then we get all these strange extra representations of the gall group of the padic numbers which can be tetrahedral or octahedral and by langland's um correspondence this ought to give you extra representations of GL2 of QP now actually pinning that down is hard work um in fact if you try and prove the langland's correspondence for GL2 of QP you spend about half your time on these funny representations of p 2 just trying to trying to sort out what's going on in fact that that that was the last case of it that um that that that was the last case that had to be done to prove langland for um over pic Fields um finally I'll just say a little bit about g2r and g2c and again to find their representations you first look at the cartan subgroups and for GL2 are there are two cartan subgroups we take the diag matrices or we can take a B minus ba a so this is really a sort of copy of the complex numbers for GL2 C there's only one sort of c and subgroup optic conjuga C Group which is just diagonal matrices so GL2 C the representation theory is particularly simple all we do is we get the principal series most of which are irreducible um for R we get some principles series and we also get some discrete series related to representations of the nonzero complex numbers I mean it turns out the discrete series aren't really new because if you take the principal series most of these are irreducible but some decompos as either finite do discrete series or as discrete series. finite in other words they've got a they might have a finite dimensional sub representation so the quotient is a discrete series or a finite dimens or a discrete series sub representation such as the quotient is finite dimensional and you can spend many happy hours writing down all representations of these two groups and inducing them and working out exactly how they decompose and um this is the result you get um and um again just as for finite Fields if you're looking at SL2 R um some of these representations tend to decompose so um discrete series representations decompose as a direct sum of two representations and again you can spend many pages of calculation um working out exactly which representations decompose and how they decompose