Representation theory through the lens of categorical actions: part III

Transcription

1 Reminders Representation theory through the lens of categorical actions: part III University of Virginia June 17, 2015

2 Reminders Definition A g-action on an additive category C consists of: A direct sum decomposition C = λ C λ over the integral weights λ of g. Adjoint functors E λ : C λ C λ+αi and F λ : C λ C λ αi for i I that satisfy the relations E λ α i i Fi λ = F λ+α i i E λ i id α i (λ) (α i (λ) 0) E λ α i i Fi λ id α i (λ) = F λ+α i i E λ i (αi (λ) 0) The functor ( i,λ E λ i )n carries an action of R n compatible with horizontal composition.

3 Reminders Fix a field k. We call an object C in a categorical module C highest weight if E i C = 0 for all i. Theorem (Rouquier) For each simple highest or lowest weight representation V of g, there is a unique graded categorification V of V with a categorical action of g such that the category of highest weight objects is the category of k-vector spaces. But, unlike representations of a finite-dimensional g, categorical modules are not semi-simple, so understanding simples is not enough.

4 Definitions Fix a commutative ring R, with κ, s 0,..., s l 1 R. Definition The rational Cherednik algebra H s is the algebra generated by copies of R[x 1,..., x n ], R[G n = Z/lZ S n ] and R[y 1,..., y n ] with relations wx i w 1 = x w(i) wy i w 1 = y w(i) [y, x] = y, x + y, α s αs, x α 2κs s S s, α s 0 + s S 1 y, α s α s, x α s, α s l 1 κ det(s) j (s j s j 1 1/l)s. j=0

5 Definitions For me, the most interesting thing about an RCA is that it has a category O. Definition Category O n (s) is the category of modules over H s which are generated by a finite dimensional subspace which is invariant under R[G n ] and R[x] with the latter acting nilpotently. One obvious source of objects in this category are the Verma modules (V) = H s R[Gn,x] V for V a G n representation with x i acting trivially. By a PBW theorem, these are isomorphic to R[y 1,... y n ] R V as vector spaces. Every simple is a quotient of (V) for V a simple G n -module.

6 Definitions Furthermore, these Verma modules behave like the Verma modules of Lie algebras: there is a partial order on simple modules V over G n, such that Hom( (V), (W)) = 0 unless V W. Furthermore, every projective object P in O has a filtration P 1 P 2... P m = P with subquotients P i /P i 1 = (Vi ) with V 1 > V 2 >. Proposition These observations together mean that O is a highest weight category.

7 Definitions This order is pretty important, so let me describe it: first, recall that representations of G n are in bijection with l-multipartitions of n. Theorem Every irreducible representation of G n (over C) is of the form V λ = Ind Gn G n1 G nl (V 1 λ 0 V ζ λ 1 V ζ 1 λ l 1 ) where V η λ is the S λ -representation associated to λ, with each τ i acting by the scalar η.

8 Definitions Given a multipartition λ = (λ (0),..., λ (l 1) ), its diagram is the triples {(a, b, k) 1 b λ (k) a }. I ll always draw Young diagrams in the Russian style with rows and columns running diagonally. If κ, s 1,..., s l R, we draw the boxes of the diagram as squares with diagonal length 2 κ l, centered at (κl(s k + b a) k, κl(b + a)). The c-function ordering of partitions is by the center of gravity of this diagram: the further left, the higher in the partial order.

9 The categorical action Assume R = C. There are induction and restriction functors E O n O n+1 F Furthermore, E and F define a level l categorical action, with Dynkin diagram given by U = {κ(s i + m) m Z} C/Z. That is, if κ = a/e Q and s i 1 e Z, then we have an action of ŝl e (we call these Uglov parameters). On the other hand, if κ / Q, then we get sl (or many copies of it). The induced g U action on the Grothendieck group is a Fock space action.

10 The categorical action The Grothendieck group of n O n is easy to understand: it has a basis u ξ = [ (ξ)] given by classes of Vermas. The functor F( (ξ)) has a filtration whose terms are all the Vermas (ξ ) with one box added to ξ. Similarly E( (ξ)) has a filtration that removes a single box in each possible way. The Jucys-Murphy eigenvalue on the piece (ξ ) with new box (a, b, k) is exp(2πiκ(s k + b a)). Hence F u, E u only picks up boxes with certain contents (mod e).

11 The categorical action The Grothendieck group of n O n is easy to understand: it has a basis u ξ = [ (ξ)] given by classes of Vermas. The functor F( (ξ)) has a filtration whose terms are all the Vermas (ξ ) with one box added to ξ. Similarly E( (ξ)) has a filtration that removes a single box in each possible way. The Jucys-Murphy eigenvalue on the piece (ξ ) with new box (a, b, k) is exp(2πiκ(s k + b a)). Hence F u, E u only picks up boxes with certain contents (mod e). -2

12 The categorical action The Grothendieck group of n O n is easy to understand: it has a basis u ξ = [ (ξ)] given by classes of Vermas. The functor F( (ξ)) has a filtration whose terms are all the Vermas (ξ ) with one box added to ξ. Similarly E( (ξ)) has a filtration that removes a single box in each possible way. The Jucys-Murphy eigenvalue on the piece (ξ ) with new box (a, b, k) is exp(2πiκ(s k + b a)). Hence F u, E u only picks up boxes with certain contents (mod e). 0

13 The categorical action The Grothendieck group of n O n is easy to understand: it has a basis u ξ = [ (ξ)] given by classes of Vermas. The functor F( (ξ)) has a filtration whose terms are all the Vermas (ξ ) with one box added to ξ. Similarly E( (ξ)) has a filtration that removes a single box in each possible way. The Jucys-Murphy eigenvalue on the piece (ξ ) with new box (a, b, k) is exp(2πiκ(s k + b a)). Hence F u, E u only picks up boxes with certain contents (mod e). 3

14 The categorical action The Grothendieck group of n O n is easy to understand: it has a basis u ξ = [ (ξ)] given by classes of Vermas. The functor F( (ξ)) has a filtration whose terms are all the Vermas (ξ ) with one box added to ξ. Similarly E( (ξ)) has a filtration that removes a single box in each possible way. The Jucys-Murphy eigenvalue on the piece (ξ ) with new box (a, b, k) is exp(2πiκ(s k + b a)). Hence F u, E u only picks up boxes with certain contents (mod e). -1

15 The categorical action The Grothendieck group of n O n is easy to understand: it has a basis u ξ = [ (ξ)] given by classes of Vermas. The functor F( (ξ)) has a filtration whose terms are all the Vermas (ξ ) with one box added to ξ. Similarly E( (ξ)) has a filtration that removes a single box in each possible way. The Jucys-Murphy eigenvalue on the piece (ξ ) with new box (a, b, k) is exp(2πiκ(s k + b a)). Hence F u, E u only picks up boxes with certain contents (mod e). 2

16 The categorical action Theorem (Rouquier-Shan-Varagnolo-Vasserot, W.) The categories O n (s) form the unique collection of categories (+minor technical details) which: carry a categorical action of g U for the U defined before, categorifying the Fock space and possess highest weight structures with the c-function order, with standards passing to the standard basis in the Fock space, with these structures being compatible, in the sense that categorification functors preserve the standard and costandard filtered subcategories, and all these features deform in a family which is generically semi-simple. Note that the parameters only enter in one place: they affect the c-function order.

17 The categorical action The proof uses the relationship to the Hecke algebra H n (κ, s) with relations (q = e 2πiκ ) (T i +1)(T i q) = 0 T i T i±1 T i = T i±1 T i T i±1 T i T j = T j T i (i j±1) X i X j = X j X i T i X i T i = qx i+1 X i T j = T i X j (i j, j + 1) p κ,s (X 1 ) := (X 1 e 2πiκs 1 )(X 1 e 2πiκs 2 ) (X 1 e 2πiκs l ) = 0 Theorem (Ginzburg-Guay-Opdam-Rouquier) The functor KZ = Res n : O n (s) H n (κ, s) := H n (κ, s) -mod is fully faithful on projectives. That is, the category of projective modules in O is equivalent to a subcategory of H n (κ, s). For generic κ, s, the functor KZ is an equivalence, but if H n (κ, s) is not semi-simple, then it ceases to be one.

18 The categorical action This perspective lets us come up with radically different models for understanding Cherednik algebras: RSVV,L O n is equivalent to a truncated parabolic category O for an affine Lie algebra; here s tells us the parabolic that appears, and κ tells us the level. W. O n is equivalent to a diagrammatic algebra generalizing the KLR algebra. In both perspectives, we can see that O n has a Koszul graded lift, which gives a q-deformation of Fock space we can understand the characters of simples/projectives in O n in the graded Grothendieck group: they correspond to the (dual) canonical basis in this space in addition to the action of B e coming from Chuang-Rouquier equivalence, we obtain an action of B l which acts on s D b (O n (s)) changing the parameters.

19 The categorical action In fact, there s a hidden symmetry between e and l. Definition Assume the parameters are Uglov. Let O(s, t) be the t = (t 1,..., t e ) weight space for the ŝl e-action in O(s). Theorem (Shan-Varagnolo-Vasserot) The categories O(s, t) and O(t, s) are Koszul-Ringel dual. That is, there is a tilting generator T in one category and a simple generator S in the other such that End(T) = Ext (S). This is a categorification of rank-level duality for affine Lie algebras.

20 The categorical action One philosophical point you might take away from this is: Cherednik algebras are well-behaved and have a uniqueness theorem because they categorify a nice representation. What s nice about Fock space? Well, it s not irreducible, but it s a restriction of an irrep over sl l. In fact, if we take any parameters for the Cherednik algebra, and deform them a little, then O n (s ) becomes the unique categorification of this simple over sl l. This suggests that O is a restriction of this simple module to the smaller algebra ŝl e (for Uglov parameters). But this requires some idea of what restricting a categorical module means.

21 Categorified tensor products Another very important class of representations that arise as restrictions are tensor products. For a list λ = (λ 1,..., λ l ) of highest weights for g, we have the tensor product V λ = Vλ1 V λl. This is the restriction of a simple representation of g n via the diagonal inclusion g g n. We ll want a graphical calculus for elements of V λ. A downward black line on the left means acting by F i. A red line at the left labeled by λ corresponds to v λ, where v λ is the highest weight vector of V λ. So, we obtain a spanning set of V λ consisting of vectors like F i (F j v λ1 v λ2 ) λ 1 j λ 2 i

22 Categorified tensor products Another very important class of representations that arise as restrictions are tensor products. For a list λ = (λ 1,..., λ l ) of highest weights for g, we have the tensor product V λ = Vλ1 V λl. This is the restriction of a simple representation of g n via the diagonal inclusion g g n. We ll want a graphical calculus for elements of V λ. A downward black line on the left means acting by F i. A red line at the left labeled by λ corresponds to v λ, where v λ is the highest weight vector of V λ. So, we obtain a spanning set of V λ consisting of vectors like F i (F j v λ1 v λ2 ) λ 1 j λ 2 i

23 Categorified tensor products Let T λ be the graded algebra whose elements are k-linear combinations of immersed 1-manifolds with black components oriented downward, dotted with labels I red components have no intersections with each other, and are labeled with the weights λ in order modulo the KLR relations and the red/black relations: λ i = = λ i λ i λ i j i i i = = j i i i + a+b=λ i 1 a i i b any diagram with a black line at the far left is 0. i λ i λ = =

24 Categorified tensor products You can think of this as adding in maps between some of the vectors of the tensor product, making a category. Theorem (W.) The category T λ -mod has a categorical g-action by induction and restriction functors. The GGG of T λ is the Lusztig integral form of V λ as an U q (g)-module, sending the functor F i to the action of F i, and the functor I λ (adding a red line) to the inclusion V v high V V λ. Thus, this map matches the vector defined by our graphical calculus for an idempotent e with the projective [T λ e].

25 Categorified tensor products In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easily described as follows: = k: spanned by the diagram. T (1,1) 2 T (1,1) 0 is spanned by,,,, This is a regular block of category O for sl 2. = End(k 3 ). T (1,1) 2

26 Categorified tensor products In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easily described as follows: = k: spanned by the diagram. T (1,1) 2 T (1,1) 0 is spanned by,,,, This is a regular block of category O for sl 2. = End(k 3 ). T (1,1) 2

27 Categorified tensor products In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easily described as follows: = k: spanned by the diagram. T (1,1) 2 T (1,1) 0 is spanned by,,,, This is a regular block of category O for sl 2. = End(k 3 ). T (1,1) 2

28 Categorified tensor products In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easily described as follows: = k: spanned by the diagram. T (1,1) 2 T (1,1) 0 is spanned by,,,, This is a regular block of category O for sl 2. = End(k 3 ). T (1,1) 2

29 Categorified tensor products This construction is analogous to the Cherednik one in several ways: the simple/projective modules correspond to Lusztig s (dual) canonical basis for the tensor product we have an action of B l on w Sl D b (T w λ -mod). We can extend this braid group action to a knot invariant categorifying the Reshetikhin-Turaev invariants. in the case of finite type A and fundamental representations we actually get a special case of the Cherednik picture, and a Koszul duality symmetry, categorifying skew Howe duality. deforming the relations to = (y z k ) λi k i λ k gives an algebra Morita equivalent to T λ 1 T λ l, that is, the categorification of the corresponding simple over g l.

30 Uniqueness As in the Cherednik case, we need ordering information in order to pin down a choice: the different orders of the tensor product lead to different categories, though they give the same representation (as usual, the quantization hints that we need to be careful about this). Definition An abelian category C is standardly stratified by the categories C λ for λ Λ if there are exact fully-faithful functors p λ : C λ C such that every simple module is a quotient of p λ (M) for some simple M, with kernel filtered by quotients of p µ (N) with µ < λ and every projective has a finite filtration P P λ indexed by Λ such that P λ /P >λ = pλ (Q) for some projective Q C λ.

31 Uniqueness We also have an analogous uniqueness theorem: Theorem (Losev-W.) The categories T λ -mod form the unique category which: carry a categorical action of g, categorifying V λ and possess standardly stratified structures compatible with the order k k (µ 1,..., µ l ) (ν 1,..., ν l ) µ i k [1, l], i=1 i=1 with standards passing to the pure tensors, with these structures being compatible, in the sense that categorification functors preserve the standard and costandard filtered subcategories, and all these features deform in a family which is generically T λ 1 T λ l. ν i

32 Graph coverings So, we ve seen two examples of very nice categorifications of restricted simple modules. What s the general framework here? Actually there s a very specific form to the inclusions that appear: they all come from graph coverings. Definition Let X, Z be oriented graphs; a graph homomorphism f : X Z is a covering if for every vertex z Z, and preimage x f 1 (z), the map f induces a bijection between edges in X starting (ending) at x and in Z starting (ending) at z.

33 Graph coverings If g X and g Z are associated Kac-Moody algebras, then a graph covering f : X Z induces an embedding: g Z ĝ X F z x f 1 (z) F x If X = n Z, then this is the diagonal embedding g g n. A restriction of a simple under this map is a tensor product. If Z is an n-cycle, we can take X to be infinite linear with the obvious covering (or several copies of this quiver). This is my favorite inclusion ŝl n sl. Restriction under this map gives examples like Fock spaces.

34 Graph coverings One natural question in categorification-land: what is the categorical version of this inclusion? Fix a field k; consider a 1-cocyle (weighting) w on the graph Z. This defines an action of the fundamental groupoid Π 1 (Z) which assigns a copy of the field k to each vertex and e a = a + w e This gives k Z = z Z k a graph structure of its own, with subgraphs subrepresentations. 1 0

35 Graph coverings Let X be a subrepresentation/subgraph of k Z equipped with projection covering X Z. Proposition Let C be a category with a categorical action of g Z with Q ij (u, v) = (v u + w e ) v + w e ), i j j i(u such that the element y acts on F i with minimal polynomial whose roots lie in X i. Then C has an induced categorical action by g X. This shows a relationship between degeneration and restriction.

36 Categorical restrictions The examples we know of suggest that the restrictions of simples should have a best class of categorifications. Allow me to make a proposal. Let Z be a graph, w a weighting in the field k, and X the covering attached to a subrepresentation of the fundamental groupoid: Definition Let V be an irreducible representation of g X, and let V be its unique categorification. Choose a order on the nodes of X which is invariant under the action of Π 1 (Z). Call a categorical g Z module W with GG V gz restricted (from V) if: W has a standardly stratified structure for which is trivialized by a deformation with W d = V k k((t)). The spectrum of y on F i lies in X i over d in the deformation, and this induces the categorical g X action on W.

37 Categorical restrictions Let V be an irreducible representation of g X which is highest weight. Theorem For each (V, Z, X, w, ), there exists a unique restriction. The proof is looking inside the categorification of the Cartan components. The existence is proven by direct construction. The uniqueness is proven using the theory of 1-faithful highest weight covers (due to Rouquier). Not 1-faithful over the categorification of the Cartan component but over something bigger. A lemma of RSVV tells us that knowing how much bigger comes down to controlling simples with Ext 1 with tiltings. This also works for a tensor product of factors i C or its dual if X is a union of infinite linear quivers.

38 Weighted KLR algebras So, what do these algebras look like? The existence proof is using weighted KLR algebras. These are KLR algebras with a twist: we weight each edge e: i j with a real number ϑ e. As before, this gives an action of the fundamental groupoid on the real numbers, and any invariant subset gives us a graph cover X Z R. A weight of an g X -irrep V of highest weight λ is of the form µ = λ X µ α x where W µ is a multi-subset of X. We order these by comparing the average of the real numbers corresponding to the elements of W µ.

39 Weighted KLR algebras We represent this visually by giving every j-colored strand a ghost ϑ e units to the right (or -ϑ e to the left). We keep the relations for like colored strands (assuming there are no loops) from KLR, but now those between differently colored strands happen not when the strands cross, but an i-colored strand forms a bigon or triangle with the ghosts of j-colored strands! = - i j i j i j

40 Consequences So, why worry about this? Hopefully you think the categories above are interesting. But also, restrictions have a lot of beautiful properties: The most important for us is that any restriction has a canonical graded lift (which seems to be Koszul if V is minuscule for g X and char k = 0). Thus their GG s have a q-deformation over the quantum group; this recovers things like the tensor product of quantum group representations, and twisted higher level Fock spaces. Note this depends on the choice of order. Theorem The projectives of a categorical restriction are a canonical basis on this q-deformation; in particular, its decomposition numbers can be computed with a Kazhdan-Lusztig type algorithm.

41 Consequences The other motivation for me is to compare different orders. Theorem Any two categorical restrictions of a given representation (but different orders) are derived equivalent. Just on the level of derived categories, there is a single best categorification, but with a lot of interesting t-structures. This equivalence is not canonical. Instead, there are many different ones that induce an action of π 1 of a hyperplane complement on the derived category.

42 Generalizations I don t think this is the last we will hear about deformation and restriction. You can try to play the same game whenever you have a map of Lie algebras that sends a Chevalley generator to a sum of Chevalley generators Work in progress w/ Bao, Shan and Wang defines a categorical action of certain symmetric subalgebras in U(sl m ), related to type BCD categories O and other representation theory. These categories also have a geometric identity, which we ll discuss in the next talk.