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What I Learned Today

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What I Learned Today What I Learned Today Tom Gannon September 2021 (9/1/2021) Today I learned that if µ is an antidominant weight, then there is some simple root α ∼ _ ≥0 and some minimal IC sheaf I(w) 2 Hw·µ,−µ such that I(w) ?L(µ) = L(w · µ), hα ; w · µi 2 Z . I also learned the proof: any antidominant weight µ, by definition, has the property that ≥0 + _ ≥0 Ξµ := fβ 2 Φ : hβ ; µi 2 Z g is nonempty. We induct on the minimal height of an element ≥0 ≥0 in Ξµ . For the base case, note that if there is an element of Ξµ of height 1, then by definition ≥0 there is a simple root α 2 Ξµ , and so (3) holds with w = 1. Now assume the minimal height is ≥0 larger than one, and let β be a minimal root in Ξµ . By assumption on the minimal height, β is not simple, and therefore, the theory of root systems gives that there exists a simple root α such _ + that hβ; α i > 0, so that sα(β) 2 Φ and the height of sα(β) is smaller than the height of β. Note that this in particular implies that β − α is a positive root, and therefore, by the minimality of β, we see that hα_; µi is not an integer. August 2021 (8/1/2021) Today I learned some more categorical descriptions for certain representations of the quantum group. Specifically, one can take the mixed version by regarding it as relative Drinfeld center of the category of representations of the Hopf algebra of the Kac-De Concini quantum universal enveloping algebra for N − (relative to the fact that we defined that as a Hopf algebra in the braided monoidal category Repq(T ), so categorically you need the natural morphisms you can get from that to agree). The formalism of the center of the representations of a Hopf algebra says a representation of a quantum group (and the fact that Lusztig and KD quantum universal enveloping algebras are dual) says that a mixed representation of the quantum group has a compatible action of the KD quantum enveloping algebra for N − and the Lusztig one for N +, but, for the Lusztig one, the action has to be locally nilpotent. (8/2/2021) Today I learned some useful facts to prove that the Whittaker sheaf in D(G=N) is equivariant with respect to the Gelfand-Graev action. Specifically, one can use the fact that the action is G-equivariant to show that the Whittaker sheaf goes to a sheaf which is also Whittaker invaraint. This allows one to write out the explicit diagrams and compute base change with respect to the big cell of both copies of G=N. (8/3/2021) Today I learned a lemma of Sam Raskin and David Yang, which says that if one has a functor of two G-categories with t-structures which are compatible with the G-action, then one may check whether a functor is t-exact if the functor is t-exact on the N-invariant subcategories. (8/4/2021) Today I learned the full statement of the Jacobson-Morozov theorem. Specifically, the theorem says that given any nilpotent element, there exists an sl2 triple which takes h to a semisimple element and e to that nilpotent element, but, furthermore, I learned a bonus result of Kostant which says that this embedding is unique up to choice of element in the unipotent radical of the centralizer of the nilpotent element you chose. Therefore, as a sanity check, we can see that if 1 - Tom Gannon 2 we chose the nilpotent element zero in a reductive group, since reductive groups have no unipotent radical by definition, the sl2 embedding is unique, and if we choose a regular nilpotent element, the centralizer is abelian, and so in particular is also reductive and so the embedding is unique. (8/5/2021) Today I learned a possible more intrinsic definition of degenerate and nondegenerate G-categories. Specifically, we can view G-categories by assumption as having a commuting action of D(G=G), and we have a monoidal functor D(G=G) !H . In particular, we can define the degenerate G-categories as those G-categories C for which C ⊗D(G=G) H ' 0, and hopefully it turns out that the more naive [P; P ]-vanishing definition for CN agrees. (8/6/2021) Today I learned the above potential alternate definition might not deal with the t- structures incredibly well, and a better alternate definition may be to require that the horocycle/Harish- Chandra functor vanishes upon averaging to some Q (instead of N). This allows still for the fact that the N-invariants of a given G-category have the same nondegenerate category as before, while maintains that nondegenerate G-categories are intrinsically G-categories themselves, i.e. not by force. (8/8/2021) Today I learned the results of some quantum Hamiltonian reduction computations. Specifically, I learned that one can identify the T × T -fixed points of global differential operators ∗ ∗ on NnG=N with Sym(t ) ⊗Zg Sym(t ). (8/9/2021) Today I learned a result of Ginzburg's which states that the isomorphism D(G=N)N −; −!∼ D(T ) is compatible with the Gelfand-Graev action! The proof uses an isomorphism of the respec- tive rings, which we can use since every quasicoherent sheaf in the category D(G=N)N −; is globally generated (since N − × B is affine). (8/10/2021) Today I learned that the fact that the compact objects of the category of ind- coherent sheaves on an ind-scheme can be written as the pushforwards of some compact object from some closed subscheme is actually a more general categorical fact, and that for any two objects in the closed subscheme you can explicitly describe the maps between them as a colimit of the `inclusions beyond'. (8/11/2021) Today I learned a way to define a t-structure on the nondegenerate category, finally. Specifically, assuming that I can show that on the heart of the t-structure that the only objects in the kernel are the monodromic objects, the kernel of a t-exact functor by design kills subobjects, ! and the condition on J∗J being exact only requires a condition that the heart is closed under subobjects. (8/12/2021) Today I learned that the compact objects in the category IndCoh(Γ) on the union of graphs of the affine Weyl group Γ has an (alternative) explicit description of the compact objects. ` ∗ Specifically, since Γ admits an ind-proper surjection w2W aff t ! Γ, we can realize the sheaf as a totalization of the cosimplicial object. Using a convenient lemma about limits and colimits of categories, this is also the colimit over the category of right adjoints, and Gaitsgory and Drinfeld showed that any filtered colimit has compact objects given by the `pushforward' of the compacts in the diagramatic categories. (8/13/2021) Today I learned that holonomic D-modules are Artinian, so that in particular one may find a Jordan normal form of them. I also learned of the existence of a condition of when a localization of a category gives a t-structure on it in Higher Algebra. (8/14/2021) Today I learned why the nondegenerate category, defined as the subcategory cut out by a bunch of kernels of the appropriate averaging maps, is actually closed under the braid group action. Of course, it suffices to check this for the generators, and the key fact comes from the fact that any nondegenerate sheaf in D(NnG=N) is also nondegenerate on the other side too. This is because you can reduce to showing that the averaging associated to a simple refleciton s vanishes on each left s coset, and you can use the (right) symplectic Fourier transform to bring - Tom Gannon 3 that coset all the way down to the coset f1; sg for free. (8/16/2021) Today I learned a very explicit statement about how compact objects generate. Specifically, given a set of compact objects in a compactly generated category, then all compact objects can be obtained by finite colimits and retracts (which, in the world of DG categories, are direct summands) of the compact generating objects, even if this is an infinite set of generators. (8/17/2021) Today I learned a heuristic explanation for the abstract Cartan that I like. Specif- ically, it is known that any two choices of Borel in a semisimple Lie group are conjugate. One might ask{must this isomorphism respect the choice of tori? The answer is yes, because once you choose the tori in the respective Borels, the isomorphism gives an identification of both tori with the respective `abstract Cartan', and thus identifies them with no extra choices. In particular, in the Lie algebra case one can identify the sl2 embeddings, since these are determined by the weights up to scalar multiple. (8/18/2021) Today I learned that there is a partially defined left adjoint to the symplectic Fourier transform given by the symplectic Fourier transform for j!. Therefore, we may equivalently show that our usual symplectic Fourier transformations preserve the nondegenerate subcategory by arguing that their associated left adjoints preserve the category of degenerate objects, i.e. send the various !Qα=N to the structure sheaves. N (8/19/2021) Today I learned a way to possibly compute what object Av∗ (!t∗ ) is at each central character without using the fact that the W action gives an equivalence.
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