Selberg trace formula

this video will be about the selberg trace formula so let's start by looking at an example of it so the cburg trace formula is given by this expression here um if you look at it more closely you see that it really looks like a rather complicated and gruesome mess um so what I want to do is explain um the underlying idea behind this which is really quite simple and how you um how you would get such a complicated expression out of this so it's used for studying um quotient spaces such as you might take SL2 of R which is 2x two matrices over the reals and quotient out by SL2 of Z and you want to understand this space here um well let's start by looking at a simpler example so I'm just going to take G to be a finite group and acting on finite set s let's suppose the action is transitive so s is equal to g modulo h where H is a subgroup of G and the problem is the following um what we want to do is um understand the action of G on the space of all complex functions on S so this is a finite dimensional Vector space with a basis of points of S and G acts on S so it acts on this and we want to know um what is this representation of G in particular we want to know its character where the character of G is just the trace of G on um this space c of s and this is actually rather easy to work out um suppose G is an element of of the group G where it acts on S and it acts with various Orbits for instance might have an orbit of size one and it might have an orbit of size three here and if we write out the matrices for these orbits you see the Matrix here is one The Matrix here looks like not one one not n n not one n and what you notice is that this is Trace equal to one and this is Trace equal to zero and what you see is that whenever there's an orbit of size one we get a contribution of one to the trace whenever there's an orbit of size greater than one we get a contribution of zero so this is just equal to the number of fixed points um of G on S which is just the cardinality of um the fixed points are sometimes indicated by s with a a g up there um and you can rewrite this in various ways for instance you can write the number of fixed points as um 1/ H time um the number of elements a in G with um a g a minus one in H or you can rewrited as a sum of over conjugacy classes of H because um for each element of H you can look at the number of points a such that this is that particular element of H so um um anyway this formula here for Kai of s is essentially the cburg trace formula for a finite Group G um and there are various generalizations of this um so the action of um G on um C of s um well this is an example of an induced representation of um of G so um it's induced from the trivial representation um one of H where H just acts on a one-dimensional vector space um and in general if h acts on V then we can obtain a a representation of G in several ways for instance one is to notice that g acts on um we can take the group ring of G and tensor over group ring of H with with um so with v and this will be an induced representation um we can interpret this as um sections of vector bundle over um the the space g over H so the special case where V is just a one-dimensional vector space c this Vector bundle would just be the trivial Vector bundle and sections are just functions on G over H and there's a formula for um the character of this it's given by fenus um and it's very easy what you do is you just take the character of um the space V so this is a function on H um and you extend it to be zero for elements that aren't in h and you just take the sum over all elements G and G modulo H of all conjugates of this um so um roughly speaking this function here is not invariant under conjugation by G and if it's a character of something you want it to be invariant under G so you just make it invariant in the most obvious possible way just by taking the sum of all conjugates under G and this is fenus as formula for um the character of an induced representation it's a generalization of the formula I had on the previous page where you just take the number of fixed points um and um now the BG Trace formula all we do is we take G to be SL2 of the reals and we take H to be our favorite subgroup for instance H might be SL2 of Z would be a typical example or we might take H might be the fundamental group of a rean surface compact rean surface and if you've got a compact rean surface of genus greater than one then its fundamental group is naturally a subgroup of SL2 of R and you you you you you can take you you you can look at um this case so what we're really doing is we're looking at all functions on g modulo h and trying to understand this as a representation of the group G and all the all the sellberg trace formula consists of is um working out the character of this in much the same way as we did before um however there are certain complications um that there are three cases if G over H is finite this is essentially trivial it's the case we've done before where you just count number of fixed points if G over H is compact then this is sort of easy um well it's not easy but it's easy by comparison to the noncompact case which is um sort of the hard case of the cellb trace formula um so what I'm going to do in the rest of the lecture is explain what the various complications you get are when G is infinite and how you deal with them so the first problem is G is not a discrete group in general um well this is a problem because for the fenus formula we quite often have to sum over all elements of G and you know if G is infinite um then the sum is generally in infinite especially if G happens to be a topological space well you do the you fix this in the obvious way you instead of taking a sum over G you have to integrate over G and if G is a locally compact group there's a sort of reasonably well defined left invariant integral so that's okay however we get several problems so first of all um the integral might be left invariant so the integral over F of G is the integral over F of a of G but not right in variant so if you're summing over um sorry SP V if you're summing over a group then um summing is both left invariant and right invariant one of the extra complications you get in the nondiscrete cases that the integral might be left invariant but not rice invariant um this doesn't actually happen for SL2 of over the reals but it does happen for um groups like the subgroup um of all matrices like this inside SL2 of R here left integration over this group is not the same as right integration which is an extra complication you have to deal with um the second thing is that the integral has to be normalized um in that there are various ways of choosing the integral so if you summing it's obvious how to normalize you just say each elements of G has weight one but if you're integrating you know you can multiply the intergal by constant it's not quite clear what to do um for SL2 R this isn't too difficult um in more General cases like higher rank groups over the adels um normalizing the integral is actually quite complicated there's um there's a standard way of doing this um due to tamagawa which is called tamagawa measure whose definition isn't actually too difficult the TR is it's quite hard even working out basic properties of the tamagawa measure like you might want to know what is the measure of G overh and actually working at the tamagawa measure of G over H is a rather hard problem known as the V conjecture which just took several decades to solve um also if G isn't discreet um you can't really use matrices so um for discrete groups you can represent a linear transformation as a matrix consisting of elements m j and then you can take things like b i is sum over m j of AJ where where you're summing over all elements J inside a group or inside some finite set um if you are working with nondiscrete groups you need to use a kernel um which is really just a matrix except X and Y are not elements of a discrete set they're elements of a topological set space and then you replace this expression by um integrating kxy times um f of Y Dy and this gives you a function G of X so um kernels behave just like matrices in fact kernels and matrices are really the same thing that they have different names and different notation for historical reasons because they were invented before leay integration um and and when you do a leag integration you realize that these are both just integrating over a set which is discreete in this case and continuous in this case um so um so the next problem is um we need to know what are the conjugacy classes of G and H and if G and H are finite they're a finite number of conjugacy classes um the problem for things like SL2 of R is there are infinitely many conjugacy classes so you need to divide these conjugacy classes into families um for example there's one obvious one which is this Matrix here then you can have the Matrix like that but then you could have a conjugacy class consisting of matrices that look like this for this element non zero um well these form a single conjugacy class but then you can have um conjugacy classes represented by matrices like this and here we have a full family of conjugacy classes because um this element is not conjugate to another element like that unless aals b or B to the minus1 so we have a family of an infinite number of conjugacy class and then there are still more conjugacy classes for instance we could have a conjugacy class that looks like this um where C is cosine of some angle Theta and S equal s of theta um and we also want to know what these conjug what the conjugacy classes of the subgroup H are and when H is equal to SL2 of Z it gets even more complicated although that's not too bad we can handle SL2 of Z but the problem is um H might be some sort of congruent subgroup like it might be the matrices a b c d the C congruent to zero mod n so this subgroup turns up quite a lot and the conjugacy classes of this group here are really rather painful to describe or think about in fact even describing a set of generators for this group is not very easy um so in order to deal with that um well the problem is if we look at SL2 over a field we can sort of handle the conjugacy classes if we look at cell 2 over a ring that isn't a field it's rather difficult to see what the conjugacy classes are so one way of dealing with that is moving to adels so instead of looking at SL2 R modulo SL2 Z what you do is you look at SL2 of the adels modul SL2 of the rationals and it turns out these two quotient spaces are sort of Fairly closely related if you understand this one you understand this one and now um the great thing about this is this number here is a field so the conjugacy classes can be described they're a bit of a mess but it's easier than doing it for for z um the problem is you've replaced here or we've replaced R by the Adel so the conjug classes of of this group here and now more complicated this is somehow the price you have to pay for changing Z to the the rational numbers um what are the adels well the adels are more or less the product of R times um um sort of restricted product of all pic numbers so now instead of working at the conjug classes of SL2 of R we also have to work at the conly classes of SL2 of the pic numbers for all primes p um so we have rather large numbers of different sorts of conjugacy classes and this is one of the main reasons why the cellb trace formul is rather complicated that we we have a terminate for each sort of conjugacy class and if you look at the conjugated mess I had here then roughly speaking each line corresponds to a different sort of conjugacy class in in SL2 of the reals or SL2 of your discrete subgroup um so the next complication is the character of a representation of G is a distribution not a function on G um so what's going on here well we what we want to know is what is the trace of G on B for G and element of our group G and the answer is very easy this will usually just be infinite or undefined or something it's very difficult to make make sense of it if G is an infinite group and V is an infinite dimensional Vector space so what we do instead is we look at um action of some sort of group um algebra on um on V so we might take um f to be a um a smooth function of compact support on G and if we lucky then F will act on V what you do is you sort of integrate um um the action of G over G um according to F so so what we can do if if V is a well- behaved representation is is find an action of smooth compactly supported functions on G and this might have a trace so what we do is we get a map from smooth compactly supported functions on G to um real numbers which are just the trace of f on this representation and linear map from smooth compactly supported functions to real numbers is distribution um and sometimes if you're lucky this distribution is actually really a function there's a famous rather deep theorem of Harish Chandra which says that um irreducible representations of semi- simple Le groups are very often um represented by distributions that are really just locally integrable functions but sometimes it's not a locally integral function and it really is a distribution so as long as these functions are of Trace class you're okay and if G over H is compact um these functions often are compact operators and are of Trace class so um we're sort of okay we get well defined distributions um the real problem in the cell Trace formula comes um when when um G over H is not Compact and then the problem is that the action of f on the space V is also not a compact operator so it usually doesn't have a trace or Trace is infinite or something it's very difficult to make sense of it so how do we deal with this um well um let's first of all take a look at a couple of examples suppose you take the group ring of R over z um now this can be if if we take all nice functions on R over Z say we take you know smooth functions or L2 functions or something I don't really care under reasonable conditions we can write this as a direct sum of um um periodic um so that should be integer n for n in Z so we we can write a reasonable function as a linear combination or an infinite linear combination of periodic functions and this is just a Furious series expansion if we try and do this for functions over the reals we can sort of try writing them as a as a furier integral so we we get a sort of integral over e 2 pi ixy * something and this is the fuer um this is essentially the fua transform and the trouble is this space is not a direct sum of the subspaces generated by this it's really a sort of integral of them um and when you decompose SL2 R modulo SL2 Z suppose we take L2 of this then this is a sum of a discrete part plus sort of integral of a continuous part so the continuous part is sort of like a Furia integral and the discrete part is sort of like a Furia series um and both of them actually occur here um now at the discrete part um the functions on G tend to have Trace class so um we can Define uh the character of action on the discrete part so the problem is that we need to figure out what the continuous part is and kind of remove it um now the continuous part um is given by things called eisenstein series um well if you've done modular forms you've come across an eisenstein series would look something like this you Su over all c d a non zero over C to plus D um to the k um now the eisenstein series that occur um um here are a slight variation of these so these are holomorphic functions what we really want a sum Over CD not equal to Zer 1 / C to + d 2 s and you notice these are now real analytic but not holomorphic and these converge for the real part of s sufficiently large and there are two problems here first of all we need the eisenstein series um in a region where they where where they don't actually convert so we need to analytically continue them now this is the eisenstein series for for SL2 Z and it's not very difficult to analytically continue that one analytically continuing them for more General groups as much Harden was done by selberg uh for SL2 uh for higher rank groups it gets to be a bit of a nightmare and the analytic continuation was done by langlands in this in this notoriously difficult manuscript so the first step is um to analytically continue the eisenstein series you can then use them to find the continuous part of um the uh decomposition um and you sort of use them to subtract off the continuous part and you're left with a discrete part where operators have a trace and the selberg formula now comes by finding the trace of continuous of smooth comply support operators on this discrete part so to summarize um what the cellbone trace formula consists of you take the fenus formula for an induced representation and you make the following modifications for it first of all you change all sums to integrals because your group is not discreet um secondly we have to work with distributions rather than functions because the character of representation is um usually a distribution third we use we tend to use adels um to handle the conjugacy classes actually if you're just doing sr2 you can usually get away without using adel's but for higher rank groups um using adel's is essential otherwise trying to deal with the congy classes is just hopelessly complicated and finally we have to use eisenstein series to get rid of the discrete part to get rid of the continuous part um so that's all you have to do and the good news is that this is all been written out in detail by people by several people for instance hedg Hall has a book giving the um detailed description almost all possible cases the bad news is that selberg that hedg hell's book looks like this um it's two volumes and together they cover about 1,300 Pages um giving you the selberg formula in its full um generality where you um allow non-trivial representation of H and allow um H to be non-compact and so on