CONFERENCE
Recording during the thematic meeting : «Automorphic forms, endoscopy and trace formulas » the September 18, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France)
Filmmaker : Guillaume Hennenfent
Find this video and other talks given by worldwide mathematicians on CIRM’s Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: – Chapter markers and keywords to watch the parts of your choice in the video – Videos enriched with abstracts, bibliographies, Mathematics Subject Classification – Multi-criteria search by author, title, tags, mathematical area
Okay so uh thank you very much for the introduction and thanks for uh to the organizers for inviting me it’s a great pleasure to speak here um so in the previous talk Professor Arthur said uh he’s known Professor Laz for 50 years um as the youngest speaker at this
Conference I cannot make that claim um but I have definitely spent uh many hours in graduate school reading Professor lees’s many many long papers and oftentimes wishing they were written in English uh just because I’m so much slower at reading French but um thank you Professor laes for the Decades of
Wonderful math and thank you for being the director of this uh wonderful Research Center that brought many mathematicians together together and happy birthday okay so today I’m going to talk about um uh a series of papers I wrote uh some of them joined with Amar o and some of
Them joined with uh Suzuki um okay I need to figure out which board so you said is this this board but oh this one ah ah okay right yes okay so I was start with um okay this is a good height for me to start with so um since there are many
Young people in the in the audience I think I will start with um a quick intro to uh what I mean by local Lance correspondence uh firstly is this uh font size good people can see you right okay okay so um so local lons conjectures um um they are they give a
Certain relation between two sides so on the one side okay maybe I should raise it a bit higher um on the one side is what people often call the group side uh which are uh which consist of irreducible smooth representations of okay some connected reductive algebraic group so this is a
Connected uh reductive algebraic group over some non archimedian local field so f is a non archimedian local [Applause] field and we consider uh irreducible smooth representations of GF on the other side uh is what people in classical lons often called the galloo side oh maybe these are slightly too far
Apart sorry let me bring okay so this is good okay so gide so these are uh consist of L parameters so what I mean by L parameters is that these are um continuous homomorphisms from okay often times denoted by Fe from the V group of f cross SL2 C to so
Usually you need to put the L group of G here but for Simplicity I’m going to assume my groups are split so I’m just going to put uh G check the complex dual group so this is the group whose root data uh is like the co- root datm of G
Okay so um if uh so there many people in the audience are more familiar with geometric l so I think people have different terminology for the group side in geometric lons this is I think called the geometric side or the automorphic side whereas the gwa side is often times
Called a spectral site okay so um the conjectural local lons cor correspondence predicts a subjective map going this way where if you start with some uh irreducible smooth representation uh you’re supposed to get some sort of L parameter so this is the uh most naive formulation of the
Conjecture and um firstly this map is predicted to always have um finite fibers and I believe uh very early on people actually thought this naive formulation of of course mod isome on this side and mod G check com conjugacy so early on people believed that this naive formulation would just give a bje
On the nose but it turns out that’s not true because the fibers um are not Singletons in general so in fact um they there are not even singl tons for I think uh SL2 and I believe this was mentioned in the cor vales for volume of uh L sha and in
Professor LA’s article there um and of course for uh the fiber is not Singleton so it’s um it’s pretty much only uh Singleton for gln and of course for Taurus okay so the fibers are uh now called L packets so fibers they’re L packets okay I guess I don’t need an
Exclamation okay um so mathematicians often times want to have BS uh because that’s a much more elegant way of formulating anything so um motivated by this I guess pursuit of Aesthetics people uh considered doing an uh enhancement on the gallawa side to make this a bje so what you do is to
Consider enhanced L parameters on the gallowa side and an enhancement is just adding a certain uh irreducible representation row Pi uh which is an ER in uh a certain component group so starting from each L parameter you can attach a certain component Group which uh parameterizes the internal structure
Of the L packet so here uh for Simplicity I can tell you what this group looks like for uh split groups so this uh this component group is just uh okay it’s given by the centralizer of Fe inside G check and you mod out its identity component
Okay um so uh so when you add the enhancement and you consider enhanced parameters then uh the conjecture is that you do get a bje between the group side you consider e g and the gwa side you consider uh L parameters with an enhancement so you put a subscript e
Here to denote the enhancement and this is the conjectural bje okay so many people in in the audience have worked towards um proving the local the classical local onance conjectures and uh let me just ah sorry ah yes uh is that okay okay yeah sorry I I I
Did say that I I yeah okay um and my arrows should really be horizontal to the board but okay okay okay so many people in the audience have worked towards the local LW conjecture let me just give uh like a very uh incomplete brief uh overview of the
History okay so so these ones are manual uh adjust okay so um okay brief history so the local all for uh okay so I guess the first case was uh due to uh so it was for gln and it was first proved in the local function field Case by lamon rapor and
Ster um their proofs use uh similar follow similar strategy as D dils proof for GL2 and uh later uh penart I believe this is in the 1990s I think okay or 80s um where uh henar gave a proof of local anons for GN using uh his earlier proof of the
Numerical local onance if I got the history right um and then later on Harris Taylor um gave a different proof of local onance for GN using ah okay so okay thanks for so they proved it first and then okay and then henard gave a different so Harris Taylor
Proof uses geometry of shimura varieties um and some sort of uh local Global compatibility um to obtain the local L and then uh henard uh gave a different proof uh simplify sort of simplifying the the proof given in Harris Taylor of local on for GN um and then uh after that and then
Schela came along and also gave a different proof but uh slightly closer in spirit to Harris Taylor’s proof of local ons for GN okay um so local LS for sln is also known um so uh there’s the work of hiraga cyle uh for characteristic zero fields and
Abps uh that’s OB so I really should write it or uh bomb pan Sol ofelt uh for positive characteristic okay and um for more General groups there’s also the work of okay there’s the work of um Arthur mlan and possibly many other people for uh quasy split classical groups [Applause] um for uh
Characteristic zero okay so pretty much each group uh has uh some authors sitting in the audience which is great um okay so um uh in my joint work with Amar we’re so so far like all these uh papers are about classical groups so uh Amari and I were the first people to
Do local onance for exceptional groups so we did it for G2 um but today I’m hoping to illustrate that our strategy is actually General and can be applied in many General cases okay so I think that’s the historical overview part so okay what is the general strategy um going to raise this Bo
So how high does it have to go okay maybe so now this one is okay so the general strategy is that um okay over is BAS it’s very simple it’s just um sort of doing local long L correspondence piece by piece so earlier um I said we want to prove a certain
Conjectural projection between the group side and the gall side so um a very natural idea is to okay I can is to sort of decompose both sides and do it piece by piece so the group side uh by the work of Burnin has a decomposition into Burnin Series so uh these
Are uh things indexed by certain Bernstein inertial classes which I will expand on in a second um and what what these inertial classes are so this uh this is really if you read papers this is really Mass Frack s and these things are uh given by a pair consisting of okay
Uh consisting of a levy of G and some superal Sigma of the levy and then we’re taking G conjugacy Class of uh such pairs Levy and uh orbit of the super Casal under twisting by unramified characters so when I write Sigma I really mean anything in this orbit uh so any sort of
Unramified character Twist of Sigma is allowed and this is also why sometimes uh people also just write a pair M and O So o stands for orbit okay and uh what are these uh e s G so this okay is this color visible by the way or should I use a different
Color um so so these are uh representations uh such that uh people often say uh that occurs in so occurs in parabolic induction of Sigma and of course we always allow twisting by some unramified character so what does a curse mean just means that any subquotient of Pi is
Equivalent to some subquotient inside this parabolic induction so in other words you can also say that this is saying that the cpal support of Pi lies inside this burns the inertial class Mass frag s okay um so on the gallawa side uh there is a completely analogous
Decomposition okay so now I wish I had moved this down here um okay so there is a an analogous decomposition okay let me extend it here okay it’s extended um I just want to move that okay okay so if I were giving a zoom talk I I could easily extend but
Now okay um so there is a an analogous decomposition due to AMS so OB musai and selt um into the galloo side analog of Burnstein Series so uh here first we have some gide analog of The Bernstein inertial classes denoted by S Che in side some sort of Gallow aide
Burnin variety um whose points will give uh yeah whose points consist of these gide Burnin classes and we can consider enhanced L parameters with cpal support living inside the scalas side analog of the Burnstein class so this is completely analogous to that so now I need to tell
You what are these s check so so just like on the group side um it’s a certain pair with a levy so here I need to take M check because I’m on the Galla side and it’s a certain orbit um of so it’s an orbit of a cpal
Enhanced L parameter under some group of unramified characters on the gall side of okay so this needs to okay so this is is a certain cital Enhanced L parameter and I’m taking G check conjugacy on this side so um what does cpal really mean on the gwa side so now um let me should I write it here so um okay I I’ll try to write it here okay so um so starting from an L parameter uh
Firstly an enhanced L parameter fee and its enhancement you can consider um a certain unipotent class inside G check which is just Fe evaluated at this standard unip poent of SL2 c um and lo stick oh so uh starting with such a pair so u v
And row here you this row is an enhancement of the L parameter but you can also really view it as an irreducible representation of um a certain uh Central uh component group of the centralizer of this unipotent inside uh some suitable Gallow aide group that’s attached to this fee so this this
Such a pair uh so so this pair essentially corresponds to uh a local system um so often times denoted by E sub Fe so it’s a local system and it’s not so important to know where it’s supported but if you really want to know it’s on the unipotent conjugacy
Class of this this U okay but the point is um so ltic formulated a notion of cospaly for these local systems and when I say this pair is cpal the official definition is that this local system is a cpal local system but that’s not very intuitive and
Instead I want to give you a more intuitive um notion of what cpal enhanced L parameter really means so um basically starting from this local system um and when it’s cut hospital for example I can consider It’s associated IC sheath attached to this local system and then I can take parabolic
Induction so there’s a certain uh parabolic galloy analog of the parabolic on the group side that’s attached to Fe and then there’s also a certain bigger group also attached to Fe and you can look at the parabolic induction and whenever I have some sort of simple perverse sub sheet so this is
A inside this parabolic induction then I can say that um the cpal support of this simple perverse sub sheie which is going to firstly this shei corresponds to some sort of enhanced L parameter um and the cpal support is uh is this IC shift attached to the local
System so in other words you can say the Casal support of this enhanced L parameter lies inside this s check so um and this uh this is what it means for for for an enhanced L parameter to be cpal okay so uh this decomposition on the gallowa side essentially captures
Okay maybe I want to use an even different color no this is not okay so this captures um basically enhanced L parameters so F and row inside Fe e g uh this is not enough space such that okay the cpal support is in s check so as you can see this is completely
Analogous to what happens on the group side and here this this IC is sort of playing the role of Sigma or Sigma tensor Kai and on this side when I take the cpal support of Pi I get uh Sigma tensor Kai okay so the so so the cidality on
The Gallo side is sort of uh the analog of super cidality on the group side okay so so now we have this decomposition so uh and the idea becomes very simple we just need to match these uh blocks on group side and G side piece by piece so guess I okay so
Uh this might work okay so the point is I want to match this one okay okay piece by piece um so now let me give you the example of uh in the case of for example G2 uh what are the blocks and uh how to match them okay so um okay this one
Okay so example uh so in the case of G2 um because G2 is quite small there are essentially three types of blocks um okay okay so first we can consider caspit support in the smallest possible Levy which is just the Taurus and consider some character of the Taurus um
And in this case the Burnstein series are just principal series of G2 okay in particular if you take Kai to be an R ramify characters uh this uh Burnstein series consists of uni poent principal series okay and on the other extrem we have um you can take no this
Is uh three you can take the largest possible Levy which is just G itself and then take super cpal of G and these are so these bursting series are not so complicated they’re their super hospitals ofg and of course you allow uh twists by unramified characters um so now there’s something
In between the smallest Levy and the largest Levy uh which is uh for G2 you can take the levy to be GL2 and then uh take a super Hospital on the GL2 so there are essentially two different kinds of G2 there’s one Associated to Long root and then there’s another one
For short root and these burin series are uh we call intermediate series well the name really makes sense because it’s in between the smallest and the largest Levy caspit support so um in the case in the first case for principal series there’s the work of
Um maybe I can use okay maybe I’ll draw it here so so there is the work so for principal series there’s the work of Raj where he starts with the group side um he considers a bje between certain simple modules of some hecka algebra attached to this principle series block um
And on the gwa side we consider enhanced L parameters with certain uh correct Hospital support and then by the work of reader so uh I think in the 1990s uh reader constructs projection between uh the enhanced L parameters and the uh simple modules of these heck algebras
And when you compose these two together you get a projection like this um except reader work is uh only for gsit with connected Center so there’s some restriction and then later on uh by the work of abps um I already wrote it out there so I
Don’t need to wrote out the full name here so by abps um they established this projection between simple modules on the group of the group side he algebra and simple modules of certain galloo side analog of the group sitech algebra and uh combining this with the work of
AMS okay so OB mus Sol did I write it yeah I wrote the full name somewhere okay anyway okay so um they were able to basically generalize readers uh principal series local allance uh for General G and not just G split with connected Center okay so uh in my first
Paper on HEC algebra with Amari OED what we did was essentially uh doing this for intermediate series and our result not only holds for G2 in fact it’s a general result for uh nons super Hospital Burns need blocks um for arbitrary reductive groups so let me uh say what we proof
There um ah I can just erase I guess I can just erase this just modify it here so okay so um oh to raise the word principle as well okay so for intermediate series um the story is a little bit complicated so let me first put it as Dash um so first uh
There’s a conjecture due to abps that uh intermediate series bursting block BLS are in bje with certain Twisted extended quo but here I’m going to assume like all the Cycles vanish so there is just going to be yeah there’s not going to be any Cycles um but it’s going to be injection
With something that looks like a Taurus and it’s some sort of cpal thing on the levy uh with uh Twisted extended quotient modulo this uh certain finite extended y group attached to the spin inertial class s so here uh I want to say that this is a finite extended by group
Um so it looks like some sort of finite V group uh semidirect with some sort of R Group okay not sure if the direction is it’s the other way or is this it’s okay um and okay so what this Twisted extended equ really is is you can think
Of it oh now I regret putting it here because okay you can think of it as um some sort of it’s actually some sort of disjoint Union over uh certain irreducible representations of uh okay something that I will explain in a second mod this uh finite extended y
Group and you take uh the you can take the union so when you take the union this really becomes a bje but um what what this uh object is here is you’re taking the stabilizer of your Sigma which is your super caspit on the levy Twisted by Kai and then you’re taking
The stabilizer of this Sigma tensor Kai inside this finite extended by grou wgs okay and when you uh take the union over all the unramified characters Kai of uh the levy M you really get a bje uh ignoring the cycle okay and um there is a completely analogous story on the
Galwa side which is that uh firstly there’s the work of uh AMS where uh they have this projection between uh enhanced L parameters with cital support in S Check and uh some sort of Levy version of the enhanced L parameter on the levy Twisted extended quotient with
Okay so this is gallawa Si so it’s very easy you just add a check everywhere so I’m considering w g check S Check um so maybe let me say uh in case you’re wondering what this wgs really is this which board is this this is the third one
Okay so um this is essentially the normalizer of like this bursting inertial class um inside G and then mod the levy M so on the gwa side this is very easy to Define again you just add check everywhere um and now um this Twisted extended quotient is um also just the union over
Irreducible uh representation of uh okay again it’s essentially adding the Dual to to everything on the group side so what is the Dual of the sigma tensor Kai it’s uh my cpal enhanced L parameter attached to the super cpal Sigma so some sort of Hospital enhanced L parameter
And Twisted by uh essentially the image of the sky uh which is inside this group of unramified characters um of M and then I take the image in the gide analog which is this uh xnr M check okay and then this mod out by um the Gallas side analog of the
Finite extended V group so um fortunately uh this first dotted Arrow was proved and now so it’s not dotted anymore by solal quite recently and so we’re able to sort of complete this picture by working on this side which is um a much easier side because you’re like really looking at levies and
Something smaller and we’re able to uh establish this projection so this this is in my paper with uh amario uh our HEC algebra where we establish a bje moreover we actually give an axiomatic setup for when this is true and uh if one wants to prove local one can just
Like verify our axm and identifying these and then we just glue uh all the pieces together so gluing okay um it always takes me a few seconds to figure out which button to press okay and then here uh we’re also gluing um so this implies so establishing a bje here implies that uh
We have this projection here so this was in 2022 um and then this implies this solid projection here here okay and this is for for G arbitrary reductive group okay um so we’re able to use this uh General result we proved on HEC algebra uh to deduce a bje for intermediate series of
G2 as a as a coraly or special case exemplifying our general result okay so that’s that’s for intermediate series and uh let me just make a remark that uh if you’re wondering what these like magical maps are they’re essentially um some sort of I would say twisted version
Of the cpal support map for example on the galwa side uh it’s some sort of twisted uh cpal support map um um there’s a twisting because the if you are familiar with the usual cital support map uh constructed using generalized Springer correspondence it goes to some sort of Taurus module or
Some sort of V group and here we’re like okay we do some Shenanigans and there’s this is a twisted extended quotient um but but there is some sort of generalized Springer correspondence hiding in the background okay which makes me think this is very geometri or like if you’re more into the
Geometric uh Theory and I think people some people have recently start uh picked up on this and are like working on it okay um so that’s for intermediate Series so now I need to uh tell you about principal series how am I sorry super hospitals how am I doing on time ah
So super hospitals um which one uh this one oh but then I need to maybe do I need to raise everything ah okay that’s a good point given certain axioms which you can verify by hand and we verify it for G2 and also for GSP 4 for example and many
Other groups it’s just I think we ran out of steam so we didn’t bother do so like some sort of functoriality on the cital local Lance on the levy for example compatibility with for the C for so so for the super cpal super yes yes and we we impose two aums on the
Super Hospital local onance on the levy which are satisfied for like many cases and definitely expected to satisfy for all cases um we we have this so we give some axium so you just need to verify the axium and you can do loal allance for arbitrary reductive groups yeah
Oh it’s a different abps abps there are many yeah there are many many APS yeah okay I think if you look in the list of references in our papers they’re probably abps 1 two 3 4 or something like that okay I’m not sure yeah I forgot which which year this is I
Guess Amar knows for yeah um we didn’t Ah that’s a that’s a good question so so like we do it like series by series I guess um we do write down what the enhanced L parameters are but um I’m trying to think yeah we write out for example the
Cach and ltic triples for those L parameters so Che you you can view it that way that’s at least how we wrote it but like I don’t know if we like we specifically wrote down all the s checks like we just wrote down in just enough for for our proofs
But oh you’re okay okay now I understand your question so yes um let me just bring this board down is this the second one okay okay I understand your question you’re wondering how to match S and S check right so so here’s how we match S
And S check so s is M Sigma uh G conjugacy so uh we match them via local lons uh caspit local lons on the levy and we get an S check which is uh okay I’m assuming everything is split because I really don’t want to deal with L
Groups um and then this is the caspit local onance image of Sigma so it’s you can write it as V Sigma R Sigma and then you take G Che conjugacy in the simplest possible form and the the the axium you need to verify is is on this uh this Association
From Sigma to V Sigma R Sigma question yes so there is a list of properties yes uh there’s a list of properties and we uh so for our local L for G2 we have some list of properties and hopefully by the end I will say something about
Stability um but we also use for example uh formal degree so there is a property that says that formal degree divided by the dimension of enhancement is constant along ANL packet this is I think due to Shahidi so that’s also a property so there are many many properties but
Perhaps I won’t have time to get to them yes yes yes but but there could be maybe a different list of properties that you know some other people might prefer to use like uh but we do have our list of properties is um for example I think maybe taso Kalisa has some uh
Characterization purely in terms of uh Atomic stability or something which is uh I’m not sure how to compare like yeah other other people’s characterization was ours but we we we do have a list of properties yes and we don’t use l functions or yeah we don’t use anything to do with L
Functions our our list does not it does not include L functions in our in our list currently yes and somehow at least for G2 we don’t need it so we have enough other other properties yeah so okay uh maybe let me continue um so I’m going to
Move oh right I I remember I was going to use that board before I okay um so first uh let me say how I think about super casals for Simplicity I will only give you examples of deps zero super hospitals um you can uh look at higher depth super hospitals by considering
This like Twisted Levy sequence and then anyway I will only look at a g0 piece so uh the way I think about a dep zero super hospital is basically okay you start with some finite field irreducible representation okay so what is this F finite field Group so what happens is if
I’m looking at a super custom at depth zero on my group G what I do is um okay I will find some parahoric subgroup and I will look at I will look at the reductive quotient of my parahoric this is just the parahoric mod the pro unipotent radical it becomes
Something over finite field um and I will look at some irreducible representation on on this uh finite finite field uh reductive quotient of my parahoric and I will take the inflation and then compact induction and this gives me my depth zero superal so now uh
I just need to tell you um in order to describe this super cpal I just need to tell you firstly uh which parahoric or which reductive quotient of the parahoric I’m looking at and exactly uh which irreducible repres entation of this this uh reductive quotient that I’m
Looking at so how do I think of this space now um what I like to do is okay so I really like the listic theory so by the linguistic um this is 1970 six or eight um I there’s a decomposition uh into of this uh space of irreducible representations into the linguistic series
Um the listic series uh indexed by some semi- simple element so s is some semi simple element inside this dual group um and so each each one is a the linguistic series but this is still not uh explicit enough or at least not easy to to really get
Your hands on so what I do next is I look at this uh each ding L stick series and I apply Lo sticks equivalence uh which brings uh this the lingustic series with non-trivial semi- simple element to something unipotent so it goes to the centralizer of the semi simple
And now uh my I can look at all the unipotent uh representations of this the centralizer so if I’m trying to find uh some representation tow of this finite uh field Group uh that is uh that gives me a super cital after I take inflation and compact induction I just need to
Find something okay cpal here and something cpal here so it now I just need to find cpal unipotent representations of this centralizer and this thing is usually quite small at least it’s much smaller than your original group so that makes the task very doable okay so um in this uh in the
Case of super caspit maybe before I give you uh examples of some very interesting L packet that we found I should first say uh I should mention a series of very very important works so uh this is okay so in the case of super Hospital there’s first the work of the
Backer uh reader so I mean I mean case three here okay case three the backer reader in 2009 where they consider super Hospital El packets uh at depth zero okay let me make more space for myself so so okay dep zero super Hospital L packets so these packets are purely
Superhospital in the sense that uh each packet only has super Hospital members you cannot have a super hospital with a non super Hospital in the same L packet and actually in today’s terminology the what the backer reader considered uh are actually regular super Hospital L packets so what regular super Hospital
Means is roughly speaking okay if you’re looking yeah here they’re looking at depth zero so it just means that your okay roughly that your RT Thea the listic virtual character is irreducible okay um and later on uh there’s the work of tasho [Applause] Kalisa who’s in the audience and I guess
Tasha can go to sleep now um is so uh okay I will not write down the year because I’m not going to remember so Tasha first did um Regular super Hospital L packets at positive depth so this generalizes uh the backer reader to all depth and then in a later paper uh taso
Did uh what’s called non- Singular superal L packets okay uh so uh all of these works are for arbitrary reductive Group G okay so if you want to do arbitrary group sometimes you have to sacrifice by only considering super casals um and um um maybe so the relation between all these different adjectives that you put
In front of superpo is roughly like this so you have okay regular superal and then uh you have a larger class which is called non singular so regular means your RT Theta is irreducible and non singular means your RT Theta can be reducible but it’s still you know nice enough that it
Doesn’t destroy your your life um and then there’s like more Super hpal that’s uh not non singular so these are the ones that we call singular super hospital so I believe uh Amar and I were the first people to kind of say to give this a name to call it singular super
Hospitals so these are the ones that are not covered in in this um and so singular super hospitals are uh very peculiar in the sense that uh they have to mix with um nonr hospitals so I will say these ones have to mix you can track that by by definition mix with nonr
Hospital yeah but it’s it’s really I I think I think yeah I can give it as an exercise to to a PhD student yeah yeah okay have to mix with no uh okay depends on the the the PHD student depends on what I’m oh I’m assuming so for Simplicity everything in
This talk is all split so I don’t want to worry about inner forms and other things okay so um yeah so perhaps one needs to refine some of the statements for for more General groups um but I what you’re saying is that they have to have the same gam and that’s nobody
Knows it’s it’s implied by some other expected property you can that’s an exercise like you take some famous person’s property expected like let’s say listed in some Bor or whatever deat and then this this one can deduce this no I okay we we can debate that after the
Talk but this is definitely true for for all the cases I’ve examined so far um and I think one can say something general uh this is okay this is a vague approximation of the yeah but one can say something in general as well about
The mixing um so okay so at least in the cases that I will discuss like all the singular super hospitals will have to mix with non super hospitals and um let me just give you an example um so one example is um okay okay is so I can look at a certain
Family of 3×3 uh we like to call this 3×3 packets because uh I will have uh three different principle series uh indexed by three ramified cubid characters so there’s uh three different principle series okay uh Prime double and um we will have two families of thetic so there’s an RT
Theta that’s going to break into three pieces so there’s going to be one two and three and uh there is another RT Theta which is sort of dual to this one so maybe like in a very vague sense let me just say some sort of r t Theta check and
Um and this will also break into three pieces so I will have um two families of six singular super hospitals and three different principal series and um so these uh what’s deceptive about these is that uh like all these representations will have the same formal degree so just formal degree
Alone is not not enough to distinguish them so the formal degree is given by okay um okay let me so this is Q uh divide Q over 3 * Q ^ 2 + Q + 1 okay and oh and I did all this work about telling you about how I think
About super casals I should say what the parahoric is in this case or rather it’s reductive quotient so this example has uh reductive quotient of the parahoric being sl3 over fq and then um the centralizer where I’m uh trying to find cpal unipotent on uh is given
By uh so this is T so it’s some torus semidirect product with mu3 okay and uh here I guess I need Q congruent to one mod 3 uh in this case so there is another uh example that’s very similar to this maybe in the lack
Of board space I will write it here um let me bring it up a bit so so another case is we have like two family of four representations so I have uh a principal series attached to certain quadratic ramified character and then another principal series attached to a different quadratic ramify character and
Then there’s a a single RT Theta ding L stick that’s going to break into two pieces and I get two such L packets like this and the for degree in this case again you cannot distinguish these four representations by formal degree or at least you cannot tell if you should put
It this way or rotate these like switch these two um and that’s because they all have the same formal degree and it’s given by q^ square okay uh depending on your normalization but in our normalization it’s q^ sare over 2 * Q + 1 squ and uh in this case my super
Custom is given by um the parahoric reductive quotient so4 and uh the centralizer is given by okay can I squeeze it in here okay centralizer is given by uh Taurus semidirect product mu2 okay so how do we actually determine uh whether it should look like this or
If I should permute these three dots and permute these three dots what to do okay so it boils down to basically stability so what I mean by stability is I’m always looking for space to write things uh I will write it here because I’m running out of time so by
Stability I mean uh okay so I’m going to assume okay yeah so there exists a nonzero uh C linear combination I there’s not much space but okay C linear combination uh so some sort of coefficients with uh the character of of Pi and you sum it over
All the members of a given L packet and you say that the senior combination is stable okay moreover uh it’s expected that you can take this coefficient to be the dimension of the enhancement and moreover no proper subset uh of this packet should have this property um so let me give you some
Divider so it’s clearer so it’s okay and so it turns out that if you try to compute compute the character sums of these uh representations um firstly um the most simple computation one could do is to just compute it on topologically unipotent Elements which form a b a neighborhood of one and it
Turns out that’s not enough because uh on topologically unipotent neighborhoods all of these no matter how you permute them they are still stable which is very frustrating and the same thing is happening here like on topologically unipotent neighborhoods you can permute them and they they’re still stable so we
Had to work harder and we had to compute them um okay um maybe I can squeeze it in here we have to compute them um what we call uh okay so first this is a divider line okay I’m going to write here um so uh firstly neighborhood one
Is not good enough because it’s doesn’t give you much information it’s just stable no matter how you permute them uh but if you do neighborhood of some other non-trivial semi simple so by this this is really kind of like an abuse of terminology what I mean is I’m Computing
Characters on uh elements of this form s and U where U this is a uh Jordan topological Jordan decomposition and U is is a topologically unipotent element uh in the centralizer of s okay so for this 3×3 case um for this this case I’m taking s
To be an order three element such that the centralizer is sl3 and then it just boils down to uh Computing stable distributions on sl3 and then in this case uh I’m taking some sort of of uh order 2 s a neighborhood of s where the centralizer is so4 and then this boils
Down to some computations about stable distributions on so4 and fortunately uh this was good enough for our situation we were able to pin down these things okay I think I’m out of time right sorry sorry for for going over and thank you very much for your [Applause] attention