 za invitacij, ki v tem vse komforti sezivimo na Frančke vsezivne vsočenje, na vsezivnih vsezivnih vsezivnih vsezivnih vsezivnih vsezivnih. Zelo sem boj, da sem vsezivni, ker je to, da je prijevna, Moj P.H.D. tezis je na svetljičnih funkciji, in zelo spremljamo nekaj počke, nekaj matematikovih življenja, vzvečenje z diskretnih serijs, in tudi se vse koncepti vzvečili v zelo v godmah. Zato videlam godmah, zelo sem všeč v Michel Di Flossa, v ročnih GCA v 1985, kako sem vsečočila Unitary Jewel, GLN. Zato vrštje različo sem jazem. Vseč je to zbavljivost, ko je zbavljivost. Korin, in je to zbavljivost, da se pravamo o historijstvenih. V mojem življenju sem jazem izgledan. to nekaj učin, nekaj spraveno, kak je s njimi se zelo zelo zelo zelo. Na prveno zelo se prejšel, da se je se v pariči, od biloče, počuji, od autobimofiku, prijezi so v Parisu na 83 km. Prijeznam, da je to, da je počuji, prijezna je, da je to, da je to, ki so počuji v GLN, in zato vidim, da je to, da je to, da je to, kaj je to, da je to, da je to, da je to, da je to več frutko v nekaj delovosti. OK. Značno. Tukaj bilo, nekaj delovosti v unitori delovosti. Tako, klasice delovosti v unitori delovosti klasikovosti, vzout, klasku vgrup G, vzout sem tudišel začel na razredno 3, na razredno 3. Vzout, koncepti harmonijski analizist tudi, that important representations of G, unit representations of G, should be understood in terms of irreducible ones. En z nekaj komutivnih grupov v matu je GLI. Zelo smo prišličili v nekaj klasikovih grupov. OK, zelo to je ovo v lokalj stranju F, akimidio, nekaj za vrštje. Tako klasičnih grupov, imamo nekaj limitirati vzvečenje in unitaritiri. Tako tukaj sem zelo vzvečen, tako zelo vzvečen. počke vzivno. OK, zato občaj bi vzivno odrečiti potenč, kako je nekaj nekaj drža. Zato tega vzivno. Vzivno ta različim vzivno. Zato zelo podeljamo potenč z Borstvenim Zelovenski na parabolic induction. Vzivno to je. je zelo veliko, zelo pa je zelo 1, zelo pa je 2, zelo 0, in tudi je zelo 3, zelo je zelo zelo. Tako, z dfom, zelo zelo, v vsej klasičnih izgledajših reprezentacijov, in prišličnji pobof, nekaj je oplezjno, nekaj je tudi prišličnji prišličnji pobof. kako je. Kako je, sem se gre. Positiv. Representacijo. Tukaj. Zelo sem srečit, da ne bi mi. Čačo so poslutila? Tukaj. Zelo sem dobro. Representacijo. kaj je 1,5, zelo delta, kaj je kaj je izgleda kaj je 1, zelo delta, kaj je n-1,5, zelo delta. Taj je taj reprezentacija, kaj je vzelo ljubno, unik in reduzible, zelo delta posljajte, kaj je taj reprezentacija. So, Okay, so we will denote by BF, rigid, the set of all spare representations. and by BF, the union of this rigid part and times mu to alpha pi times mu to minus alpha no, pi, when pi is stray representation. So, okay, and alpha is greater than 0 less than 1 half. Okay, so, now we can write classification theorem. Which tells you the mapping. p1 to pk in to p1 cross pk, je obježekšnja z vsej finajtnih multisets In bf in bf on to union of gln, f unit reduce and greater or equal to zero. Ok, so this very simple theorem to state of Solve unitizability question for gln groups in Akimedian and non Akimedian proof. So the proof is not that simple like expression, but also it's not too complicated. So this theorem applies to Akimedian and non Akimedian case. So in my experience, even if I'm interested in all the non Akimedian case, somehow the simplest expression of unitizability is one which applies also to Akimedian case. For example, with lapiden muč, we classified generic unit reduce of classical groups and theorems applies in both cases. Also with muč, we classified spherical unit reduce of split classical groups and the proof is only for spherical case, but we expect that the formulation makes sense in the Akimedian case and we expect to hold there. So, but today I'm going to talk about possible strategy for unitary for other classical groups, but it will be exclusive for non Akimedian case. Ok, this may sound some kind of contradiction that I said the best is to have uniform and now I'm just switching to one. Ok, so there are some reasons for this. The simplest reason is that I do not know, I do not have idea of approach which would, let's say, make sense in both cases, some effective. So I can do something, but not really, one can try to do, but there is no, I do not know, have idea to go a little bit further. Ok, second reason is that I started to think about this approach, which I will talk today. After Lapid and Alberto published few, actually, two years ago, I think a paper, which just, which give even simplification to this, the proof. But let me just maybe before tell one additional thing. So, this theorem also applies to GL of division algorithms. So this was proved somewhere in 1984, beginning of 1984. So, and this case was proved by Badolesk, Renard and Secher about 20 years later after this. So, one of reason that it took so much time was that proving irreducibility of unitary parabolic induction was hard and it was not able there to apply strategy of, I would say, galfant, nijmark, and Kirilo, which uses mirabolic subgroup and properties of invariant distributions. Actually, the theorem of Badolesk, Renard and Secher is, more or less, everything is same except one need to, in these definitions, switch, put instead of nu, some power of nu, which can be explained, appropriate power, I am not going to talk about this. Okay, so Secher solved this problem of irreducibility of unitary parabolic induction just using types. So his proof is actually very interesting, but also very, very complicated. So, what Alberto and RS did is that they essentially showed one property of spare representations. I think they call this, which is property of non, just in non-interducible. I think that they call this, that these representations are saturated, that if you take pi, spare representation, and then multiply with any irreducible GL representation, that this representation has unique irreducible subrepresentation. Is Alberto here? Okay, so I hope this is right. Okay, so this is something, which doesn't have much things with unitary, it has with spare representation, but from this, essentially using the strategy of this, they were able to get this theorem also in division algebra case, but also in field case. It's now significant simplification, but you do not need invariant distributions, and their proof is basically using the very well-known theory of Jacques Modules, which was developed very well by Berznan and Zelivinski. Okay, so now that we essentially, their proof, you can now go to this classification, just understanding only first reducibility point, so if you have a role irreducible hospital GL representation, you need to know this exponent, okay, new to alpha, the role, when this is reduced, alpha greater or equal to zero, and then from this you can do everything, so. Okay, so this open possibility that you, so you just, these hospitals are black boxes, and you need to know this alpha, and alpha is equal to one in the field case of all this. Okay, so I said this open, and this just motivated me to think, can you do this for the classical groups? Okay, so, but we need to know ingredients, these ingredients, so hospital representations, and hospital reducibility points. Okay, so classification of R-tour of discrete series, more general temperature representations, and Maglaine work on this, that she singled out hospitals, parameters among them, and also she gave a simple formula for a disability points in this case. So, thanks to R-tour and Maglaine, we have now ingredients, these hospitals, representations, and the reducibility points, just to think about such kind of approach, which let's say, now it's possible in GL case, thanks to this, thanks to Ares and Alberto. Okay, so, now we can go to classical groups. So, and there is also one reason, not real reason, but there is some historical experience, so it's interesting that this theorem was first proved in non-akimidian case. And just immediately after, you start to think, okay, what happens in Akimidian, and then it was pretty soon realized that it works also in Akimidian case. Okay, so one may ask why it was not proved in firstly in Akimidian case, because here this set, okay. So, this set, let me see, did I wrote, okay. This case here is much smaller, and this case was much more elaborated. And as, let's say, Bushnell knows, this is much, much more complicated in Piedi case. And even it's interesting that this was not conjecture in Akimidian case, because it's very simple, but the truth is that Gelfand and Neimark in their books from 1950, and I think maybe even in some paper in 1947, they claimed for, let's say, SLNC. But just this is essentially same as GLNC. This to be interdual here. And what later turned by Stein that this, their list was incomplete, actually not, some complimentary series were missing, in dimensions bigger than one, so this is. Okay, even Neimark published in beginning of 50's proof, which was completely wrong, that this is unitary dual, and somehow this was later abandoned, this idea. Okay, so I said the first was solved non Akimidian case and then Akimidian case. Now it's question, puzzling question. So is it going to be this the case also in other classical groups. So I do not know. Also it's interesting question why it was solved also in first in non Akimidian case and then in Akimidian case, even this GL case. Okay, so these are just my unorthodox thinking. So let me, let us now go to classical groups. So BF is all spare representations, spare representations. Okay, so for simplicity, I will consider SP 2N plus 1F split. So these are all GL 2N plus 1F, such that G tau, where this is transpose, but with respect to second diagonal, is identity matters. Okay, F is in the rest non Akimidian and characteristic zero. Okay, so also we can, I will look at symplektic groups. So using realization by matrix, this 1, 1, 1, minus 1, minus 1. So what is here important that levi factors, isomorphic to, thank you very much. Okay, I do not want to invent odd symplektic groups. Okay, so I was writing this and I was already thinking about this, so this was the problem. So okay, now levi factors isomorphic to GLK. So such groups I will denote by Sn. So I will fix one series, I think then just denote. So fix one of the series, doesn't need to be just also some other. So then levi subgroup isomorphic to Sn minus K. And actually isomorphism, which we will put, here is just that gh is going to gh to minus 1 tau. So this is something like this. Okay, so now when you have a representation pi and sigma, pi of GL, sigma of classical group. So you can make tensor product and induce extending trivial here. And just like Bernstein-Zelovinsky, you can define now multiplication of GL representations and classical group representations. Okay, now we will say here what are hospitalary disabilities. So we will fix now role irreducible hospital representation, GL representation and sigma irreducible hospital of classical group. So we will assume that it's self-contragradient advice. We will not have reducibility. Then by Silberger there is unique exponent alpha rho sigma greater than or equal to zero, such that nu to alpha rho times sigma is reducible. Okay, so this is Silberger result that there is unique this and it's not that complicated, but what is complicated is the fact that this exponent's integers, so half integers, so this was proved by Mark McGlenn. Now from Artur admissible homomorphism attached to hospital representation sigma, one can read the reducibility point. I will just go the other way. So this is nice fact proved by McGlenn is that if you fix sigma as above hospital irreducible of classical group, then you can read admissible homomorphism attached to sigma from all reducibility points, which are one or bigger than one. So it's following to sigma corresponds to direct sum, running of a self-contragradient rho, such that alpha rho sigma is greater than or equal to one. Now you have direct sum of k is greater than or equal to zero, less than or equal to alpha rho sigma minus one, such that k is congruent to two alpha rho sigma minus one, modulo two. And here you have phi gl rho, there's a product of e tk, this is local language correspondence for gl and this is unique analytic irreducible k in dimension of representation of SL2C. OK, so from classical reducibility greater than one, you can read admissible homomorphism attached to sigma. Actually it's obvious that you can go other way. So read from this, if you know this, what are these? And this is what Maglian calls fundamental assumption. OK, so there is also, so this relates this reducibility points. So you can also read when it's zero, when it's one half from the type of this representation. If it's not greater than one half, it's zero and one half and you just need to look at type and it depends on the group, just which one you will have. OK. So this is the local language for gl. So now, so these are reducibility points. So this is what I said, very simple formula for reducibility points. So second thing is that Maglian singled out hospital representation in art of classification. I will not go in description of, precise description, but there are three very simple conditions for them. So no gaps in admissible homomorphisms. Second is corresponding character of component group is alternated. And last is if you have in admissible homomorphism piece like this form, so character is equal minus one. OK, so my idea is not just to explain this, but just to give impression that it's very simple conditions which distinguish hospital representation in art of classification. So next thing which we have is reduction. One simple reduction is the following. So if you take now row, essentially score integrable. OK, so actually, OK, so, do I need, OK, so, hospital gl representation irreducible. Then there is unique row unitary. And alpha, OK, e, row, real number, such that row is isomorphic to new to e, row, row u. So actually you just take non character, you twist by character to get unitary representation. And then minus one, so this defines this. And this actually works for us. It's interesting now only this. So now we have definition. We call admissible irreducible representation pi of classical group, weakly real. If you have, if you embed pi into some product. So this is a hospital representation, this gl, this classical group. Then this unitary parts are self-contregredient. So in the case of, let's say, when this are really characters, so it means that these are real valued. OK. So maybe about one decade. Before I proved following simple result, take pi in Sn. So this is irreducible unitary. So then exist tau irreducible unitary gel representation. And pi one in Sn minus k hat, weakly real, such that pi is isomorphic to pi one plus tau. So since we know by there, so what irreducible unitary representation, so this reduces prone to weakly real unitary, irreducible unitary representations. Actually, here's also claim about uniqueness. So this is unique, but I will not talk about here. OK. So next thing, what is, here is the people who work. I worked with GLN, hospital of GLN, usually very soon somehow turned to look at representation supported by specific particular hospital lines. And actually, in this way, you using simple parabolic induction, you write pi as product, where this representation supported by hospital lines, we called representation supported by, and then important questions for representation theory about pi, just simply reduce this in lines. Actually, and then I was computing something seriously, I was computing in lines. OK. So in the case of, so what is here for us interesting is it's almost, it's pretty easy to prove that for example pi is unitary if and only if pi i are all unitary. And actually one direction is very well known that parabolic induction carry unitary to unitary, and the other is, I think this was first noted by Spe, but it's pretty easy to show. OK. So in the case of, in the case of classical groups, we do not have, we cannot do, use parabolic induction for such induction, but there is something which corresponds to this and this is Jansen, the composition, I will just. Sorry, SO2n is not included? What is it? You said SO2n plus 1 and SP2n, but what about SO2n? I prefer O2n. No, no. If you work with SO2n, then, OK. So O2n is included, so this is, but I feel more comfortable for it, symplactic and orthogonal groups. So this. OK. Now we have Jansen, the composition. Essentially, I think that for SO2n, 2n plus 1 is, SO2n plus 1 is equivalent to O2n plus 1 because it's just a simple product, we sent it, so maybe, maybe just. Easy to mention the note, to be noted. Yes. OK. We will denote by C, all caspel, reducible caspel, GL representations. Now take some X subset of C and we will suppose here that it's as set, it's same as it's contragredient. It's not point wise. Now, as you know, it's not what is supported in GL case, but this similar, it's similar defined support in, in, OK. You say that some GL representation is supported in X if pi is, y is in X. Now, at caspel, reducible representation, sigma of classical group, then you say that representation pi of classical group is supported in X union tau if pi embeds into, OK, rho1 times rho k times sigma, rho i in X and, OK, so this is this. So now, I said what caspel lines of GL representations fix. OK, so now, take pi, weakly real. OK, we could consider not only weakly real, but this is only interesting and take some caspel line, we will take just self-controredient. Then you can find sigma, irreducible caspel representation of classical group, then pi L, irreducible representation of classical group supported by L and L C, irreducible GL representation supported out of L such that pi embeds into pi C L cross pi L. So what is interesting, this is completely determined by pi and L. So now what is also easy, it's obvious that there exist finally many caspel lines let's say L1 to Lk such that for other caspel lines pi L is equal sigma caspel lines L so essentially nothing except this caspel representation show up. Then just the composition is just by ejection pi just attaching to pi to pi L1 to pi Lk and now there is by ejection between just you can like by ejection I will not talk about this, but simple pi is described in this way by representation supported in this caspel lines so here sigma is missing by this caspel lines and this sigma so so actually this what Jansen has shown that this correspondence this by ejection has very nice properties just irreducibility square integrability temporanese multiplicities either preserved or multiplicities you just you need to multiply to get n and so on so I will not talk about this but now I will ask so this question is pi unitary and only if all pi Li unitary there is very this was no less trivial and in this case no implication is obvious actually this is quite hard question and there is some evidence for this very limited so this is generic if pi is generic unitary then spherical then rank less than equal 3 and then also some partial results in one direction ok so we do not know this but if we would know this this would reduce problem of unitaryzability to the case of unitaryzability and Casper lines I mean representation supported by this actually even if it doesn't reduce when you will start look at reducibility you will probably not go to more than Casper lines so question which arises here is does I have defined before this row, Casper and self-pontragoneral row and sigma reducibility point doli reducibility point among them ok, next question is does unitaryzability depends only on alpha, row, sigma not on row and sigma itself so I will not this question can be very precisely stated but it's using Langlin's classification but later you will get impression what I mean by this that it doesn't just depends only on this so last year I tried to check is this approach works for in generalized rank 3 case and actually I will now talk about this so my approach is not very clever and it's a lot of work and so on and I think Michelle DiFlo knows you are looking at complex groups of rank 2 and this approach by brutal force is pretty painful because you need to compute everything but I'll not just there is this paper on my web page but I'll not I'll give only idea what is happening here and then after this we will discuss just general case not general case but just thinking of general case so we come to unitarizability rank less than or equal to 3 actually rank 3 is new but this was no smaller so so it means I will ask for unitarizability of representations subquotions here it is a hospital where k is less than or equal to 3 first now question 1 so that Jansen the composition preserves very simple it's not completely trivial but in this case have positive answer so now we are left with representations supported by hospital lines so the case k equal 1 is obvious you have unitarity up to first the disability point so case equal 2 is very simple although the proof is not completely trivial so namely we are just working with role and sigma and what we are using only is a reducibility point no nothing about them so reducibility point alpha role sigma I will just for sure to write simple as alpha so I will write so general case will be combination of these lines when k is equal 1, k is equal 2 and k is equal 3 I said k equal 2 is very simple and it's known for longer time I'll write proposition maybe just maybe I can I can skip this I can go to k equal 3 case to save little bit time so now let me go to k equal 3 so in k equal 3 so you need to lines you need to treat this ok so you should have you have infinite many cases and just so good thing question 2 is important just that you do not have additional just informations ok so here you have 4 separate cases so this case you can treat as 1 simple answers are different so actually strategies from this but just somehow it degenerates and you need a little bit to adapt arguments and this gives also different answers and also different little bit proves technical different but same as ideas like in this case I will now describe this case because it's most interesting and just let me say that this case these 2 cases show up infinitely more often than this case whatever it means infinitely more often I can talk little bit but let me just now say what is answer in this case I will just recall what are generalized timebar representations when alpha is greater than simple strictly greater than 0 then look at alpha is a reducibility point look at representation alpha plus n n is greater than equal to 0 for all you decrease by 1 rho sigma so here you have unique irreducible supper representation and I will denote this delta nu alpha rho nu alpha plus n rho sigma so and this is score integral and I call this generalized timebar representation so now we can tell how looks answer in this case so this proposition let alpha be greater than equal to 3 half pi irreducible unitary sub portion of nu to x1 rho times nu to x2 rho times nu to x3 rho times sigma where x1 is greater than equal to 0 less than equal to x2 less than equal to x3 always you can come to this situation ok I hope you know this family by star then pi is one of the following irreducible unitary sub portions 1st delta nu to alpha rho nu to alpha 2 rho sigma or it's ober schnider du 2nd irreducible sub portion of nu to x1 rho times theta where x1 is greater than equal to 0 and less than or equal to alpha minus 1 and theta is delta nu to alpha rho nu to alpha plus 1 rho sigma or it's du ok, so this is 2 3 is unitary reducible quotient of nu to alpha rho times nu to alpha minus 1 rho times delta nu to alpha rho nu to alpha nu to alpha rho ok, so here we have only one of ok, so this and number 4 just to complementary series irreducible sub quotient of star xi satisfying at least 1 of following conditions ok, so conditions are a is x2 plus x3 less than 1 b is x1 plus x2 less than or equal to 1 x2 plus 1 less than or equal to x3 less than or equal to alpha then x1 plus x2 less than equal to 1 and x3 is less than 1 minus x1 less than equal to 1 plus x1 and last is x1 plus 1 less than or equal to x2 and x2 plus 1 less than or equal to x3 less than or equal to alpha ok, so these are complementary series, little bit complicated I think first these conditions you can find in your papers and then later in barbaš paper there are such conditions for complementary series ok, so the theorem is not too complicated but unfortunately the proof is I said just it is by brutal force and I'll maybe just say only few words later about the proof so let me just say that unitary duels are topological spaces in natural way so what is what are most delicate and usually most interesting parts of this unitary duels are isolated representations so in this theorem isolated representations are mostly discrete series or duels of discrete series so we understand them pretty well so there is one additional isolated representation this is this is a representation this is self-dual representation so it is unitary but it is not discrete series definitely not a duel of it and unitary of these representations was proved by Meg Lenn so using Arthur packets and her work on Arthur packets so this is first really interesting example so you do not get in generalized rank 2 let me say for GLN first representation for PLD GLN which is isolated module center but not discrete series or duel of discrete series you get for GLNite so this is so maybe just last few minutes I will tell you what can be expected in general I only want to say that in this low ranks situation is very simple so let's say if you think about SLN and now you look at spherical unitary duel only isolated point is trivial representation so it is ok, must be different from 2 so Mujč and me made classification of circle case in circle interview for classical split groups so later I spent some time to single out isolated representations there in the right combinatorial formula for number of isolated representations so for example for SP 200 so there you have isolated representations 16,051,857 so you have a little bit more and actually the number when you look at this low ranks it's 2,2,3 then maybe 5 it's going very slowly and then at some point it start to go very fast ok so just let me say how you in our classification paper is based on Mujč paper where he show that these representations are automatically namely the unitary coming from because they are local factors in residual spectrum so this is how it's proved but you can come to this representation in different way so denoted by X all generic if a chore fixed discrete series of SP 200 ok so then then there exist X prime subset of X such that dual of X prime is equal to all isolated spherical representations in this case so ok, so actually me glen did construction of discrete series from casperals and what this really on what this really depends is this reducibility point so I'm just ok maybe just one or two minutes so when did we start no I'm very well known for being late with that but today we have been late with the beginning so at what time did we start so not 5 there was more than 5 ok, we will not negotiate I will not tell you ok, sorry ok, so what is just I'm saying if you now take ok, so now component of Bernstein component of this type this is generic you can you can essentially using this you can copy these representations but just we didn't here you are working with 1F and 1 ok sp0 here you can put a row and sigma and you can get in the same way discrete series you apply duality and from my glen we know that this duals are unitary and it's natural to expect that you will have that you will have that isolated in unitary duals so you will have also at least this number but then you actually do not need only to take generic so you will get much more but just looking at here and you know what are isolated here points here you get that most of isolated points in this Bernstein component will be not of discrete series or bare duals but also you will get another one so this is another one I hope you will get them from R2P packets so ok so main problems in general are just construct isolated points and then how understand how they decompose when you induce and last problem which is most mysterious even from the time of Gelfand Neimark is once when you will have some kind of list to prove completeness and this is just what we have so going to non-unitarity is very painful and this is this was my approach because I didn't have idea for this rank 3 but in general this is let's say here there is no chance ok thank you very much and sorry for being clean with you