 Very good. Yes, so we will have three talks today. We'll come everybody and all talks today will be by graduate students and they'll be a very tight schedule. So yeah, with no breaks. So just get started. And so let's start by introducing one point you who will speak about bill is some billions and dream fill super varieties of parahoric group schemes and twisted global them as your modules. Please go ahead. Okay, thanks for the invitation and thanks for the organizer. Today. Let me introduce myself first. My name is one point you. I'm a fifth year grad student in UNC at trouble here. Today, I'll talk about the bedding syndrome for sugar ideal for parahoric group skins and twisted global them or two modules. This is my joint work with my advisor to do. Okay, let's start from the finite dimensional case. The GBA simply connected the same simple algebra group over C. For example, the group to end, and we fix a boreal separate subgroup and a maximum torus T inside the boreal subgroup. We can define a flag variety G portion B. For example, when G is john based the group of our triangular matrices is the group of diagonal matrices. Then there is a bijection between the G portion B and the set of legs of the n dimensional space C and that's why we call it a flag writing. And let you be a unipotent radical of the boreal subgroup B. For any weight meal, which is a map from the maximum torus to C star. We can extend this map to a map from braille subgroup to C star. I didn't know it by to the meal. This is the composition map. It can be used to be quotient you. It can be identified with the maximum torus T, and then T maps to C star. We know by C sub meal, the one dimensional B sub B module journey by one vector V. This is a B module given by defined by this action, the boreal subgroup add on this vector V, we are the map to the meal. And then we can define a line bundle on the flag variety G portion B. The line bundle with the flag variety G point B it is G times the one dimensional B module over the boreal group B. This is a line bundle over the flag variety G point B. And then we have a boreal theorem. It says that the, the space of global sessions of the line bundle L mu on the flag variety G portion B is isomorphic to the deal of the high weight representation. V of meal of of high weight meal. And question. This is the final case. And moreover, for for this group G, we have a blue heart decomposition. Here, W is the wire group of G. This uses a decomposition of the flag variety into be obvious G quotient B is the disjoint union of BWB quotient B. This is a be orbit. You can think B act on this flag variety. This is a be orbit with denoted by X sub W. This is also called supercell. And it's closure is called a super ID. For the highest weight module V of meal, we can define a B sub module inside the view of meal. It is the UB span of the better V sub W mu. The B sub W mu is the better of weight. The view here W is in the in the white group. This B sub module is called a demazoo module. And we have a isomorphism. The global space of global sessions of the line bundle on the super ID is isomorphic to the deal of the demazoo module. This is the Boreal weight theorem in the final dimensional case. It has analogous for often should variety. Let the GBA are simple the algebra and a capital G be the joint group of G. We can define often super often grass manian of G. The quotient G of K quotient G of O. Here K is the ring of a long series and all is the ring of former power series. I denote this often customer and bike up good G good subject. We fix a maximum torus T in the group. For each dominant co weight lambda. We can define a point in T of K. This is an image of T under the map from K star to T of K. And for for for this point T T to lambda in T of K. We didn't know by E to lambda the associate point in the often cross morning good subject. And then we can define a often sugar cell to be the geo orbit E to lambda. This is called a often sugar cell. And similar to the final dimensional case. This is very similar to the previous case that often cross morning. This is a disjoint union of the often super cell. It is called a candy composition. The often super cell are parametrized by the dominant co weight of T. If we take the closure of the often super cell, we get often super ready. And there is a breakaway type serum due to Kuma and the measure. We have an isomorphism between the space of. Global sessions of the level one line bundle. This power of level one line bundle on often super ready and often level C often them a zoom module. I'm not defined them a zoom module here. Because this is a super ID similar. We only focus on the super IDs. Okay, any questions over. This is also the untested case. The twisty case, but we have a similar. A boy with a type zero. We first introduce a group skin. Curly G. Let us sigma be a standard or more efficient of order M on G preserving a barrier subgroup and a maximum torus T. Now, you can just think about the case when G is not a to L. In this case, Sigma is defined to be a diagonal or more efficient when G is a to L. Things are a little bit complicated. It is autumn autumn for automorphism for for this standard or more efficient on for this standard automorphism on G, we can define on the ring of long series and the ring of former power series by letting Sigma act on T to be Yipson inverse times T. Here the Yipson is a primitive MC rodeo of unity. And we do not buy a bar the Sigma fixed point of the of the ring of a former power series. It is C of T to M. Sigma fixed point in the ring of phone power series. We define the we define a group skin to be the Sigma of fixed point of the restriction of the skin to all bar. The definition is a little bit complicated. But here, we just want to define a twisted often cross morning of this group skin G. The definition is a little bit common, but the basic idea is very simple. The C point of this twisted often cross morning. Go sub curly G is exactly the Sigma fixed point of G of K quotient now Sigma fixed point of G of O. In this case, the twisted often cross morning also has a cadenic composition into twisted supercells twisted often supercells. Those are parametrized by the Sigma co invariant dominant the Sigma co invariant. The Sigma invariant dominant co weight of T. This said it is image of the dominant co weight under the projection map. The co weight to the co invariance of the co weight. Okay, as before we take the closure of the twisted often supercell. To be the twisted often super ready. And we let L be the level one line bundle on the twisted often cross morning, similar to the artistic case. We have a body with types here and due to come on and the measure. There is an ice movies and between the space of global sessions of this power for the one line bundle on the twisted often super writing and twisted often the module. If we allow Sigma to be the trio movies and the trio diagram automorphism. Then this group of skin is exactly the constant group of skin G. So the left hand side is exactly. When Sigma is trivial diagonal morphism left hand side becomes the space of line bundle on the untested often super ideas. And the right hand side is exactly the untested often the module. Now this is a boy with types here and in the twisted the case. This two boy with types here and in either twisted case on the twisted case have a global version. That is a boy away type serum for Benison dream view. Should variety. Okay, any questions so far. From now on I'll introduce my work with my advisor to soon. Okay. I'll give those definition of global demo through modules. Okay, we define a paracordic blue heart T's group skin G. Let a one bit often line with coordinate T and it has an action. Sending P to eat some P. A one bar the quotient, the quotient line a one question now the Sigma action. It is ice movie to a one. And let pie be the projection map from a one to a one bar. Five minutes. Okay. And denote by P bar, the image of P under the projection map. We can define a group skin over a one bar. Which is the Sigma fix point of the way your institution. Of the constant group skin from a one to a one bar. This is a paracordic blue heart T's group skin over a one bar in the sense of pain loss. Report definition looks complicated but there is some property that captured this group skin. For for any for any why. For any why, which is not zero. The restriction of this paracordic group skin to the formal disk around why is isomorphic to the constant group skin. G. Over over the formal disk. Around why. If why is zero then the restriction of the group skin to the formal disk around why is isomorphic to the paracordic group skin, the courage we defined before. This is the case of for. For this paracordic group skin. G, we can define a bit in some dream field. This is a collection of triples, a point in a and a G torso over a one bar and a trivialization of the torso outside those PI. This looks complicated but let's just look at the fiber for each fiber. P such that the coordinates are in different sigma orbit, then it is just the product of and copies of often question money. If PI is not zero. Then good GP I is just the untested often question money when PI is zero then it is a twisted often question money. And for for this paracordic group skin we can also define a penance and dream field should variety. Good G and lambda arrow. This penance and dream field often question money. Penance and dream field should variety has a factorization probably. And it's fiber, when, when, when the coordinate are not in the same orbit. But it's just the product and copy surf often should often should varieties. Okay, this is, yeah. This is one last page. This is my result. Let L be the level one line bundle on the penance and dream field should variety. We have an isomorphism between the space of global session of this power of flying bundle and the penance and dream field should variety of paracordic group skin, and the artistic global them and some of you. This is the boy with type serum for the penance and dream field should variety. Okay. And the artist case was first for by the man ski figure and the finger burger. That is, when, when Sigma is the trivial automorphism. We get the artist case, it was first approved by the man ski figure and finger in our serum, we require Sigma to be a standard automorphism. That's all. Thank you. Thank you for a very nice talk.